## Chapter 1

## Circuit Variables and Circuit Elements

##### CHAPTER OBJECTIVES

*To introduce the concepts of electrostatic potential difference, voltage and electromotive force.**To introduce the concepts of ‘current density’ and ‘current intensity’ and to explain the conduction process.**To explain the ‘quasi-static’ approximation involved in circuit theory.**To explain the need for and role of quasi-static charge distributions over electrical devices.**To define and explain various idealized two-terminal element models in use in Circuit Theory.**To explain passive sign convention for two-terminal elements.**To relate power and energy in a two-terminal element to its terminal voltage and current variables.**To explain what is meant by ‘lumped, linear, bilateral, passive, time-invariant circuit element’.**To define and explain four types of linear dependent sources.*

##### INTRODUCTION

The profession of Electrical and Electronic Engineering deals with the generation, transmission and measurement of electric signals with signal power level varying from few nWs (10^{–9 }Watt) to hundreds of MW (10^{6} Watt) in various applications.

This textbook deals with one of the ‘kingpin’ subjects in the entire field of electrical and electronic engineering in detail. An *Electric Circuit *is a mathematical model of a real physical electrical system.* *Physical electrical systems consist of electrical devices connected together. *Electric Circuit *idealizes the* *physical devices and converts the real physical system into a mathematical model that is governed by a set of physical laws. *Circuit solution *is obtained by applying those physical laws on the mathematical circuit model and employing suitable mathematical solution techniques. The circuit solution approximates the actual behaviour of the physical system to a remarkable degree of accuracy in practice.

Though the term *electric circuit *refers to the idealized mathematical model of a physical electrical system, it is used to refer to the actual electrical system too in common practice. However, when we refer to *electric circuit *in this text, we always mean a mathematical model, unless otherwise stated.

Laws of electromagnetic fields govern the electrical behaviour of an actual electrical system. These laws, encoded in the form of four Maxwell’s equations, along with the constituent relations of electrical materials and boundary constraints, contain all the information concerning the electrical behaviour of a system under a set of specified conditions. However, extraction of the required information from these governing equations will turn out to be a formidable mathematical task even for simple electrical systems. The task will involve solution of partial differential equations involving functions of time and space variables in three dimensions subject to certain boundary conditions.

Circuit theory is a special kind of approximation of electromagnetic field theory. ‘*Lumped parameter circuit theory’ *converts the partial differential equations involving time and three space variables arising out of application of laws of electromagnetic fields into ordinary differential equations involving time alone. Circuit theory approximates electromagnetic field theory satisfactorily only if the physical electrical system satisfies certain assumptions. This chapter discusses these assumptions first.

##### 1.1 ELECTROMOTIVE FORCE, POTENTIAL AND VOLTAGE

*Charge *is the attribute of matter that is responsible for a force of interaction between two pieces of matter under certain conditions. Such an attribute was seen to be necessary as a result of experiments in the past which revealed the existence of a certain kind of interaction force between particles that could not be explained by other known sources of interaction forces.

Charge is bipolar. Two positively charged particles or two negatively charged particles repel each other. Two particles with charges of opposite polarity attract each other. Further, charge comes in integral multiples of a basic unit – the basic unit of charge is the charge of an electron. The value of electronic charge is –1.602 × 10^{–}^{19} coulombs. The SI unit of charge – *i.e., *Coulomb – represents the magnitude of charge possessed by 6.242 × 10^{18} electrons.

#### 1.1.1 Force Between Two Moving Point Charges and Retardation Effect

The force experienced by a point charge of value *q*_{2} moving with a velocity of due to another point charge of value *q*_{1} moving with a velocity of at a distance *r *from it contains three components in general. If the charges are moving slowly, the force components are given by approximate expressions as below.

**Fig. 1.1-1** Force between two point charges in motion

The first component is directly proportional to the product *q*_{1}*q*_{2} and is inversely proportional to the square of distance between them. This component of force is directed along the line connecting the charges and is oriented away from *q*_{1}. This component is governed by Coulomb’s law; is termed ‘*the electrostatic force’ *and is given by

where *ε*_{0} is the dielectric permittivity of free space (= 8.854 × 10^{–12} F/m) is the unit vector directed from *q*_{1} to *q*_{2}.

The second component of force is the *magnetic force *and arises out of motion of charges. This component is given by

where *μ*_{0} is the magnetic permeability of free space (= 4*π* × 10^{–7} H/m) and and
are the velocities of *q*_{1} and *q*_{2} respectively.

The third component of force is the *induced electric force *and depends on relative acceleration of *q*_{1} with respect to *q*_{2}. It is given by

Electromagnetic disturbances travel with a finite velocity – the velocity of light in the corresponding medium. Therefore, in general, the force experienced by *q*_{2} at a time instant *t *depends on the position and velocity of *q*_{1} at an earlier instant. Or, equivalently, the force that will be experienced by *q*_{2} at *t *+ *r/c *depends on *r *and at *t *where *c *is the velocity of light. This effect is called the *retardation effect. The expressions described above ignore this retardation effect and assume that the changes in relative position and velocity of q*_{1} *are felt instantaneously at q*_{2}.

Retardation effect can be ignored if (i) the speed of charges is such that no significant change can take place in *r *during the time interval needed for electromagnetic disturbance to cover the distance and (ii) the acceleration of charges is such that no significant changes in the velocity of charges take place during the time interval needed for electromagnetic disturbance to cover the distance between charges. The first condition implies that the speed of charges must be small compared to velocity of light. This condition is met by almost any circuit since the drift speed associated with current flow in circuits is usually very small compared to velocity of light. However, though the speed of charge motion is small, it is quite possible that the charges accelerate and decelerate rapidly in circuits such that the second condition is not met.

Consider the two point charges in Fig. 1.1-2. *q*_{1} is oscillating with amplitude *d *and angular frequency *ω *rad/sec. However, let us assume that *d *<< *r. *Then, neither the distance between the charges nor the unit vector between them change much with time. Therefore, the electrostatic force experienced by *q*_{2} is more or less constant in time. *q*_{2} is at rest and hence it does not experience any magnetic force. However, it will experience a time-varying induced electric force. Let the horizontal position of *q*_{1} be given by *x *= *d *sin (*ωt*) m. Then, velocity of *q*_{1} is = ω*d* cos ω*t* m/s. The time taken by any electromagnetic disturbance to travel *r *meters is *r/c *where *c *is the velocity of light in m/s. The quantity that decides the induced electric force experienced by *q*_{2} will change by in that time interval. Retardation effect can be ignored only if this change is negligible – *i.e., *only if

**Fig. 1.1-2** Pertaining to retardation effect in a two-charge system

In the context of circuit theory, *ω *is the highest angular frequency of time-varying signals present in the circuit and *r *is the largest physical dimension measured in the physical circuit. Then, retardation effect due to finite propagation speed of electromagnetic waves can be ignored in the analysis of an electrical system if the highest frequency present in the circuit and largest dimension of the system satisfy the inequality

An electrical system in which the retardation effect can be ignored is called a *quasi-static electrical system. *Electric circuit theory assumes that the electrical system that is modeled by an electrical circuit is a quasi-static system.

#### 1.1.2 Electric Potential and Voltage

The force experienced by a charge *q *in arbitrary motion under quasi-static conditions due to another moving charge has three components as explained in the previous sub-section. The second component that is velocity dependent and the third component that is acceleration dependent are put together to form the *non-electrostatic component. *Thus, the force experienced by *q *due to another charge has an *electrostatic component *and a *non-electrostatic component. *

The force experienced by a charge *q *in the presence of many charges can be obtained by adding the individual forces by vector addition. Thus, the electrostatic force experienced by the charge *q *due to a system of charges is a superposition of electrostatic force components from the individual charges. *Electrostatic field intensity vector*, at a point in space is defined as the total electrostatic force vector acting on a unit test charge (*i.e., q *is taken as 1C) located at that point. Then, the electrostatic force experienced by a charge *q *located at a point *P *(*x, y, z*) is given in magnitude and direction by *q * N. The SI unit of *electrostatic field intensity *is Newtons per Coulomb.

Electrostatic field intensity due to a point charge falls in proportion to square of distance between location of charge and location at which the field is measured. Hence, the field intensity at large distance from a system of *finite amount *of charge tends to reach zero level. Therefore, a test charge of 1 C located at infinite distance away from the charge collection will experience zero electrostatic force.

Now assume that the test charge of 1 C that was initially at infinity is brought to point *P*(*x, y, z*) by moving it quasi-statically through the electrostatic field. The agent who moves the charge has to apply a force that is numerically equal to and opposite in direction. The total work *to be done *in moving the unit test charge from infinity to *P*(*x, y, z*) is obtained by integrating the quantity over the path of travel where is a small length element in the path of travel. This work is, by definition, the electric potential (*electrostatic *potential to be precise) at the point *P*(*x, y, z*)*. *It is usually designated by *V*(*x, y, z*)*. *Then,

The unit of electric potential is Newton-meter per Coulomb or Joule per Coulomb. This unit is given the name ‘Volt’. This leads to another unit for electrostatic field intensity – namely, Volt/m.

*Electrostatic force field is a conservative force field. *Hence the value of this work integral will depend only on the end-points and not on the particular path that was traversed. Therefore, this work integral (and hence the potential at point *P*(*x, y, z*)) has a unique value that depends on *P*(*x, y, z*) only. Moreover, the conservative nature of electrostatic force field implies that the work done in taking a test charge around a *closed path *in that field is zero.

Eqn. 1.1-1 defines the potential at a point with respect to a point at infinity. The difference in potential at two points can be interpreted as the work to be done to carry a unit test charge from one point to another. Specifically, let *V*_{1} and *V*_{2 } be the electric potentials at point-1 and point-2 in space. Then, the potential of point-1 with respect to the point-2 is *V*_{1 }– *V*_{2} volts and is equal to the work to be done in carrying a unit positive test charge from point-2 to point-1. This value is designated by *V*_{12} and is read as ‘potential of 1 with respect to 2’ or as ‘potential difference between 1 and 2’. In Circuit Analysis, this electrostatic potential difference between two points is called the ‘voltage between point-1 and point-2’. The same symbol that was used to designate the potential difference is used to designate ‘voltage’ too. Thus,

If *V _{ab}* is a positive quantity, we state that there is a ‘

*potential rise’*or ‘

*voltage rise’*from

*b*to

*a.*Equivalently, we can state that, there is a ‘

*potential drop’*or ‘

*voltage drop’*from

*a*to

*b.*If

*V*is a negative quantity, we state that, there is a ‘

_{ab }*potential rise’*or ‘

*voltage rise’*from

*a*to

*b.*Equivalently, we can state that, there is a ‘

*potential drop’*or ‘

*voltage drop’*from

*b*to

*a.*

The work to be done in moving a unit positive test charge is positive when the charge is moved in a direction opposite to the direction of electrostatic field. Hence, the direction in which maximum voltage rise takes place will be opposite to the direction of electrostatic field at any point in space.

When a charge *q *is moved through a *voltage rise, *some non-electrostatic force has to work against the electrostatic force to effect the movement. The non-electrostatic force will do work on the charge in the process of moving it. This work done on the charge gets stored in the charge as increase in its potential energy. Hence, a charge *q *receives *qV _{ab}* Joules of potential energy when it is carried from

*b*to

*a.*If

*V*is positive –

_{ab }*i.e.,*if there is a voltage rise from

*b*to

*a*– then, the potential energy level of charge

*q*increases by

*qV*Joules in moving from

_{ab}*b*to

*a*.

If *V _{ab}* is negative –

*i.e.,*if there is a voltage drop from

*b*to

*a*– then, the potential energy level of charge

*q*decreases by

*q|V*| Joules. The non-electrostatic force that maintains quasi-static condition during the movement of charge from

_{ab}*b*to

*a*receives this energy.

Sustained and organised movement of charges through voltage rises and voltage drops in an electrostatic potential system is possible only if there are sources of non-electrostatic forces present in the electrical system. These non-electrostatic forces deliver and absorb the required amounts of energy to make movement of charges through voltage rises and voltage drops possible. Obviously, a non-electrostatic force can fulfil this role only if it has a component along the direction of motion of charge. A force that is always perpendicular to the velocity of a charge can not deliver energy to the charge or absorb energy from it. Magnetic force on a charge is always perpendicular to the velocity of the charge. Hence magnetic force can not be the non-electrostatic force that we want.

The *induced electric force, *a component of force of interaction between two charges in quasistatic motion, can deliver energy to moving charges and absorb energy from them. Thus, the induced electric force can be a source of the required kind of non-electrostatic force in an electrical system.

Electrical sources are sources of non-electrostatic forces in an electrical system. Some of the sources make use of the *induced electric force *to generate the non-electrostatic force. A DC Generator, an AC Generator etc., are examples of this kind of sources. However, there are other sources of non-electrostatic forces. A dry cell, for instance, converts the internal chemical potential energy into a non-electrostatic force that acts on any charge carrier that transits through the conducting material inside the dry cell. We do not concern ourselves anymore with the exact nature of the non-electrostatic force available within an electrical source. It suffices for our purpose to understand that some nonelectrostatic force is available within the electrical source.

#### 1.1.3 Electromotive Force and Terminal Voltage of a Steady Source

Consider an electrical source on open circuit as in Fig. 1.1-3. A free charge located at a point inside the source will experience a non-electrostatic force as shown in the figure. This non-electrostatic force is expressed as a force field and the quantity represents this force field. Thus, the nonelectrostatic force experienced by a charge *q *located inside the source will be *q *N where is the non-electrostatic field intensity vector. may not be constant in magnitude and direction everywhere inside the source. However, will not vary with time in the case of a *steady source. *

**Fig. 1.1-3** A steady electrical source under open-circuit condition

The source contains conducting material inside. Conductors contain free electrons. The free electrons in the conducting substance will tend to move from top to bottom (electrons have negative charge) under the influence of non-electrostatic force. The first few electrons that move so reach the bottom electrode (at B) and accumulate at that terminal. Electrons moving to B will cause an equal number of positive charges to appear at the terminal marked A. But then, such a separation of charges will result in generation of electrostatic field inside (as well as outside) the source. Thus, the remaining free electrons inside the source will experience two forces – a non-electrostatic force that tends to move them towards the lower electrode B and an electrostatic force that tends to move them towards the upper electrode A. The source reaches a steady-state soon. Under steady-state condition, the magnitude and spatial distribution of charges over the metallic electrodes at A and B are such that all the free electrons that are still within the source will experience zero net force and remain stationary (except for random thermal motion). Thus, the electrostatic field at a point inside will cancel the non-electrostatic field at that point under steady-state. The charge distribution at the terminals will arrange itself suitably such that this cancellation takes place at all points in the active region of source.

However, there is no non-electrostatic field outside the source. Therefore a test charge kept at a point outside the source will experience an electrostatic force.

**Fig. 1.1-4 **Pertaining to definition of e.m.f. and open circuit voltage of a source

Refer to Fig. 1.1-4. Let a unit test charge be carried from B to A through the path BOA – *i.e., *over a path that is outside the source. Some work has to be done for this. The required work will be positive since we are carrying a positive test charge from a negatively charged terminal to a positively charged terminal. The work that is required to be done is the voltage of A with respect to B – *i.e., V _{AB}*.

However, the work required to carry a unit test charge in a closed path in electrostatic field is zero due to conservative nature of electrostatic forces. Therefore, the work to be done *against the * *electrostatic force field *to carry the test charge in the path B-O-A-I-B is zero. That is, work to be done against electrostatic force to move a unit test charge from B to A over a path outside the source *plus *work to be done against electrostatic force to move a unit test charge from A to B over a path inside the source is zero. Therefore, work to be done in path B-O-A = work to be done in path B-I-A. But the electrostatic field vector inside the source is equal to Therefore,

But is the work done *by *the non-electrostatic force generated by the source *on *a unit positive charge when it moves through the source from negative terminal to positive terminal. This quantity is defined as the *Electromotive force *(e.m.f.) of the source. It is usually represented by the symbol *E. *Therefore, the electrostatic potential difference between positive terminal and negative terminal of a source under open-circuited condition (that is, the open-circuit terminal voltage *V _{AB}*) is equal to the electromotive force of the source.

*E *of a steady source is a constant. Such a source is called a DC Voltage Source. DC stands for ‘direct current’. The *terminal voltage *(*V _{AB }* in Fig. 1.1-3 and Fig. 1.1-4) of a practical steady source will be equal to the electromotive force

*E*only under open-circuit condition. Terminal voltage becomes less than

*E*when the source is delivering some current due to the inevitably present voltage drop in the internal resistance of the source.

##### 1.2 A VOLTAGE SOURCE WITH A RESISTANCE CONNECTED AT ITS TERMINALS

A piece of conducting substance is now connected to a steady voltage source with a steady-state static charge distribution at its terminals as shown in Fig. 1.1-3. The resulting system is shown in Fig. 1.2-1.

#### 1.2.1 Steady-State Charge Distribution in the System

There was an electrostatic field due to the terminal charge distribution of the source in the space that is now occupied by the conducting substance. This field can change only if the terminal charge distributions change. Charges have to move and reach terminals in order to change the distribution there. That takes time. Therefore, the charge distribution at the terminals remains unaffected for a brief interval even after the conducting substance is connected across the source.

But this will result in large electrostatic forces on the free electrons in the metallic substance. These free electrons start migrating towards positive terminal (A) of the source and cancel the positive charge distribution there partially. Simultaneously, electrons are pulled from terminal B into conducting substance, thereby canceling the charge distribution in the negative terminal partially. But this change in the charge distribution will affect the balance between electrostatic force and non-electrostatic force inside the source. Now the non-electrostatic force will not be cancelled completely by the electrostatic force. The remnant non-electrostatic force will propel the free electrons present within the source towards the negative terminal (and an equal amount of positive charge gets propelled towards positive terminal). This will lead to a restoration of charge distribution at the terminals as well as creation of charge distribution on the surface of connecting wire and the conducting substance.

**Fig. 1.2-1** Steady-state with a resistance connected across a DC voltage source

Soon a steady charge distribution is established as shown in Fig. 1.2-1. The non-electrostatic force and the electrostatic force at any point inside the source will cancel each other under this condition. [The conducting substance inside the source is assumed to have zero resistivity.] Therefore the terminal voltage *V _{AB }* will be equal to the electromotive force

*E*of the source. The geometry of the system will decide the amount of positive and negative charges distributed on the terminal surface, wire surface and surface of the conductor. Once established, this charge distribution remains steady (in the case of a steady source) unless spatial arrangement is disturbed or the resistance is disconnected.

#### 1.2.2 Drift Velocity and Current Density

This steady charge distribution will produce steady electrostatic field everywhere inside the conducting substance. Free electrons inside the conductor get accelerated by this electrostatic force. The velocity as well as kinetic energy of free electrons would have reached high levels in the absence of any opposing force. However, there is an opposing force. This force arises out of collisions suffered by the accelerating electrons. The electrons collide with ionized atoms in the conducting substance inelastically and lose some of the kinetic energy they acquire under the action of electrostatic force. The average effect of multitude of collisions suffered by an electron accelerating under electrostatic force is similar to that of friction. Thus, the inelastic collisions that take place between accelerating free electrons and ionized metal atoms result in a non-electrostatic force that is similar to friction. The accelerating electrons pick up kinetic energy first since they are losing potential energy by falling through a voltage drop. The kinetic energy is subsequently delivered to the lattice through inelastic collisions (or equivalently, to the non-electrostatic force that is manifest within the conducting substance as an average effect of multitude of inelastic collisions).

The free electrons inside the conducting substance reach a steady velocity in the direction opposite to that of electrostatic force under the action of electrostatic force and the non-electrostatic force arising out of collisions. This is somewhat similar to an object reaching a terminal velocity when it falls through a viscous medium. The terminal velocity attained by a free electron inside a conducting substance is called *drift velocity *and is denoted by* *. Experiments have revealed that this velocity is proportional to the electrostatic field intensity through a proportionality constant called *mobility *of the substance. Note that this is an average velocity that is super-imposed on the random thermal velocity of electrons.

Thus, the drift velocity of electrons at a point where the electrostatic field intensity is is given by * *where *μ *is the mobility of the material and has m^{2} per Volt-sec unit. *μ *of a material is temperature dependent quantity and decreases with temperature in the case of metals. The negative sign is needed to account for the fact that negatively charged electrons move in a direction opposite to that of electrostatic field.

Consider a small area element *∆A *m^{2} at a point inside the conductor where the drift velocity is* * and electrostatic field intensity is . Let the area element be taken perpendicular to the direction of (or ). Let *N *be the number of free electrons available in unit volume of the conducting material. Then, there will be *N(∆A) v _{d}* free electrons present in a volume element constituted by area

*∆A*m

^{2}and a length

*v*m along the direction of drift velocity. All these electrons will cross the area element in the next second.

_{d}Negative charges crossing a surface in one direction is equivalent to positive charges crossing the same surface in the opposite direction. Hence, the total *positive *charge that crosses *∆A *in one second in the direction of is *N q _{e} (∆A) v_{d}* coulombs where

*q*is the

_{e }*magnitude*of charge of an electron and

*v*is the

_{d}*drift speed.*

A vector quantity called ‘*current density vector*’ (denoted by ) is defined at a point inside a conductor as the total *positive *charge that crosses *unit area *in *unit time *in the direction of electrostatic field at that point with the area kept perpendicular to the direction of electrostatic field at that point. [This definition is correct only for homogeneous and isotropic materials. Metallic conductors are assumed to satisfy this requirement for all practical purposes]. It must be obvious that at a point is proportional to . The dimension of current density is Coulomb per m^{2} per second. Its unit is Ampere per m^{2} since Coulomb/sec is called ‘Ampere’.

where *σ* is defined as the *conductivity *of the material and has ampere per volt per meter as its unit. Ampere per Volt is given a special name – ‘Siemens’. Therefore, the unit of conductivity will be Siemens/m.

Reciprocal of conductivity is called the *resistivity *of the material and is denoted by *ρ. *Its unit is Ohm-m. [Volt/Amp is given the name ‘Ohm’].

The value of conductivity (and resistivity) of a *linear, homogeneous, isotropic *material is a constant at a particular temperature. Conductivity and resistivity vary with temperature. In general, conductivity decreases with increasing temperature in the case of metallic conductors.

#### 1.2.3 Current Intensity

*Electric current intensity *or, simply, *current intensity *through a surface is defined as the amount of charge that crosses the surface in unit time. ‘Current intensity’ is usually referred to as ‘current’ itself.

It is a scalar quantity. The definition begs a question – ‘charge that crosses the surface’ in which direction?

The direction of crossing implied in the definition is unambiguous in the case of a closed surface. It is in the direction of surface normal. Surface normal is drawn outwards in the case of a closed surface. Thus, the current intensity through a closed surface is the amount of charge that crosses the surface in one second *from *inside *to *outside. A positive current implies net positive charge flow out of the volume enclosed by the surface or net negative charge flow into the volume. A negative current implies net positive charge flow into the volume or net negative charge flow out of the volume.

However, there is ambiguity in interpreting current in the case of open surfaces if current is considered to be a pure scalar quantity. The surface normal for an open surface is not unique. The value of current through a surface in a given context can be positive or negative depending on the choice of surface normal. Therefore, the value of current intensity can be uniquely interpreted only if some direction is also associated with it. Therefore, it is customary in practice, especially in the case of wire conductors with uniform cross-section, to refer to the *direction of current. *The current density in such conductors will be approximately uniform and hence current density will have same direction at all points in the cross-section of the conductor. *The direction of current intensity in such conductors *(*wire conductors of uniform cross section) is taken to be same as the direction of current density itself. *

The unit of current is Coulomb/second and this unit is given a special name of ‘Ampere’. Current is denoted usually by *I *if it is a steady current (*i.e., *DC current) and by *i*(*t*) (or by *i*) if it is a time-varying quantity.

Let *q*(*t*) be the *total net charge *that crossed a given cross-section in a specified direction from *t *= –∞ to *t *= *t *and let *i*(*t*) be the current flowing through that cross-section in the same direction at *t *= *t. *Then, the relation between these two quantities are given by the following equations.

#### 1.2.4 Conduction and Energy Transfer Process

Consider the steady voltage source with resistive load across it shown in Fig. 1.2-2. Five cross-sections (A, B, C, D and E) are marked in the figure. Also, the directions of positive current flow and electron flow are marked. Electrons flow in the counter-clockwise direction in the circuit and positive current flows in the clockwise direction.

**Fig. 1.2-2** A steady-current system – current through A, B, C, D and E have same value

Consider the volume between the two cross-sections marked as B and C in Fig. 1.2-2. There is a surface charge distribution on this volume. [It is possible to show by employing equations of electromagnetic fields that there will be no charge distribution inside the volume in a homogeneous conducting substance under steady current conditions as well as under quasi-static current conditions. Charges will reside only on the surface.] This charge distribution remains stationary in time since the nonelectrostatic force within the source of e.m.f. is assumed to be steady. Therefore, the amount of charge that crosses into the volume through the cross-section B in unit time has to be same as the amount of charge that crosses out of the volume through the cross-section C in unit time – otherwise the surface charge storage within this volume will change. Hence the current through B has to be the same as the current through C. Similar argument for other cross-sections will lead us to the conclusion that current through all cross-sections will have same value in this circuit.

The surface charge distribution present throughout the system is stationary. But that does not mean that the individual electrons that make this distribution stay put. For instance, a particular electron that is part of the current flow after crossing C may cancel a positive surface charge. But that will result in an unbalance in the system and another electron will move out from surface and join the current stream leaving a positive charge on the surface. Thus, though the identity of individual charges that form the surface charge may not be preserved, the surface charge will appear stationary at a macroscopic level.

Consider an electron that is part of the current flow. The electrostatic field is oriented from positive terminal to negative terminal inside the source. The non-electrostatic field is oriented from negative terminal to positive terminal. When the conduction electron travels from positive terminal to negative terminal through the source it gains electrostatic potential energy. The non-electrostatic field does positive work to impart this extra potential energy to the electron. The conduction electron then flows through connecting wire to the negatively charged terminal of resistor. The electrostatic field inside the conductor tries to accelerate it and convert its potential energy into kinetic energy. The electron soon transfers its kinetic energy to the lattice through inelastic collisions with atoms. By the time it emerges at the positively charged terminal of the resistance, it would have lost all the extra potential energy it gained earlier to the lattice. The lattice energy appears as heat in the conductor.

Thus, electrons act as a medium for transferring energy from source to conductor. The electrostatic field present everywhere in the system is a facilitator of this energy transfer process. The nonelectrostatic field in the source transfers the source energy into charge carriers flowing through it in the form of potential energy of the charged particles in an electrostatic field. The charged particles carry this potential energy with them into the conductor. The non-electrostatic force (*i.e., *the average effect of inelastic collisions) absorbs the potential energy of charges and transfers it to the lattice. The electrostatic field that is present within the conductor facilitates this process by converting the potential energy of charged particles into kinetic energy before they can deliver it to atoms through inelastic collision process.

Thus, electrostatic field permeating throughout the system is a necessary requirement for conduction and energy transfer process to take place in an electrical system. The required electrostatic field is created by surface charge distributions on conducting surfaces everywhere in the system.

#### 1.2.5 Two-Terminal Resistance Element

Consider the steady voltage source with resistive load across it shown in Fig. 1.2-3.

Let us work out the electrostatic potential difference between *e *and *f. *

Work to be done *against electrostatic force *to carry a unit positive test charge around a closed path is zero. Therefore, the work to be done to take +1 C charge from *f * to *e *must be the same whether we move it through a path that lies inside the conducting substance or outside. But the electrostatic field is given by inside the conductor. Therefore, with *dl *oriented from *f *to *e. *The value of * * this integral will be same for any path through the conducting substance. However, evaluation of the integral to yield a closed-form result will be possible only in simple cases where the geometry of conductor has some kind of symmetry or other.

**Fig. 1.2-3** Pertaining to voltage across a two-terminal resistance

We consider a simple case of a conductor with uniform cross-section. The total current may be assumed to distribute itself uniformly throughout the cross-section in such a conductor. This results in a current density vector that has a constant magnitude of *I/A *(*A *is the area of cross-section) and direction parallel to the axis of conductor. This is a satisfactory assumption everywhere except at the connection ends. With this assumption, with *l *as the length of conductor and *A *as its uniform crosssectional area, we get,

Eqn. 1.2-2 relates the electrostatic potential difference across the connection points of a piece of conductor with uniform cross-section to the current flow through it. The proportionality constant is dependent on material property (conductivity or resistivity) and geometry of the conductor. This proportionality constant is called the *resistance parameter R. *

However, actual connection point between the resistive material and external circuit may not be accessible for observation of voltage. We measure the voltage across a resistance by connecting a voltmeter to the connecting wire on either side of the element. Assume that the voltmeter is connected across *a-c. *Then, the voltmeter will read the electrostatic potential difference *V _{ac}*. But,

Therefore, a unique voltage difference can be assigned to the conducting body only if the conductivity of connecting wires is infinitely large. However, it is to be noted that this does not imply thick connecting wires. In fact, Circuit Theory assumes that connecting wires have zero resistance *and *negligible thickness. The reason behind the assumption of negligible cross-section for connecting wires will be explained in a later section.

**Fig. 1.2-4** Two-terminal resistance

With this assumption, the electrostatic field inside connecting wires will be zero (since conductivity is infinite). Then, the electrostatic potential difference between the ends of conducting body has a unique value irrespective of which pair of points (*a *and *b*) on the connecting wire are chosen to measure it.

Now a unique voltage and current variable pair can be assigned to the conducting body and its electrical behaviour can be described entirely in terms of these two variables. This model of a conducting body is called the *two-terminal resistance element *model. The symbol and element relation is shown in Fig. 1.2-4.

Ohm’s Law, which is an experimental law, states that the voltage drop across a two-terminal resistance made of a linear conducting material and maintained at a constant temperature is proportional to the current entering the element at the higher potential terminal.

Resistivity and Conductivity are functions of temperature. If the temperature range considered is small, resistivity may be approximated as *ρ*(*T*) = *ρ*(*T*_{0})[1 – *α*(*T *– *T*_{0})] where *ρ*(*T*_{0}) is the known resistivity at temperature *T _{0}* and

*α*is the

*temperature coefficient of resistivity.*

#### 1.2.6 A Time-Varying Voltage Source with Resistance Across it

Now we consider a time-varying source of non-electrostatic force with a resistance load. Refer to Fig. 1.2-5.

We assume that the conducting matter inside the source has infinite conductivity. Hence the non-electrostatic force at every point inside the source has to get cancelled by the electrostatic force at that point at all instants of time. We implicitly assumed that electromagnetic disturbances propagate with infinite speed in making this statement.

**Fig. 1.2-5** A time-varying source e.m.f. with conductor across it

The non-electrostatic field inside the source is a function of time here. Thus is a function of space and time. Therefore, also has to be function of time and space in order to match *. *A time-varying * *inside the source calls for a surface charge distribution that is time-varying. Thus the charge distributed in the system *Q*(*t*) is a time-varying function.

But, now the current flowing in the circuit has two functions to perform – (i) supply the time-varying surface charge requirement at all points in the system (ii) transfer the source energy to the conducting substance. Therefore, the net charge crossing different cross-sections in unit time will not be equal. For instance, consider the two cross-sections marked C and D in Fig. 1.2-5. The volume between these two cross-sections has a certain quantity of charge distributed on its surface at *t. *The quantity of charge that has to get distributed in the same surface is different at *t+**∆t. *Therefore the current crossing D can not be equal to the current crossing C since a portion of the current crossing C will get used up in supplying the required change in surface charge distribution.

Thus, the current crossing various cross-sections will be different and there is no unique single value of current in the circuit at any instant. We restore uniqueness to the circuit current by resorting to certain assumptions.

First, we assume that the connecting wires are very thin. In this case it is possible to show that the surface charge distribution on the surface of connecting wires will be extremely small in value compared to the charge distribution elsewhere. Thus, Circuit Theory assumes that (i) connecting wires are made of material with infinite conductivity so that the electrostatic field inside connecting wires is zero and voltage drop in them is zero and that (ii) connecting wires are so thin that there is virtually no surface charge distributed on their surface. With only negligible surface charge on their surface, the connecting wires will not divert any portion of current flowing through them to supply the changes in surface charge distribution. Then, the current through a section of connecting wire will be the same everywhere.

The next assumption used in Circuit Theory is that the current component that is needed to supply the changes in surface charge distribution at any point in the system is a negligible portion of the current at that point. Note that this does not amount to ignoring the electrostatic charge distribution altogether. That can not be done. Charge distribution is essential for conduction and energy transfer to take place at all. It is only that we chose to ignore the diversion of current for creating a time-varying charge distribution at various points in the system. Obviously, this assumption will be satisfactory only if the source electromotive force is a slowly varying one.

With these assumptions, the currents everywhere in the circuit in Fig. 1.2-5 will be described by a unique function of time. Then there is essentially no difference between an electrical system with a steady-source and an electrical system with a time-varying source. The conductor in Fig. 1.2-5 can be modeled by a two-terminal resistance element satisfying Ohm’s law on an instant to instant basis. If *v*(*t*) is the electrostatic potential difference across the resistance and *i*(*t*) is the current entering the higher potential terminal, then *v*(*t*) = *i*(*t*)* R. *

##### 1.3 TWO-TERMINAL CAPACITANCE

Consider the electrical system shown in Fig. 1.3-1. A source of time-varying electromotive force is connected to a pair of metallic electrodes A and B. We assume that the connecting wires are of infinite conductivity and near-zero cross-section. Further, we assume that the metallic electrodes are made of material with infinite conductivity. Therefore the electrostatic field inside the two electrodes will be zero at all *t *even when there is current flow in the electrode material.

**Fig. 1.3-1** A time-varying e.m.f. source with two electrodes

The total surface charge distributed on the conducting surfaces in the system has two components – the charge distributed on the source terminal and the charge distributed on the electrode. The charge distributed on the connecting wire is negligible since the wire is assumed to be very thin.

The surface charges on the source terminals and electrodes will assume suitable magnitudes and suitable distributions such that (i) the non-electrostatic field in the source is cancelled by the electrostatic field created by the charge distributions on an instant to instant basis everywhere within and (ii) the electrostatic field everywhere inside the connecting wires and electrodes is zero at all time. Thus, *Q*(*t*)*, *the total charge distributed in the electrode and the manner in which it is distributed will depend on of the source, the spatial geometry of the entire system and material/medium dielectric properties. Therefore, *Q*(*t*) will change if the source is moved without affecting the relative position of electrodes. The voltage between the electrodes A and B – *i.e., V _{AB}*(

*t*) – will be equal to the electromotive force always; but the charge stored in the electrode system will vary with the spatial position of the source. Thus a unique ratio between

*Q*(

*t*) and

*V*(

_{AB}*t*) will exist only for a particular spatial arrangement of source and electrodes. The ratio will change with the position of source and can not be called a property of electrode arrangement alone.

All components in an electrical system will have static charge distributions at their terminals and on their surfaces. The electrostatic field at a point is the superposition of fields created by all these charge distributions. Thus, the voltage across terminals of one component will be decided by the work done in carrying a unit positive charge across the terminal pair against an electrostatic force that is decided by the static charge distributions in the entire electrical system. Thus, a simple ratio of the voltage across terminals of one circuit element to the value of charge distributed at its terminals and surface can not be defined in general.

Now we introduce certain assumptions so that we can ascribe the ratio *Q*(*t*)*/V _{AB}*(

*t*) to the electrode pair A and B without any reference to the position of other elements in the system. We assume that the distance between electrodes and the physical dimensions of the two-electrode system are very small compared to the distance between the two-electrode system and other circuit elements in the electrical system. [The reader may think of a parallel plate capacitor of large capacitance value and wonder how such a capacitor can satisfy this requirement. That is precisely why a parallel plate capacitor is found only in the pages of textbooks. A practical ‘parallel plate capacitor’ has two aluminium foils of large length rolled into a tight cylinder shape with a pair of dielectric films between them. Such an assembly of a pair of electrodes will satisfy the assumption stated above.]

Positive and negative charge distributions of equal magnitude kept close to each other will produce only negligible electrostatic field at distant points. Therefore, the charge distribution on a pair of electrodes that satisfy the assumption stated above would not affect the electrostatic field at the locations where other circuit elements are located. And, charge distributions on other circuit elements will not affect the electrostatic field at the location where this electrode pair is located. Therefore the ratio of charge stored in the electrodes to voltage between the electrodes will depend only on the geometry of the electrode system and dielectric properties of the medium involved.

This unique and constant ratio associated with an electrode pair is defined as its *capacitance *value and the electrode system that satisfies the assumptions explained above is termed as a *two-terminal capacitor. *The magnitude of charge stored in one of the electrodes in a *linear capacitor *is proportional to the voltage across it. The symbol and variable assignment of a two-terminal capacitance is shown in Fig. 1.3-2.

**Fig. 1.3-2** A two-terminal capacitor

In fact, Circuit Theory extends the assumption of ‘*locally confined stationary electrostatic field’ *to all elements in the circuit. It assumes that the electrostatic field created by the charge distribution residing on a particular element (remember that there is no charge distribution on wires; they are of near-zero cross-section. Therefore, charge distributions can be ascribed to elements uniquely) is significant only near that element and is negligible at the location of other elements. This makes the electrostatic field around a circuit element a function of its own charge distribution alone. Therefore, the potential difference across terminals of one element will be proportional to the charge distributed on it. Thus assumption of ‘*locally confined stationary electrostatic field’ *amounts to neglecting electrostatic coupling between various elements. With this assumption, the voltage across a circuit element becomes proportional to the total charge distributed on its terminals and conducting surfaces. The proportionality constant depends on the geometry of the circuit element as well as on material dielectric properties. The fact that there has to be a certain amount of charge distributed on the surface of a circuit element for a voltage difference to exist between its terminals is equivalently described as *the capacitive effect *present in the component. *Thus every electrical element has a capacitive effect inherent in it. *

Therefore, a piece of conductor too has a capacitive effect associated with it. We ignored the current component that is required to support a time-varying charge distribution across a resistance in the previous section (Section 1.2) in order to define a two-terminal resistance. This is equivalent to neglecting the capacitive effect that is invariably present in the resistance. There is no pure resistance element in practice. All resistors come with a capacitive effect. However, if the capacitance that is present across a resistor draws only negligible current in a given circumstance, then, it may be modeled by a two-terminal resistance.

The capacitance that is present across a two-terminal resistance is called the *parasitic capacitance *associated with it. The adjective ‘*parasitic*’ gives us an impression that it is some second-order effect that has only nuisance value. That is not true – it arises out of the charge distribution that is required to make conduction possible in the resistance. Without this parasitic capacitance the resistor will not carry any current at all!

The relation between the charge stored in a capacitor and voltage across it is given by *q*(*t*) = *Cv*(*t*)*. C, *the capacitance value has ‘Coulomb per Volt’ as its unit. This unit is given a special name – ‘Farad’. One Farad is too large a value for capacitance in practice. Practical capacitors have capacitance value ranging from few pFs (1 pF = 10^{–12} F) to few thousand *µ*Fs (1*µ*F = 10^{–6 }F). The value of *C *is a constant if the geometry of capacitor does not change with time and the material that is used as the dielectric between the metallic electrodes is linear, homogeneous and isotropic. If the value of *C *is a constant, it is called a *linear capacitor. *

The current that has to flow into the positively charged electrode of the capacitor is given by rate of change of the charge residing in that electrode. Therefore, the voltage across a linear capacitor is related to the current flowing into the positive electrode as below.

The current through a capacitor depends on the first derivative of voltage appearing across it. Therefore, the current flow through the parasitic capacitance that is inevitably present across any electrical element can be neglected in the circuit model for that element only if the rate of change of electrical quantities involved in the circuit is small enough. Thus, a two-terminal resistance will model a piece of conducting substance with sufficient accuracy only if the frequency of voltage and current variables in the circuit is sufficiently small.

We have seen that there is no purely resistive two-terminal element in the physical world. A parasitic capacitance always goes along with a resistance. However, is there a pure two-terminal capacitor in real world?

**Fig. 1. 3-3** Pertaining to the discussion on resistive effect in a capacitor

Consider a parallel-plate capacitor with a current *i*(*t*) flowing into its positive plate as shown in Fig. 1.3-3. The current entering the positive plate from the left has to deposit charge all along the plate. Therefore the current has to flow through the cross-section of the plate from left to right. The magnitude of current comes down with length traveled towards right. Specifically, the current crossing the cross-section of the plate at mid-point will be about 0.5*i*(*t*)*. *Thus, there is a linearly varying current crossing the cross-section of metallic electrode at any instant. This current flow meets with the impeding resistance of the metallic plate. Thus there will be a resistive voltage drop along the length of the plate and the plates will no longer be equipotential surfaces. This resistive effect will produce power loss and heating in the capacitor.

There is yet another resistive effect present in a capacitor. A practical capacitor may use some dielectric material (like paper, polyester film, polypropylene film etc) between the electrodes in order to increase the capacitance value. The dielectric substance in between the electrodes has a non-zero (though very small) conductivity; resulting in leakage current that flows from positive plate to negative plate. Thus the current entering the capacitor gets partially diverted for supplying this leakage current and only the remaining portion is available to create the charge distributions on plates.

A two-terminal capacitor model can model a physical capacitor only if these two resistive effects – one in series and one in parallel – can be neglected. The first resistive component can be made negligible by increasing the thickness of the electrodes and using a high conductivity metal to construct them. These two resistive effects are called the *parasitic resistive effects *in a capacitor.

**Example: 1.3-1 **

The current flowing through a capacitor of 0.5F is given by Find the voltage across the capacitor at *t* = 5 s.

**Solution**

Voltage across a capacitor at *t *is proportional to the area under the applied current waveform from –∞ to *t. *No current was applied to the capacitor until *t *= 0. Therefore, the voltage across capacitor is 0 V for *t *< 0.

The current through the capacitor is 2A from *t *= 0. Hence, the voltage across capacitor increases at the rate of 2A/0.5F = 4V/s from *t *= 0. The expression for the voltage will be *i*(*t*) = 4*t* V as long as 2A is maintained through it.

The value of voltage across capacitor at *t *= 2 s is 8V. The current through the capacitor is 0A from *t *= 2 s onwards. There is no area addition to the area under the current waveform after *t *= 2 s. Therefore, the capacitor voltage continues at the same value that it had when the applied current went to zero. Note that a current source of 0A is an open-circuit. A non-zero voltage can remain in an *ideal *capacitor forever.

**Example: 1.3-2 **

A resistor of value 5Ω and a capacitor *C *are connected in parallel. The voltage applied across the parallel combination is *v*(*t*) = *V _{m}* sin

*t*V. The current through the resistor and capacitor are seen to have equal amplitude. Find the value of

*C.*

**Solution **

The current through the capacitor is given by

The current through resistor is 0.2*V _{m }* sin

*t*A. If these two currents have same amplitude, the value of capacitance must be

*C*= 0.2F.

The current through a capacitor of 0.1F is known to be 0.5*π* sin 2*πt* A in the time interval [0,0.5 s]. The voltage across capacitor at *t *= 0.5 s is seen to be 0 V. What was the voltage across capacitor at *t *= 0?

**Solution**

The change in voltage across a capacitor over a given time interval is (1/*C*) times the charge delivered to it over the same time interval. Charge delivered in a given time interval is given by the area under the current waveform over the same interval.

The voltage across capacitor at the end of a given time interval = voltage at the beginning of the interval *plus *the change in voltage over the given time interval.

The voltage across capacitor at *t *= 0.5 s is given as 0 V. Therefore, the voltage across capacitor at *t *= 0 must have been –5 V.

##### 1.4 TWO-TERMINAL INDUCTANCE

A moving charge that is part of a steady-current flow (*i.e., *DC current) in a circuit is, in general, acted upon by non-electrostatic forces contributed by source regions, electrostatic forces contributed by charge distributions and frictional forces within conductors. A new force called *induced electric force *that acts on such a moving charge makes its appearance in circuits carrying unsteady (*i.e., *time-varying) currents. This new force component gives rise to a circuit element called *inductance. *

#### 1.4.1 Induced Electromotive Force and its Location in a Circuit

Consider a source of electromotive force with short-circuit across it as in Fig. 1.4-1. The conducting material inside the source is assumed to be of infinite conductivity. The shorting wire is assumed to be of material of infinite conductivity and the cross-section of the wire is taken to be of near-zero dimension. These assumptions imply that there is no net force needed to make charged particles move inside the source as well as inside the shorting wire. Further, the static charge distribution on the surface of shorting wire is negligibly small since the wire is very thin.

**Fig. 1.4-1** A shorted source of electromotive force

Therefore, there can be no resistive voltage drops inside the source and the shorting wire. Hence, the current flow in the system will reach infinitely large level if the electromotive force of the source is a steady value – *i.e., *if the source is a DC voltage source. Of course, the small resistances that are inevitably present within the source and in the shorting wire will limit the current in practice.

Now consider the situation with a time-varying electromotive force in the source. The nonelectrostatic field produced by the source in the source region – *i.e., *– is a time-varying quantity. Therefore, the charge distribution on the source terminals has to be time-varying in order to generate a time-varying electrostatic field inside the source for exact cancellation of time-varying . This time-varying charge distribution will, in turn, produce time-varying electrostatic field inside the shorting wire, resulting in a time-varying current flow in the wire. However, the conductivity of wire material is assumed to be infinity. Hence, we should expect a time-varying current of infinite magnitude in the wire. But, the current is observed to have finite amplitude in practice. If there is no resistive effect in the wire (conductivity is taken to be infinity), then what is the mechanism responsible for preventing the current from reaching infinitely high value?

The required mechanism arises out of the third component of force of interaction between two charges in arbitrary motion. We had observed in Section 1.1 that this component is dependent on relative acceleration of interacting charges and is given by is the component of force experienced by charge *q*_{2} due to charge *q*_{1} and is the velocity of *q*_{1}. We had termed this component of force as the *induced electric force.*

Thus, a moving charge can experience four kinds of force in general – (i) the force due to the non-electrostatic field inside a source acting on it (ii) force due to electrostatic field (iii) magnetic force due to other moving charges (iv) induced electric force from other charges which are accelerating with respect to the location of this charge. The first kind of force will be present only if the charge is inside a source region.

Magnetic force on a moving charge is in a direction perpendicular to the velocity of charge. Hence magnetic force can not change the energy of a charged particle. Therefore magnetic force can not affect the current flow in a circuit though it may produce mechanical forces in current carrying systems. Hence, we need to consider only the remaining three forces on a moving charge in circuit analysis.

The net induced electric force experienced by a charge located at a certain point in a circuit carrying *steady current *(*i.e., *a DC current) due to all the other moving charges in the circuit will be zero. We accept this statement without proof.

Hence, moving charges in a DC circuit do not experience any induced electric force provided there are no other circuits carrying time-varying currents in its vicinity.

However, the net induced electric force experienced by a charge located at a certain point in a circuit carrying *time-varying current *due to all the other moving charges in the circuit *will not be *zero. It will experience an induced electric force that will be proportional to its value. Thus, we can define induced electric field * *at a point as the net induced electric force experienced by +1 C charge kept at that point. This field exists everywhere in space (including the source region) unlike the nonelectrostatic field that is present only within the source.

Now, the force balance on charges inside the source requires that since the material inside the source has infinite conductivity. Similarly the force balance condition inside the shorting wire requires that since the shorting wire is of infinite conductivity.

Electrostatic field is a conservative field. Hence the work to be done against the electrostatic force in carrying a unit test charge around a closed loop is zero. Induced electric field is non-conservative. Hence the work to be done against the induced electric force in carrying a unit test charge around a closed loop is non-zero.

The electrostatic field inside the source and the shorting wire can be expressed in terms of induced electric field and the non-electrostatic field generated by the source as follows.

It must be noted that all the field quantities appearing in these equations are functions of space as well as time and that these equations are valid at all points inside the source and wire. Strictly speaking, they should have been expressed as below.

Let a +1 C charge be taken around the circuit from B to A through the source and from A to B through the shorting wire. Then,

But inside the source and inside the shorting wire. Therefore,

The quantity on the left-hand side is the electromotive force of the source – that is, it is the work done *by *the non-electrostatic force provided by the source when +1 C is taken through it from its negative terminal to the positive terminal. Similarly, the quantity on the right-hand side is the *negative *of work done *by *the non-conservative induced electric force when +1 C is taken through the loop in clockwise direction. That is, it is the *negative *of electromotive force due to the induced electric field in clockwise direction in the loop. We will term this electromotive force as the *induced electromotive force. *Obviously, the current in this circuit will have suitable magnitude at all instants such that the source electromotive force and induced electromotive force meet each other without leaving any net electromotive force in the circuit loop.

#### 1.4.2 Relation between induced electromotive force and current

The induced electric field at a point in a circuit is the superposition of terms of the form where *q *is the charge per carrier and is the carrier velocity and *r *is the distance between the carrier and the point. All moving carriers in the circuit are to be considered in the vector summation. Since current in the circuit is related to carrier velocity, we expect the summation to turn out to be proportional to in the circuit. Thus, induced electric field at all points in space will be proportional to Electromotive force of a force field is defined as the *work done by the field *when a unit test charge is taken through a closed path lying in that force field. Hence the induced electromotive force in any closed path will be proportional to The proportionality constant will depend on the spatial geometry of the circuit and the magnetic properties of the medium involved. This proportionality constant is termed as the *inductance *of that closed path. Inductance is designated by the symbol *L. *If the geometry of the circuit does not vary with time, the value of *L *will be a constant.

#### 1.4.3 Farady’s Law and Induced Electromotive Force

Faraday’s law of electromagnetic induction states that the induced electromotive force in a closed path in a circuit is equal to the time rate of change of flux linkage through that closed path. If the closed path is traversed in counter-clockwise direction and positive flux linkage is defined according to righthand screw rule, then, this law states that induced electromotive force is the flux linkage through the closed path in ‘Weber-turns’ unit.

Faraday’s law gives the total induced electromotive force in a closed path. *However, Faraday’s law can not tell us where exactly this electromotive force is located. *The discussion in the previous subsection has shown that the induced electromotive force is distributed all around the closed path.

Determining the polarity of induced electromotive force by using Faraday’s law can be confusing at times for a beginner in circuit analysis. Lenz’s law is a better option for this purpose. Lenz’s law, in effect, states that the induced electric field will be in such a direction that it opposes the change in current that is the cause for appearance of the induced electric field. Refer to Fig. 1.4-1. Assume that the current *i*(*t*) is increasing at some time instant. This means that all the charge carriers are accelerating in the direction of current flow at that instant. This acceleration is the cause of induced electric field in the wire and elsewhere. The direction of induced electric field inside the wire will be such that the induced electric force on a positive charge will tend to *deccelerate *it. Thus the induced electromotive force will work against the source electromotive force.

Thus, the induced electromotive force in a closed loop in a circuit with time-varying current is given by (for static circuits; see the previous sub-section) as well as by (by Faraday’s law) with the direction of electromotive force as per Lenz’s law. Therefore,

Thus, inductance of a closed path is the flux linkage in that closed path for unit current. The unit of inductance is Weber-turns/ampere. This unit is given a special name – ‘Henry’and is represented by ‘H’. Since yields an electromotive force, inductance gets another unit – volt-sec per amp. It follows that volt-sec and weber-turns refer to the same physical quantity.

#### 1.4.4 The Issue of a Unique Voltage Across a Two-Terminal Element

Refer to Fig. 1.4-2. A time-varying source of electromotive force is connected to a conductor by thin connecting wires of infinite conductivity. The charge distributions at the source terminals and load terminals produce electrostatic field everywhere in space. The electrostatic field inside the source cancels the non-electrostatic field available inside the source and the induced electric field inside the source exactly. (The conductivity inside the source is assumed to be infinity). Electrostatic field inside the connecting wires cancels the induced electric field inside them. Electrostatic field inside the conductor meets the frictional force arising out of collisions of charge carriers with atoms in the lattice and the induced electric force manifesting inside the conductor.

**Fig. 1.4-2** Pertaining to uniqueness of terminal voltage of a two-terminal element

Three issues arise in this context.

- The voltage across two points is the electrostatic potential difference between the two points. The voltage across the resistance is given by the potential difference between
*e*and*f.*This voltage can be obtained by calculating the work*to be*done in carrying a +1 C charge from*f*to*e*through the inside of the conductor. But, the electrostatic field inside the conductor is equal to –(frictional force field + induced electric field). Therefore the terminal voltage of resistance will contain a resistive voltage drop plus a term that depends on (*i.e.,*an inductive voltage drop). Therefore, the conductor can no longer be modeled as a pure two-terminal resistance. - The voltmeter connected on the right of the conductor attempts to measure the terminal voltage of the resistance right across its terminals. However, the voltmeter connection creates a closed path comprising the resistance element, connecting leads and the meter. This closed path will have induced electric field everywhere inside the connecting leads as well as within the meter. Thus the meter ends up reading the terminal voltage plus the induced electromotive force in the voltmeter leads and in the meter internal circuit. Thus, the reading is in error. The amount of error will keep changing with geometry of voltmeter connection – that is, the reading will be different when the leads are disturbed into a new spatial configuration. The amount of error is dependent on the time- rate of change of flux linkage of the voltmeter loop.
- The voltmeter connected on the left of the conductor reads the actual terminal voltage of the resistance element plus the induced electromotive force in the path
*f-c-*VM-*a-e.*Thus, the reading includes the induced electromotive force in the voltmeter leads and portions of connecting wire in the circuit.

Thus, no unique voltage can be assigned to the resistance by measurement. Therefore, we bring in certain assumptions. The first assumption is that the induced electric field (and, hence, the induced electromotive force too) inside the connecting wires everywhere in the circuit is negligible. The second assumption is that the induced electric field inside the conductor (or inside a capacitor) is negligible. Obviously, this is equivalent to ignoring the inductive effect present everywhere in the circuit.

No circuit can satisfy these assumptions exactly (except in DC circuits). Induced electric field is proportional to rate of change of current. If the rate of change of current is low, then, the strength of induced electric field inside the sources, resistances, capacitors and connecting wires will be low compared to electrostatic field. Thus, the assumptions stated above can be employed if the rate of change of current in the circuit is sufficiently small.

With these assumptions, it becomes possible to model a physical resistor by an ideal two-terminal resistance model and a physical capacitor by an ideal two-terminal capacitance model. Further, voltages across a source, resistor and capacitor become unique even with time-varying current in the circuit.

But, this does not mean that we will not make use of induced electromotive force in a circuit at all!

#### 1.4.5 The Two-Terminal Inductance

An electrical device, in general, can have four kinds of force fields that can affect current flow at every point inside the device. They are:

- Some non-electrostatic field arising out of some kind of potential energy stored within the device – for instance, the non-electrostatic field generated by chemical potential energy in a dry cell.
- Electrostatic field created by charge distributions on this device as well as other devices nearby.
- Induced electric field created by time-varying current flowing in the circuit containing this device as well as in neighboring circuits.
- Non-electrostatic force field due to the collisions between moving charged particles and lattice atoms during conduction.

The model used in circuit theory for a device will depend on which of these are strong and which are negligible.

Electrostatic field will be present in all devices in an electrical system and can not be ignored in any device. Electrostatic field inside *any *device is a function of charge distributions on *all *devices in the system. However, if the physical dimensions of the devices are small compared to spatial distance between the devices, then, the electrostatic field inside a particular device is determined uniquely by the charge distribution on its surface alone. Then, there will exist a unique ratio between the electrostatic potential difference across its terminals and the total charge stored on its surface. This is how a two-terminal capacitance can be defined at all.

Thus, a two-terminal capacitance is a model for an electrical device that has only electrostatic field inside in it and the electrostatic field inside depends only on its own charge distribution. The nonelectrostatic field existing in the metallic electrodes when current flows in them is ignored in an ideal two-terminal capacitance. The induced electric field that exists inside the device due time-varying currents everywhere is also ignored in an ideal two-terminal capacitance.

A piece of conductor with finite conductivity carrying a current will have electrostatic field, non-electrostatic field arising out of frictional forces and induced electric field due to time-varying currents in the circuit as well as in other circuits. An ideal two-terminal resistance models this piece of conductor by ignoring (i) the current component that is needed to build a time-varying charge distribution on its surface and (ii) the induced electric field inside the conductor.

Circuit Theory models a piece of connecting wire by ignoring all fields that exist within the wire and taking all of them to be zero at all instants. Thus, Circuit Theory assumes that there is no resistive drop across connecting wire; there is no induced electromotive force in connecting wire and there are no charges distributed on the connecting wire. Such an element is called the *ideal short-circuit element. *

An electrical source will have all the four kinds of fields inside. However, the *ideal two-terminal source model *of Circuit Theory attempts to model such a source by (i) ignoring the non-electrostatic field arising out of friction within conductor (ii) ignoring the induced electric field inside in comparison with electrostatic field and (iii) ignoring the component of current needed to build a time-varying charge distribution at its terminals.

And, the *ideal two-terminal inductance model *of Circuit Theory is a model for an electrical device in which there are only two fields – the induced electric field and the electrostatic field. It is not a source and hence there is no source field. It uses conducting substance and hence there is a non-electrostatic field arising out of collisions of charge carriers with lattice atoms when a current flows through it. But this field is ignored in comparison with the other fields. Further, the component of current needed to build a time-varying charge distribution on its surface is assumed to be negligibly small.

Consider a long piece of round conductor carrying a time-varying current as shown in (a) of Fig. 1.4-3. This wire is not a connection wire. It has a non-zero cross-sectional area. But it is indicated by a line in Fig. 1.4-3. The current entering the conductor is *i*(*t*) and the same current leaves the conductor at far end. The value of current crossing any cross-section at a particular instant will be the same everywhere since we neglect retardation effect as well as the current that is required to build the surface charge distribution.

There is induced electric field at all points within this conductor. The induced electric field at a point inside is the sum of terms of the form where *q *is the charge per carrier and is the carrier velocity and *r *is the distance between the carrier and the point – as many terms as there are moving carriers in the conductor. All the charge carriers will be moving with same instantaneous velocity that is proportional to *i*(*t*)*. *But the distance between the point at which the induced electric field is calculated and the location of carrier (*i.e., r*) will be large for all those carriers that are moving at a far away location at the instant under consideration. Therefore, only those carriers that are presently moving within the immediate vicinity of the point at which field is being calculated will contribute to the induced electric field significantly. Thus the induced electric field will be relatively low everywhere, and, correspondingly the total induced electromotive force in the long conductor will be relatively low. The induced field as well as the total induced electromotive force will be proportional to that appears in the equation for induced electric field due to a moving charge is directly related to

The conductor is assumed to be of large conductivity. Then, the net force experienced by a charge carrier inside must be zero. Therefore, the induced electric field at every point within the conductor will be cancelled exactly by the electrostatic field created by the surface charge distribution. This charge distribution is shown in Fig. 1.4-3 assuming that is positive at the instant under consideration.

**Fig. 1.4-3 **Towards a two-terminal inductance

A physical inductor is constructed so as to strengthen the induced electric field and the induced electromotive force inside the conductor forming the inductor.

Refer to (b) of Fig. 1.4-3. The same conductor is wound into a coil of 4 turns. Now the relative distances between moving carriers in various sections of the wire are reduced considerably. Hence the induced electric field at any point in the conductor will have a value greater than the value when the entire conductor was stretched out in a straight-line as in (a). Therefore, the total induced electromotive force will also be higher. Obviously, the value of induced electromotive force will go up further if the turns can be kept closer.

Refer to (c) of Fig. 1.4-3. The same conductor is wound into a coil of lower diameter and higher turns. And the turns are kept closer. This structure will have still higher induced electric field everywhere. The total induced electromotive force will also be higher. If the wire has an insulation cover the turns can touch each other.

Thus, winding a long length of wire into an optimally sized and layered coil with turns touching each other will result in large induced electric field everywhere in the wire and large induced electromotive force over the length of the wire when the current through the coil is time-varying. The induced electric field everywhere inside will be cancelled by the electrostatic field created by the surface charge distribution all along the wire surface.(The conductivity of wire material is assumed to be very large). Therefore, the electrostatic potential difference between the ends of the coil – *i.e., *the voltage difference between coil terminals – will be equal to the total induced electromotive force in the coil. The polarity of voltage will follow Lenz’s law.

What we have described here is an air-cored coil. Air-cored inductor is essentially a long piece of wire that is arranged to occupy a small region of space of dimensions that are very small compared to its length. Such a spatial confinement of a long wire results in strengthening of induced electromotive force in it. Further strengthening of induced electric field inside the wire can be attained by winding it around a core made of magnetic material (usually iron). If the core made of magnetic material is a closed structure, the induced electric field will be enhanced further. Moreover, a closed core structure confines the time-varying magnetic field to the core itself and reduces the magnetic flux linking rest of the circuit to negligible levels.

A physical inductor that is designed to strengthen the induced electric field within itself, while confining the time-varying flux-linkage to predominantly within itself, can be modeled by an ideal *two-terminal inductance model *provided the resistive voltage drop in the coil can be neglected and the capacitive effect due to surface charge distribution over the coil surface can be neglected.

The value of inductance depends on the geometry of the coil and core assembly and the magnetic properties of the core. *Inductance of a coil is proportional to the square of number of turns of the coil, area of a turn and magnetic permeability of the core material. *

The symbol and variable assignment for an ideal two-terminal inductance is shown in Fig. 1.4-4.

The governing equations of a linear two-terminal inductance are:

**Fig. 1.4-4** A two-terminal inductance

where *ψ*(*t*) is the flux-linkage at *t *in Weber-turns, *v*(*t*) is the voltage across the inductance and *i*(*t*) is the current entering the higher potential terminal. *i*(0) is the current in the inductor at *t *= 0.

A coil can have induced electric field and induced electromotive force present in it due to accelerated motion of charges (*i.e., *time-varying current) in the circuit in which it is connected and/or due to accelerated motion of charges taking place in another physically separated circuit. The electromotive force induced in the coil due to its own time-varying current is termed as *self-induced electromotive force *and the electromotive force induced in it due to current in another circuit is termed as *mutually induced electromotive force. *Self-induced electromotive force is associated with an inductance value called *self-inductance. *Eqn. 1.4-1 describes the governing equations of self-inductance.

There is no region without induced electric field and induced electromotive force in any circuit carrying time-varying current. All devices and components of such a circuit are affected by electromagnetic induction. Thus, all devices have inductive effect associated with them. The associated inductance will be called the *parasitic inductance *of the two-terminal element (unless it is a two-terminal inductance). Ideal two-terminal element models ignore the parasitic inductance in a resistor, capacitor, source and connecting wire.

**Example: 1.4-1 **

The voltage applied across an inductor of 2 H is given by* * Find the current in the inductor at *t* = 5 s.

**Solution **

Current in an inductor at *t *is proportional to the area under the applied voltage waveform from –∞ to *t. *No voltage was applied to the inductor till *t *= 0. Therefore, the current in the inductor is 0 A for *t *< 0.

The voltage applied across the inductor is 2 V from *t *= 0. Hence, the inductor current increases at the rate of 2 V/2 H = 1 A/s from *t *= 0. The expression for the current will be *i*(*t*) = *t*A as long as 2 V is maintained across it.

The value of current at *t *= 2 s is 2 A. The voltage applied across the inductor is 0 V from *t *= 2 s onwards. There is no area addition to the area under the applied voltage waveform after *t *= 2 s. Therefore, the inductor current continues at the same value that it had when the applied voltage went to zero. Note that a voltage source of 0 V is a short-circuit. A non-zero current can circulate in a shorted *ideal *inductor forever.

A 1 H inductor and a 10 Ω resistor are connected in series. The current that flows through the series combination is such that the voltage across the resistor is equal to the voltage across the inductor for all *t. *The voltage across the inductor at *t *= 0 is seen to be 1 V. (i) Find circuit current at *t *= 1 s. (ii) Find the energy dissipated in the resistor in the time interval [0, 1 s].

**Solution**

- Let
*i*(*t*) be the circuit current. Then, the voltage across resistor is 10*i*(*t*) and the voltage across the inductor is These are stated to be equal for all*t. i.e.,*for all*t*.The only function that can satisfy this requirement is an exponential function since only exponential function has the property that the waveshape remains the same on differentiation and integration. (Note – Sinusoidal functions are special cases of a generalised complex exponential function). Hence, we try the solution

*Ae*Substituting this function in the above equation, we get,^{at}.*Aαe*^{αt}= 10*Ae*^{αt}for all*t*∴*α*= 1Inductor voltage at

*t*=*Aαe*^{αt }|_{t=}_{0}= 0*Aα*=*A*. This value is 1 V. Therefore*A*= 1.∴

*i*(*t*) =*e*A for all^{t}*t*is the current through the series combination.Hence, current through the circuit at

*t*= 1 s is 2.7318 A. - Energy dissipated in the resistor over

##### 1.5 IDEAL INDEPENDENT TWO-TERMINAL ELECTRICAL SOURCES

Electrical sources are devices that are capable of applying a non-electrostatic force on a charge that moves through the source region. They can deliver energy to the charged particle or absorb energy from it.

#### 1.5.1 Ideal Independent Voltage Source

A two-terminal voltage source will have non-electrostatic field at every point inside the source region. The charge distribution on the terminal surfaces of the source will create an electrostatic field at all points inside the source. The two fields cancel each other at all points at all instants under all conditions if the material inside the source is of infinite conductivity. The terminal voltage (which is an electrostatic potential difference) will always be equal to the internal electromotive force in that case.

The conducting material inside the source will have finite conductivity in practice. Charge carriers moving inside such material require net non-zero force to work against collisions with lattice atoms. This will call for a difference between the internal non-electrostatic field and the electrostatic field. Then, the terminal voltage will be different from the internal electromotive force. It will be less than the internal electromotive force if the source is delivering positive current out of its positive terminal and it will be more than internal electromotive force if it is absorbing positive current at its positive terminal. The difference between terminal voltage and internal electromotive force is termed as the voltage developed across the *internal resistance *of the source.

A practical voltage source with time-varying internal electromotive force will require a time-varying current flow component to support the time-varying surface charge distribution on its terminals. That is, a practical voltage source has a parasitic capacitance right across its terminals.

**Fig. 1.5-1** Approximate equivalent circuit of a practical voltage source

A practical voltage source will have induced electric field inside due to its own time-varying current as well as due to time-varying currents elsewhere in the circuit and in neighboring circuits. This will affect the voltage appearing at its terminals. That is, a practical voltage source has internal parasitic inductance too. Thus a detailed circuit model for a practical voltage source will be as shown in Fig. 1.5-1. *L _{i}* is a lumped parameter approximation for the inductive effect distributed within the source.

*R*is a lumped resistance parameter that approximates the distributed resistive effect within the source.

_{i}*C*is a lumped capacitance parameter that approximates the distributed capacitive effect within the source and at its terminals.

_{i }*E*(

*t*) is the internal electromotive force of the source. The + and – signs do not signify the polarity of charges at the terminals. Rather, the + sign indicates the point at which the potential difference is specified and – sign indicates the reference point for specifying the potential difference. Thus

*V*(

*t*) is the voltage of the terminal marked with +

*with respect to*the point marked with – sign at the time instant

*t*in Volts.

**Fig. 1.5-2** Ideal independent voltage source

An *ideal voltage source *is one in which all the three elements *R _{i}* ,

*L*and

_{i}*C*are assumed to be negligible. Thus the terminal voltage of an

_{i}*ideal voltage source*is always equal to its internal electromotive force quite independent of magnitude or waveshape of current delivered or absorbed by it. Such an ideal voltage source is called

*an ideal independent voltage source*if the electromotive force is a function of time only and does not depend on any other electrical or non-electrical variable. An ideal independent voltage source is specified by the following terminal equations.

*v*(*t*) = *E*(*t*) , a specified function of time

*i*(*t*) = Arbitrary, decided by the rest of the circuit in which this source is connected.

The symbol of a constant ideal independent voltage source (that is, a DC source) is shown in (a) of Fig. 1.5-2 and that of a time-varying ideal independent voltage source is shown in (b) of Fig. 1.5-2.

#### 1.5.2 Ideal Independent Current Source

An ideal independent current source delivers or absorbs a current at its terminals that is a specified function of time. Rest of the circuit in which it is connected decides its terminal voltage. The current delivered or absorbed by it does not depend on the voltage that appears across its terminals.

Practical current sources will have a parallel resistance and capacitance at its terminals representing the effect of finite conductivity within the source and charge distribution on its surface and terminals. These parasitic components are neglected in the ‘ideal independent current source’ model.

The symbol of a constant ideal independent current source (that is, a DC source) is shown in (a) of Fig. 1.5-3 and that of a time-varying ideal independent current source is shown in (b) of Fig. 1.5-3.

**Fig. 1. 5-3** Ideal Independent Current Source

An ideal independent current source is specified by the following equations:

*i*(*t*) = *I _{s}*(

*t*) , a specified function of time.

*v*(*t*) = Arbitrary, decided by the rest of the circuit in which this source is connected.

There are no ideal independent voltage sources and ideal independent current sources in practice. These are only models of practical sources that give reasonably accurate results provided they are not applied under extreme loading conditions. The ideal model will undoubtedly fail for a practical voltage source that is shorted or for a practical current source that is open-circuited. The short-circuit current in a DC voltage source is limited by its internal resistance while that of a time-varying voltage source is limited by internal resistance and internal inductance. Similarly, the open-circuit voltage that appears across a practical current source is limited by its internal resistance in the case of a DC source. It is limited by internal capacitance and resistance in the case of a time-varying current source.

In fact, ideal model for a voltage source models a practical voltage source accurately only when the current delivered/absorbed by it is a small fraction of its short-circuit current. Similarly, ideal model for a current source models a practical current source accurately only when the voltage appearing across its terminals is a small fraction of its open-circuit voltage.

#### 1.5.3 Ideal Short-Circuit Element and Ideal Open-Circuit Element

Ideal two-terminal short-circuit element is the element that is used to model a piece of connecting wire in Circuit Theory. It is also used to model an ideal switch in closed condition. It has no resistance, no inductance and no charges distributed on it. The voltage across its terminals is constrained to remain at zero. It can carry an arbitrary current that is decided by rest of the circuit. Thus, definition of ideal short-circuit element parallels that of an ideal independent voltage source. *Hence an ideal short-circuit element may be thought of as a special case of an ideal independent voltage source with E*(*t*) = 0 *for all t. *It is described by the following equations.

*v*(*t*) = 0 V,

*i*(*t*) = Arbitrary, decided by the rest of the circuit in which this source is connected.

Similarly, *an ideal open-circuit element *is equivalent to *an ideal independent current source with I _{s}*(

*t*) = 0.

It is described by the following equations.

*i*(*t*) = 0 A,

*v*(*t*) = Arbitrary, decided by the rest of the circuit in which this source is connected.

In practice, a short-circuit element has a little resistance and inductance in series. A practical open-circuit has a small capacitance shunting its terminals.

##### 1.6 POWER AND ENERGY RELATIONS FOR TWO-TERMINAL ELEMENTS

An ideal two-terminal circuit element has a unique voltage variable assigned at its terminals and a unique current variable assigned to its terminals. The electrical behaviour of such an element can be described in terms of these two variables at all instants. Electromagnetic disturbances are assumed to travel instantaneously to all parts of such an element. This results in an electrical description that is independent of space variables for the element. Such an electrical description for an element is termed as *lumped parameter description. *

Further, ideal two terminal elements have only one kind of electrical phenomena taking place inside them. The capacitive and inductive effects in a practical resistance are neglected in order to arrive at an ideal two-terminal resistance model. The resistive and inductive effects in a physical capacitor are neglected to model it by an ideal two-terminal capacitor. The capacitive and resistive effects in a physical inductor are neglected to arrive at the ideal two-terminal inductance model.

**Fig. 1. 6-1** An ideal two-terminal element

Moreover, lumped two-terminal elements confine the electromagnetic fields associated with them to the space inside them and in the immediate vicinity.

Such a two-terminal element can be represented in general by the symbol below in Fig. 1.6-1. The variable assignment for the element is also shown in the figure.

#### 1.6.1 Passive Sign Convention

Current in a wire has a direction associated with it. The actual direction associated with a current – *i.e., *the *current direction *– is the direction in which positive charges move.

Consider a wire section A – B with a cross-section identified at C as in Fig. 1.6-2. Four possible kinds of charge motion are depicted in (a) to (d) in this figure. The *direction of current *in (a) and (d) is from left to right (from A to B). This is so since negative charge crossing a cross-section in one direction is equivalent to positive charge crossing the same cross-section in opposite direction. The *direction of current *in (b) and (c) is from right to left (from B to A) for the same reason.

However, it is not possible to decide the direction in which current will flow in an element that is a part of a circuit before we actually solve the circuit analysis problem. The actual direction in which positive charge flow through the element can be ascertained only after the circuit solution is obtained. But, despite this we need to assume some direction for current flow in each and every element in a circuit so that we can prepare the circuit equations needed for solving the circuit.

This is where the ‘*reference direction for current*’ comes in. We assign a particular direction along the element as the reference direction for current. That is, we assume that positive charge moves through the element in the direction chosen as the reference direction even before we arrive at the circuit solution. The circuit solution will either confirm our assumption or reveal to us that actual current direction is opposite to the direction we assumed. If the circuit solution returns a positive value for the element current, then positive charge flows through the element in the assumed reference direction (or negative charge flows in opposite direction). If the circuit solution results in a negative value for the element current, then positive charge flows in a direction opposite to reference direction (or negative charge flows in the reference direction).

**Fig. 1. 6-2 **Pertaining to the discussion on direction of current

**Fig. 1.6-3 **Four different choices for reference directions and corresponding statements of Ohm’s law

A similar issue comes up in the case of voltage across a two-terminal element. We can not determine which terminal is the higher potential terminal before we actually solve the circuit. Hence, we choose one of the two terminals to be the higher potential terminal prior to solving the circuit. The circuit solution will either confirm it by returning a positive value for element voltage or correct us by returning a negative value for that element voltage.

But, the circuit solution can reveal the correct state of affairs for an element only if the chosen reference directions for the element variables are consistent with the element voltage-current relationship that was used in solving the circuit. Consider the four different ways of selecting the reference directions for current and voltage of a two-terminal resistance as shown in Fig. 1.6-3. The correct statement for Ohm’s Law is shown by the side of each choice in Fig. 1.6-3. We can use any one of these four reference direction choices provided we employ the correct statement for Ohm’s Law.

However, we would like to avoid the confusion that may result from multiple choices available for reference polarities. We settle the matter once and for all by choosing one set of current and voltage reference directions for all two-terminal elements. The chosen reference directions will be as per the scheme marked as (i) in Fig. 1.6-3. *That is, we assign positive polarity of voltage variable to one of the two terminals and then assign positive current flowing into that terminal from outside. This choice of reference directions for current and voltage of a two-terminal element is called the ‘Passive Sign Convention’. *See Fig. 1.6-4.

**Fig. 1. 6-4** Passive sign convention

The choice shown in (iv) in Fig. 1.6-3 is also per passive sign convention. *The choice of the terminal to assign positive polarity of voltage variable is arbitrary in passive sign convention. *

The *v *– *i *relation for a two-terminal resistance with passive sign convention is *v*(*t*) = *R i*(*t*)*. *It is for a two-terminal* * inductance and for a two-terminal* * capacitance with passive sign convention.

#### 1.6.2 Power and Energy in Two-Terminal Elements

Consider a general two-terminal element shown in Fig. 1.6-4. Assume that at the instant *t *both *v*(*t*) and *i*(*t*) are positive.

This implies that positive charges are flowing into the element at that instant. Positive charges moving through a voltage drop will lose their potential energy to the element. If the element is a resistance, then, the energy lost by positive charges moving from higher potential end to lower potential end will appear as heat in the resistive element. If the element is a capacitor, then, the energy lost by these charges will get stored in the capacitor as electrostatic energy storage. If the element is an inductance, the energy lost by charges will appear as increase in energy stored in the magnetic field in the element. If the element is a source, the energy lost by the charges will be absorbed by the source element and stored inside in some other form of energy.

Let us assume that *∆q *coulombs of positive charge crossed the left terminal into the element in a time interval *∆t * centered around the time instant *t. *Then, the energy lost by these charges will be *∆E *= *v*(*t*) × *∆q *Joules since 1 C passing through a voltage drop of *v*(*t*) V will lose *v*(*t*) Joules of energy. *Energy lost by the charge is equal to energy delivered to the element. *Therefore, the energy delivered to the element over the time interval *∆t *is *v*(*t*) × *∆q *Joules. The average rate at which this energy is delivered to the element is given by * * Joules/second. The instantaneous rate at which energy is delivered to an element is defined as the *instantaneous power delivered to the element *and is denoted by *p*(*t*)*. *Therefore,

The unit Joules/sec is given the name ‘Watts’ and denoted by ‘W’.

Thus, the instantaneous power delivered to a two-terminal element is given by *p*(*t*) = *v*(*t*) *i*(*t*) Watts where *v*(*t*) and *i*(*t*) are the voltage across the element and current through the element as per passive sign convention.

Since instantaneous power *p*(*t*) is the instantaneous rate at which energy is delivered to the element, total energy *E*(*t*) that was delivered to the element from *t *= –∞ to the current instant *t *is given by [Note: We are using the symbol *E*(*t*) to denote the time-varying electromotive force of a voltage source as well as the total energy delivered to a two-terminal element. The symbol will have to be interpreted contextually.] The relation between energy delivered to a two-terminal element and power delivered to it is summarized below.

Power delivered *by *a two-terminal element is obviously the negative of power delivered *to *it. Therefore,

Instantaneous power delivered by a two-terminal element = –*v*(*t*) *i*(*t*) W, where *v*(*t*) and *i*(*t*) are instantaneous voltage and instantaneous current of the element as per passive sign convention.

A circuit can get coupled to the surroundings by electrostatic/electromagnetic coupling with other physically separate circuits in the vicinity or by mechanical, thermal or optical interaction with the environment. Consider an *isolated circuit *that has no energy coupling of any kind with the surroundings. Obviously, the total energy in that circuit has to remain constant in time. That is, the sum of energy delivered to all the elements in the circuit must remain constant. Let there be *n *two-terminal elements connected in such a circuit. Some of them may be electrical sources. Then,

*E*

_{1}(

*t*) +

*E*

_{2}(

*t*) +…+

*E*

_{n}(

*t*) = Constant

Differentiating this equation both sides with respect to time, we get,

But each term in this equation is nothing but the instantaneous power delivered to the corresponding two-terminal element. Therefore,

Thus, the sum of instantaneous power *delivered to *all elements in an *isolated *circuit is always zero. Or equivalently, the sum of instantaneous power *delivered by *all elements in an *isolated *circuit is always zero. This implies that total power *delivered by *the elements that deliver positive power at *t *must be equal to the total power *absorbed by *the elements that absorb positive power at that instant. This principle can be employed to check the solution of a circuit analysis problem.

Note that ‘*power delivered to an element*’ and ‘*power absorbed by an element*’ mean the same.

The instantaneous power delivered to a two-terminal element does not have to be positive at all instants of time. Neither does it have to be negative at all instants. It is always positive in the case of a resistance. But in all other cases, it can be positive or negative depending on the relative polarity of voltage and current in the element.

**Example: 1.6-1**

The current through a two-terminal element is given by *i*(*t*) = 10(1 – *e ^{–}*

^{1000t}) mA for all

*t*≥ 0 and = 0 for

*t*≤ 0. (i) Find the amount of charge that went through the element in [0, 5 ms]. (ii) Find an expression for the charge that went through the element up to the time instant

*t.*(iii) If the voltage across the element is a constant at 10 Volts and the current

*i*(

*t*) flows out of positive terminal find and plot the energy delivered by the element as a function of time.

**Fig. 1.6-5** The two-terminal element in Example: 1.6-1

**Solution **

Refer to Fig. 1.6-5. *i*(*t*) as per passive sign convention is –10(1 *– e* ^{–1000t})* *mA.

- Charge that went through the element in a time interval [
*t*_{1},*t*_{2}] is given by Substituting the time-function for*i*(*t*) and using limits*t*_{1}= 0 and*t*_{2}= 5 ms, we get, - The required expression is obtained by
- The instantaneous power delivered
*to*the element is*p*(*t*) =*v*(*t*)*i*(*t*) where*v*(*t*) and*i*(*t*) are as per passive sign convention. Therefore, the power delivered*by*the element is given by –*v*(*t*)*i*(*t*)*.*The energy delivered by the element is obtained by integrating this quantity as below.

**Example: 1.6-2 **

The voltage across a two-terminal element and current through it are given in Fig. 1.6-6. Passive sign convention may be assumed. Obtain the instantaneous power delivered to the element and the energy delivered to the element as functions of time.

**Fig. 1. 6-6** Voltage and current waveform for Example: 1.6-2

Instantaneous power delivered to the element is obtained by *p*(*t*) = *v*(*t*)* i*(*t*)*. *This waveform will contain straight-line segments since the current waveform contains straight-line segments and voltage waveform is a symmetric rectangular pulse waveform. The power waveform is shown in (a) of Fig. 1.6-7.

**Fig. 1. 6-7** (a) Waveform of instantaneous power and (b) Waveform of energy in Example: 1.6-2

The energy delivered to the element is obtained by integrating the power delivered to the element from *t *= –∞ to *t *= *t. *The equation of *p*(*t*) in the interval [0, 2 ms] is that of a straight-line of slope 18 W/ms. Integrating this straight-line equation results in a parabolic curve for energy in that interval. The parabolic curve reaches 18 mJ value at 2 ms (since area of the triangle in *p*(*t*) curve is 18 W × 2 ms × 0.5 = 18 mJ.) Then *p*(*t*) reverses polarity and remains negative and linear in the interval [2 ms, 4 ms]. This means that the element *delivers *power during this interval. The area of triangle in the power curve in the interval [2 ms, 4 ms] is again 18 mJ; but with a negative sign. Therefore, the total energy delivered to the element at the end of 4 ms period must be 18 mJ – 18 mJ = 0 mJ and the energy curve between 2 ms and 4 ms must be parabolic again. The variation of energy delivered to the element is shown in (b) of Fig. 1.6-7.

Note that the net energy delivered to the element at the end of 8 ms is zero. The element received a total of 36 mJ of energy during the intervals [0, 2 ms] and [4 ms, 6 ms]. The element delivered a total of 36 mJ of energy during the two intervals [2 ms, 4 ms] and [6 ms, 8 ms].

**Example: 1.6-3**

In charging a storage battery, it is found that energy of 2 watt-hour is expended in 30 minutes in sending 200 C through the battery. (i) What is the terminal voltage of the battery assuming that this voltage remains constant during the charging process? (ii) What is the magnitude of average charging current?

**Solution **

- 200 C of charge went through the battery. Energy delivered to the battery is given by
The battery voltage is stated to be a constant during the charging process. Let this constant voltage be

*V*volts. Then, the energy delivered over 1800 seconds is where*Q*is the charge that went through the battery in the same time interval. Therefore,*VQ*= 2 watt-hour = 2 × 3600 watt-sec = 7200 joules. Since*Q*is 200 C,*V*= 7200/200 = 36 Volts. - The average charging current is the value of a constant current that will result in same charge flow over the same time interval. Therefore, the average charging current is 200 C/1800 sec = 1/9 Amps.

**Example: 1.6-4 **

Find the current *I *in the direction marked in Fig. 1.6-8.

**Fig. 1. 6-8** Circuit for Example: 1.6-4

**Solution**

The sum of power delivered by all elements in an isolated circuit must be zero at all instants. Power delivered by an element in a DC circuit = –*VI *where *V *and *I *are its voltage and current variables *as per passive sign convention. *

The values of *V* and *I* for 10 V source = 10 V and –5 A

∴ Power delivered by 10 V source = 50 W

The values of *V* and *I* for 20 V source = 20 V and –20 A

∴ Power delivered by 10 V source = 400 W

The values of *V* and *I* for 15 V source = 15 V and *I *A

∴ Power delivered by 15 V source = –15*I* W

The values of *V* and *I* for 5 A source = –5 V and 5 A

∴ Power delivered by 5 A source = 25 W

The values of *V* and *I* for 10 A source = 15 V and –10 A

∴ Power delivered by 5 A source = 150 W

The values of *V* and *I* for –20 A source = –5 V and –20 A

∴ Power delivered by 5 A source = –100 W

Sum of power delivered by all elements = (50 + 400 – 15*I* + 25 + 150 – 100) W
= (525 – 15*I*) W

This has to be equal to zero. Therefore, the value of *I* is 35 Amps.

##### 1.7 CLASSIFICATION OF TWO-TERMINAL ELEMENTS

Circuit elements can be classified based on different criteria. Classifying elements based on the physical dimensions of the element results in two broad classes of circuits – *lumped parameter circuits *and *distributed parameter circuits.*

#### 1.7.1 Lumped and Distributed Elements

Electromagnetic effects propagate within the circuit in the form of waves with a finite velocity. Hence, the time-variation of electromotive force taking place within electrical sources will be felt at different points in the circuit with different time delays. Therefore, the description of electrical phenomena in circuit elements, in general, will involve time and space variables. A circuit element can not be described by a unique voltage and current variable pair in that case.

However, if the circuit dimensions and element dimensions are such that the time taken by electromagnetic waves to propagate over the largest dimension in the circuit is small compared to *the characteristic time of variation *of the electromotive forces acting in the circuit, then, the retardation effect due to finite velocity of electromagnetic waves can be ignored and a simple circuit model for elements can be used.

Assume that the circuit contains many sources of sinusoidal nature and the maximum angular frequency of such source functions is *ω*_{o} rad/sec. That is, there is some voltage or current variable of the form *X *sin(*ω*_{o}*t*) present in one of the sources in the circuit. Then, this variable will complete one cycle of oscillation in 2*π*/*ω*_{o} seconds. The *characteristic time of variation *in this circuit is then 2*π*/*ω*_{o} seconds. That is, this is a measure of the minimum time-interval over which significant changes in circuit variables will take place. Now let us assume that the largest dimension of any element in the circuit (including connecting wires) is *d *meters. Then, electromagnetic waves will take *d/c *seconds to cover this distance where *c *is the velocity of light in free space. If *d/c *is much less than 2*π*/*ω*_{o}, we may ignore the travel time of electromagnetic disturbances and model all the elements in the circuit by terminal voltage-current relationships. Note that this conclusion is valid only for operation at ≤ *ω*_{o} rad/sec.

The ‘characteristic time of variation’ of a circuit depends on the waveshape of source functions present in the circuit. The source functions need not be sinusoidal always. However, it is possible to expand arbitrary time-functions in terms of sinusoidal functions under certain conditions. The highest frequency that appears in such expansions will have to be used to decide whether the circuit can be modeled by ignoring retardation effect. There is one kind of source function, which if present in a circuit, will not permit us to ignore retardation effect. That is a source function that contains sudden, instantaneous changes in values – that is, a function that has step discontinuities. Obviously, the characteristic time of variation of such a function is zero.

An element is classified as a *lumped element *if the net effect of electrical phenomena taking place within that element can be described in terms of only its terminal voltage and current variables, irrespective of its internal details and geometry. This amounts to neglecting the retardation effect in the element. If the electrical description of an element calls for voltage and current variables that are functions of space variables over the element (in addition to time variable), the element is called a *distributed element. *

An electrical device can be modeled by a *lumped model *only for a range of frequencies in the source functions in the circuit. The same electrical device may call for a *distributed model *if the source functions in the circuit vary rapidly enough to make retardation time within the device significant.

For instance, consider a solenoid coil of length 5 cm and diameter 1 cm with 100 turns of wire. One may be tempted to assume that the largest dimension of the coil is its length- *i.e., *5 cm. It is not. The largest dimension that we need here is the length of the wire and that is about 314 cm. The retardation time over this length = 3.14/3 × 10^{8} ≈ 10 ns. If *ω*_{o} is the highest frequency of sinusoidal components present in the sources within the circuit, then, the ‘characteristic time of variation’ is 2*π*/*ω*_{o} sec. If this time is 10ns then *ω*_{o} is 628 Mrad/sec. The corresponding cyclic frequency will be 100 MHz. Thus, this coil can be modeled as a two-terminal lumped inductance with good accuracy if the circuit contains source sinusoidal components at 1 MHz or below. However, it will call for a distributed model if the sources contain > 10 MHz sinusoidal components.

Consider a power transmission line of length 300 km. The retardation time over the length of the line is 1 ms. 50 Hz sinusoidal source functions have a waveform period of 20 ms. Hence, a lumped parameter model for this line amounts to ignoring 1 ms in comparison with 20 ms. But 20 ms is the time required for one full oscillation of source function. Significant change in function value takes place within a quarter cycle – *i.e., *in 5 ms. Obviously, this power line requires a distributed model even at 50 Hz.

A 1 nF ceramic capacitor typically has two leads of 1.5 cm each. The retardation time over 3 cm is 100 ps (1 ps = 10^{–12} sec). This corresponds to a frequency value of 10 GHz. Therefore, a lumped parameter model will be satisfactory for frequencies below 10 MHz. A distributed model will be necessary for frequencies > 50 MHz.

All circuit elements of arbitrary dimensions can be modeled by lumped elements if all the sources are DC sources. But, no element, of any dimension whatsoever, can be modeled by lumped parameter model to obtain *detailed *circuit solution at and around the instants at which such DC sources are either switched into the circuit or switched out of the circuit. Such switching operations represent very rapid changes in circuit variables and retardation time can not be ignored in comparison with infinitesimal intervals.

#### 1.7.2 Linear and Non-Linear Elements

Two-terminal elements are classified as *linear *or *nonlinear *based on whether the voltage–current relationship of the element satisfies the *linearity property. *

Two variables of time – *x*(*t*) and *y*(*t*) – satisfy the *property of linearity *if the relation between them is *homogeneous *and *additive *at all *t. *

Let the relation between the variables be represented by *y*(*t*) = *f *[*x*(*t*)]*. *

*y*(*t*) = *f *[*x*(*t*)] is *homogeneous *if *f *[*ax*(*t*)] = *af*[*x*(*t*)] for any *t *where *a *is any real number. That is, scaling the variable *x*(*t*) by a real number *a *results in the scaling of the variable *y*(*t*) by the same real number *a. *

*y*(*t*) = *f *[*x*(*t*)] is *additive *if *f *[*x*_{1}(*t*) + *x*_{2}(*t*)] = *f *[*x*_{1}(*t*)] + *f *[*x*_{2}(*t*)] for any *t. *That is, *y*(*t*) corresponding to sum of two variables *x*_{1}(*t*) and *x*_{2}(*t*) is equal to the sum of *y*(*t*) corresponding to *x*_{1}(*t*) and *y*(*t*) corresponding to *x*_{2}(*t*) at any time instant.

We may combine the requirements of *homogeneity *and *additivity *into a composite requirement called *superposition property. *

*y*(*t*) = *f *[*x*(*t*)] satisfies *superposition property *if

*f *[*a*_{1}*x*_{1}(*t*) + *a*_{2}*x*_{2}(*t*)] = *a*_{1} *f *[*x*_{1}(*t*)] + *a*_{2} *f *[*x*_{2}(*t*)] for any combination of real numbers *a*_{1} and *a*_{2} and for any *t. *

*Thus, a two-terminal element is linear if its v *– *i relationship satisfies the principle of superposition. *

The simplest case of a *linear *relationship between two variables occurs when *y*(*t*) is proportional to *x*(*t*)*. *

Let *y*(*t*) = *k x*(*t*) where *k *is a real number.

Then, *f *[*a*_{1}*x*_{1}(*t*) + *a*_{2}*x*_{2}(*t*)] = *k *× [*a*_{1}*x*_{1}(*t*) + *a*_{2}*x*_{2}(*t*)] = *k a*_{1}*x*_{1}(*t*) + *ka*_{2}*x*_{2}(*t*) = *a*_{1} ×*k x*_{1}(*t*) + *a*_{2} ×*k x*_{2}(*t*) = *a*_{1} *f *[*x*_{1}(*t*)] + *a*_{2} *f *[*x*_{2}(*t*)]*. *

Therefore, *y*(*t*) = *k x*(*t*) is a *linear relation *for any real *k. *

But, a relation does not have to be algebraic for it to be a linear relation. Consider the relation . Then,

Therefore, is a *linear relation.*

Similarly, it can be shown that too is a *linear relation. *

Beginners in Circuit Analysis often tend to equate the property of linearity to straight-line nature of functional relationship between the concerned variables. Consider the following relationship.

*y*(*t*) = *mx*(*t*) + *c *where *m *and *c *are two real numbers. Obviously, the graph of this function will be a straight-line with *c *as its vertical-axis intercept. But this is not a *linear relation *in the sense of linearity as defined in Circuit Theory.

Therefore, *y*(*t*) = *mx*(*t*) + *c *is not a *linear relation *in Circuit Theory. It does not satisfy the property of homogeneity. It does not satisfy the property of additivity too.

Let us examine the linearity property of various two-terminal elements we have discussed so far.

Consider a two-terminal resistance element. Its *v *– *i *relation is *v*(*t*) = *Ri*(*t*)*. *It is a linear element if the *R *parameter is a real constant or a function of time alone. The resistance of a piece of conductor is temperature dependent. It may depend on current level in certain cases. Thus a two-terminal resistance is linear if the temperature is constant and the *R *parameter is either a constant or is an independent function of time alone.

A two-terminal inductance is described by in general. If the inductance parameter* L *is a constant, then two-terminal inductance is a linear element. *L *can vary with time if the physical geometry of the device changes with time (but independent of electrical variables). The element is linear in that case too. But if *L *varies as a function of the current in it, then, the element is a nonlinear one.

A two-terminal capacitance is described by in general. If the inductance parameter *C *is a constant, then two-terminal inductance is a linear element. *C *can vary with time if the physical geometry of the device changes with time (but independent of electrical variables). The element is linear in that case too. A tuning capacitor in a radio receiver is an example. But if *C *varies as a function of the charge in it, then, the element is a nonlinear one.

A two-terminal ideal independent voltage source is described by the relations *v*(*t*) = *E*(*t*) (an independently specified function of time) and *i*(*t*) = arbitrary. Obviously, this is a non-linear relationship. Thus an ideal independent voltage source is a nonlinear element. Similarly, an ideal independent current source is a nonlinear element.

**Fig. 1. 7-1** Voltage–current relationship

**Fig. 1. 7-2** V – I Curve for a nonlinear, bilateral resistance of a diode – non-bilateral

#### 1.7.3 Bilateral and Non-Bilateral Elements

Some elements have a *v *– *i *relation that depends on the direction of current flow in them. A diode is an example. See Fig. 1.7-1. The *v *– *i *relation of this two-terminal element is not symmetrical about the vertical axis. This is a non-bilateral element. The current that will flow in the device when it is connected across a battery of *V *volts will depend on how it is connected. If the terminal marked *a *is connected to positive terminal of the battery the resulting current will be large. If the terminal marked *b *is connected to positive terminal of the battery, the resulting current flow will be small.

An element with a voltage-current relation that is odd-symmetric about the vertical axis in the *v *– *i *plane is called a bilateral element. A linear resistor is a bilateral element. A linear inductor is a bilateral element. A linear capacitor is bilateral.

A linear two-terminal element will always be bilateral too. However, a multi-terminal element (with more than two terminals) can be non-bilateral even if it is a linear element.

A two-terminal element is non-linear if it is non-bilateral. However, the converse is not true. Consider the *v *– *i *curve of a nonlinear resistor shown in Fig. 1.7-2. The element is nonlinear; but bilateral.

#### 1.7.4 Passive and Active Elements * *

The energy delivered to a two-terminal element from *t *= –∞ to *t *= *t *is given by * * An element is called a *passive element *if the energy delivered to it is always non-negative for any *t *and for any possible terminal voltage – current conditions of the device. That is, an element is passive if *E*(*t*) ≥ 0 for all *t *and for all permissible (*v*(*t*) , *i*(*t*)) combinations.

Consider a linear resistance with constant *R. *Then,

Therefore, a resistance is a passive element. Consider a linear inductance element. Then,
a positive number or zero. (We assume * * that *L *is positive). This energy is stored in the magnetic field inside the device. We have assumed that the energy storage inside the inductor at *t *= –∞ is zero. Thus, an inductor is a passive element. Similarly, it may be shown that* * for a capacitor and that a capacitor is also a passive element.

A positive-valued resistance is a dissipating element. It can not deliver energy even for a short interval. This can be seen from the equation for power delivered to the resistor. *p*(*t*) = *v*(*t*)* i*(*t*) = *R *[*i*(*t*)]^{2} = a positive number or zero. Therefore, the power delivered to a resistance can not be negative at any *t *and hence resistance will always consume power.

A positive-valued inductance or capacitance is an energy-storing element. It can deliver the energy back to other elements. But it can deliver only as much energy as that was given to it earlier. It can not generate energy and deliver it. Thus, the energy storage in an inductance or capacitance can not be employed to deliver energy to other elements indefinitely. The instantaneous power delivered to such an element can be negative; but the area under *p*(*t*) waveform from –∞ to any *t *will be non-negative. In other words, the energy storage in an inductance or in a capacitance can only be zero or positive at any instant.

An independent source is an active element (since it is not a passive element). An independent voltage source can *deliver *any amount of current for any duration Similarly, an independent current source too can do that. Therefore, independent sources are *active elements. *

#### 1.7.5 Time-Invariant and Time-Variant Elements

*An element is ‘**time-invariant’ if the values of parameters that characterize it are independent of time. *Therefore, a two-terminal resistance is time-invariant if *R *is a constant; a two-terminal inductance is time-invariant if *L *is a constant and a two-terminal capacitance is time-invariant if *C *is a constant.

A synchronous generator driven at constant speed by some prime-mover is an example system that contains time-variant elements. The inductance value of various coils in the machine varies with time due to rotation.

In this book, we deal only with circuits comprising a *finite *number of ‘*lumped, linear, bilateral, time-invariant’ *two-terminal as well as multi-terminal elements interconnected.

##### 1.8 MULTI-TERMINAL CIRCUIT ELEMENTS

Many electrical devices that are characterized by voltage and current variables at more than two terminals are in common use in Electrical engineering. Transformers in power engineering, transistors in electronics engineering etc., are some examples.

#### 1.8.1 Ideal Dependent Sources

Ideal dependent sources form a category of multi-terminal elements that we employ in circuit analysis. These models are used extensively in analysis of electronic circuits to model devices like transistors, amplifiers etc.

They have two terminal pairs. The first terminal pair *senses *either a voltage variable or a current variable at the location where this terminal pair is connected in the circuit. The second pair of terminals delivers either a voltage or a current to the location at which this terminal pair is connected in the circuit. *However, the source function delivered is a function of the variable sensed by the first terminal pair. *That is why they are called *dependent sources. *They are ideal in the sense that (i) the first terminal pair does not affect the circuit variables in any way (ii) the source function delivered by second terminal pair depends only on the variable sensed by the first terminal pair and on nothing else.

**Fig. 1. 8-1** Ideal dependent sources

There are four dependent sources depending on the nature of circuit variable sensed by the first terminal pair and the nature of source function delivered by the second terminal pair.

A *Voltage-Controlled Voltage-Source *(*VCVS*) senses a voltage variable at some location in the circuit and delivers a source voltage that depends on the sensed voltage at some other location in the circuit. If the source voltage delivered is a linear function of the controlling voltage, the dependent source will be called a *linear VCVS. *

A *Voltage-Controlled Current-Source *(*VCCS*) senses a voltage variable at some location in the circuit and delivers a source current that depends on the sensed voltage at some other location in the circuit. If the source current delivered is a linear function of the controlling voltage, the dependent source will be called a *linear VCCS. *

A *Current-Controlled Current-Source *(*CCCS*) senses a current variable at some location in the circuit and delivers a source current that depends on the sensed current at some other location in the circuit. If the source current delivered is a linear function of the controlling current, the dependent source will be called a *linear CCCS. *

A *Current-Controlled Voltage-Source *(*CCVS*) senses a current variable at some location in the circuit and delivers a source voltage that depends on the sensed current at some other location in the circuit. If the source voltage delivered is a linear function of the controlling current, the dependent source will be called a *linear VCCS. *

The symbols used for the four *linear dependent sources *are shown in Fig. 1.8-1. *k _{v }* ,

*k*,

_{i}*g*and

_{m}*r*are real numbers.

_{m}*k*is a trans-conductance,

_{y}*k*is a trans-resistance,

_{z}*k*and

_{v}*k*are dimensionless.

_{i}##### 1.9 SUMMARY

*Electric Circuit*is a mathematical model of a real physical electrical system. It is an approximation of Electromagnetic Field Theory.- Electromagnetic disturbances travel with a finite speed in electrical systems. Electric circuit theory assumes that the largest dimension in the circuit is so small that electromagnetic disturbances take negligible time to cover that distance compared to the time interval required for source quantities in the circuit to change significantly.
- In addition,
*lumped parameter circuit theory*assumes the following:- The electrostatic field created by charge distribution on an electrical device is confined to space within the device and in the immediate vicinity of the device predominantly. Thus, the terminal voltage across a device and the charge stored in that device can be related through a unique ratio.
- The induced electric field inside connecting wires and outside the devices is negligible. This makes it possible to assign a unique voltage variable to a device.
- The connecting wires are of infinite conductivity and near-zero thickness. Thus, there are no charges distributed on their surface and no current component is needed to create charge distribution on them. This makes it possible to assign a unique current variable to an electrical device.
- The component of current needed to create the time-varying charge distribution on the surface of the device is negligible except in those devices (capacitors) that are designed to make such current flow the dominant electrical phenomenon in them.
- The induced electric field inside devices is negligible except in those devices (inductors) where induced electric field is the dominant electrical phenomenon by design. Such devices are designed to confine the time-varying flux linkage to space within them.
- The conductivity of metallic conductors employed in capacitors and inductors is infinity. Hence, there is no resistive effect in them.

- A two-terminal element is a mathematical model of an electrical device with a terminal voltage variable
*v*(*t*) and current variable*i*(*t*) assigned to it. These are functions of time only. The entire electrical behaviour of the device can be characterized by these two variables and a relation between them. - Passive sign convention assigns reference polarity for
*v*(*t*) with + at one end and – at other end of the element. Reference direction for*i*(*t*) is such that it flows into the + polarity of*v*(*t*)*from outside the element.*The power delivered to the element is given by*p*(*t*) =*v*(*t*)*i*(*t*) with this sign convention. - The symbols and
*v*–*i*relations for the three passive two-terminal elements are shown in Fig. 1.9-1.**Fig. 1.9-1**Passive two-terminal elements - Ideal independent voltage source is a two-terminal element with its voltage variable specified as a function of time and current variable as a free variable. Ideal independent current source is a two-terminal element with its current variable specified as a function of time and voltage variable as a free variable.
- Circuit elements are classified into
*linear*and*nonlinear*elements depending on whether their*v*–*i*relationship satisfy the*principle of superposition*or not. Independent sources are nonlinear elements. - An element is a
*passive element*if the energy delivered to it from -∞ to*t*is non-negative for all*t*and for all permissible (*v*(*t*),*i*(*t*)) combinations.*R*,*L*and*C*are passive elements. - A circuit element is a
*time-invariant element*if the parameter of the element is a constant. - Ideal dependent sources are four-terminal elements. The first terminal pair is connected at some location in the circuit to sense a voltage variable or a current variable there. The second terminal pair delivers either a voltage or a current at the location where it is connected. The value of voltage or current delivered is a function of
*controlling variable*that is sensed by the first terminal pair. - There are four kinds of dependent sources. They are Voltage-Controlled Voltage Source, Voltage-Controlled Current Source, Current-Controlled Voltage Source and Current-Controlled Current Source. A dependent source is a
*linear*element if the source quantity is a linear function of the controlling-variable.

##### 1.10 PROBLEMS

- A fully charged lead-acid battery contains 120 Ampere-hours of charge in it. The terminal voltage of the battery is a function of charge remaining in it and is given by
*V*= 11(1 + 0.001C) V, where*C*represents the charge that remains in the battery in Ampere-hour units. The battery delivers a current of 10 A to a load. (a) Express the initial charge storage in the battery in Coulombs. (b) What is the terminal voltage of the battery when it has been discharged to 50% level? (c) A voltage-sensing relay cuts out the battery from the load when the terminal voltage falls below 11.35 V. How long can the battery power the 10A load? (d) What is the energy delivered by the battery to 10A load in Joules and kW-h units if the load is kept powered till cut out takes place? - A fully charged battery contains 100 AH charge in it. The terminal voltage of battery is a function of the charge that remains in it and the current that is delivered by it. It is given as
*V =*11 + 0.001*C*– 0.02*I*V, where*C*is the AH (Ampere-hours) remaining in it and*I*is the current delivered by the battery in Amps. This battery is delivering current to a pulsed load that draws current with a period of 5 sec as shown in Fig. 1.10-1. The load is kept powered for 8 hours. (a) What is the charge that remains in the battery after the load is switched off? (b) What is the value of constant load current that would have resulted in same charge consumption? (c) What is the energy consumed by the load (in Joules and kW-h) in 8 hours? (d) What is the value of constant load current that would have consumed same energy in 8 hours? What would have been the charge consumption with this value of load current?**Fig. 1.10-1** - The waveshape of current that is delivered to a 12 V lead-acid battery to charge it is shown in Fig. 1.10-2. The initial charge in the battery is 36000 C. The voltage across the battery is given by
*V*= 11.5 + 0.01*C*V where*C*is the charge stored in the battery in AH units. The charging current is applied to the battery for 5 hours. (a) Express the initial charge in the battery in AH. (b) What is the charge stored in the battery at the end of charging? (c) What is the constant charging current value that would have delivered the same amount of charge to the battery in the same time interval? (d) How much is the energy consumed in charging? (Hint: Assume that the battery voltage remains constant during a cycle of charging current.)**Fig. 1. 10-2** - The current through a two-terminal element is given by Find the constant
*k*if the total charge that went through the element in the interval [0,0.05 s] is zero. - The voltage across an ideal two-terminal passive element is
*v*(*t*) = 10*e*^{–100t}V for*t*≥ 0 and zero for*t*< 0. The current through the element is*i*(*t*) = 0.1*e*^{–100t}A for*t*= 0 and zero for*t*< 0. (a) Identify the element and its parameter value. (b) What is the amount of charge that went through the element in the time interval [0.01 sec, 0.05 sec]? (c) What is the amount of charge that went through the element in [0, ∞] time-interval? (c) What is the ratio of instantaneous power delivered to it at 0.01 sec to the corresponding value at 0 sec? (d) What is the total energy delivered to the element? (e) What is the time at which the energy delivered to it reached 99% of total energy delivered to it? Assume that*v*(*t*) and*i*(*t*) given are as per passive sign convention. - The voltage across an ideal two-terminal passive element is
*v*(*t*) = 10(1 –*e*^{–}^{1000t}) V for*t*≥ 0 and zero for*t*< 0. The current through the element is*i*(*t*) = 0.001*e*^{–1000t}A for*t*≥ 0 and zero for*t*< 0. (a) Identify the element and its parameter value. (b) What is the amount of charge that went through the element in [0, ∞] time-interval? (c) What is the total energy delivered to the element? (d) What is the time instant at which the power delivered to the element is a maximum? What is the value of this maximum power? What is the value of energy delivered to the element till that instant? Assume that*v*(*t*) and*i*(*t*) given are as per passive sign convention. - The voltage across an ideal two-terminal passive element is
*v*(*t*) = 600 cos (100*t*) V for*t*≥ 0 and zero for*t*< 0. The current through that element as per passive sign convention is*i*(*t*) = 10 sin (100*t*) A for*t*≥ 0 and zero for*t*< 0. (a) Identify the element and its parameter value. (b) Find an expression for the charge that goes through the element as a function of time. (c) Find an expression for instantaneous power delivered to the element as a function of time. (d) Show that the energy delivered to the element till*t*is non-negative for all*t.* - The current that flows through an ideal independent voltage source with
*v*(*t*) = 12 V is*i*(*t*) = 10 + 10 cos 100*πt*A for*t*≥ 0 and 0 A for*t*< 0. Assume passive sign convention. (a) What is the power delivered by the source at*t*= 1 sec? (b) What is the change in energy storage in the source between*t*= 0 and*t*= 1 sec? Does the energy storage in source increase with time or decrease with time? - The voltage across a two-terminal element is
*v*(*t*) = 10 sin 1000*πt*V. The current that flows into the element as per passive sign convention isFind

*k*if the energy delivered to the element in the time interval [0,0.1 s] is zero. - The voltage across a two-terminal element is
*v*(*t*) = 10 sin 1000*πt*V and current in that element isAssume passive sign convention. Find the average power delivered to the element over any time interval of width equal to the period of the voltage and current waveforms.

- The voltage across a two-terminal resistor is
*v*(*t*) = 10 sin 1000*πt*V. Find the value of a DC voltage that will deliver a power that is equal to the average power delivered by this voltage source over any time interval equal to the period of the voltage waveform. - The value of resistance of a resistor is measured to be 10Ω. at room temperature of 35° C. Temperature coefficient of this resistance is 0.004. A constant current source of 0.25A is connected across the resistance. The resistance attains a steady temperature after some time. The
*temperature rise*in the resistor after the temperature has reached a steady-state is given by 100*p*where*p*is the power dissipated in the resistor in Watts. (i) Find the steady-state temperature, corresponding resistance value and the power dissipated in the resistor under steady-state condition. (ii) Find the critical value of current at which the temperature of the resistor increases without any limit and it burns out. - A DC voltage source of 2.5 V is connected across the resistor in Problem 12. (i) Find the steadystate temperature, corresponding resistance value and the power dissipated in the resistor under steady-state condition. (ii) Find the critical value of applied DC voltage (if such a value exists) at which the temperature of the resistor increases without any limit and it burns out.
- There are only three elements in an isolated circuit. Assume passive sign convention. The terminal voltage and current of first element are given by
Corresponding variables for the second element are

*v*_{2}(*t*) =*v*_{1}(*t*) and*i*_{2}(*t*) = –2 A. The voltage across the third element is*v*_{3}(*t*) =*v*_{1}(*t*)*.*Identify the third element assuming that it is a passive element, find its parameter value and the current through the third element as a function of time. [Hint: Sum of power delivered by all elements in a circuit is zero.] - The
*v*–*i*characteristic of a passive two-terminal element as per passive sign convention is*v*(*t*)*=*100*i*(*t*) + 20*i*(*t*)*|i*(*t*)*|*V. (a) Show that this element is nonlinear. (b) Show that this element is a passive element. (c) Show that it is a bilateral element. (d) Find the current flow through the element when the voltage across it is a constant at 100 V. There are two possible values for the current. How do you choose the correct one? - The current
*i*(*t*) through a passive two-terminal element is a single pulse as shown in Fig. 1-10-3. Plot the voltage across the element if the device is (a) a resistance of 10 Ω. (b) an inductance of 0.5 H with zero initial energy storage and (c) a capacitance of 10000*μ*F with zero initial energy storage.**Fig. 1.10-3** - A 1000
*μ*F two-terminal linear capacitor had a charge storage of 10 mC across it at*t*= 0. The current delivered by the capacitor out of its positive terminal is given by*i*(*t*) = 2 cos (1000*πt*) A for*t*≥ 0. Find and plot the voltage across the capacitor terminals as a function of time. - The voltage across a 0.2 F capacitor was 10V at
*t*= 0. A current source*i*(_{s}*t*) =*Ae*A was applied across the capacitor during the time interval [0, 1 s] with^{t }*i*(_{s}*t*) flowing into the positive plate. Find*A*if the voltage across capacitor is seen to be 0V at*t*= 1 s. - A resistor of 10 Ω and a capacitor of 0.5 F are connected in parallel across a time-varying voltage source. The current drawn by the resistor and the current drawn by the capacitor are seen to be equal at all
*t.*The current delivered by the voltage source at*t*= 0 s is found to be 6A. Find the current through the resistor and the capacitor at*t*= 1 s. - The voltage across a 0.2 H two-terminal inductance is
*v*(*t*) = 10*e*^{–10 t }V for*t*≥ 0. It was kept shorted for*t*< 0 with*i*(*t*) = 0.5 A circulating in it. Assume passive sign convention and find out the current in the inductance, flux linkage in it and energy storage in it at*t*= 0.5 sec. - A current source with
*i*(*t*) = 5 sin*ωt*A flows through series combination of 4 H inductor and 1 F capacitor. The voltage across the current source is seen to be 0 V for all*t.*What is the value of*ω*? - (a) List the voltage and current values for all the elements in the circuit in Fig. 1.10-4 as per passive sign convention. (b) Find the unknown voltage
*V*. (c) The circuit is known to be a DC circuit. Can the nature of the two-terminal element across which_{x}*V*be identified?_{x}**Fig. 1.10-4** - An isolated circuit contains four elements. The
*v*–*i*values at a particular time instant for three of them as per passive sign convention are (5 V, 2 A), (15 V, 1 A) and (10 V, 2 A). The voltage across the fourth element at the same instant is seen to be 15V. (a) Find all possible*v*–*i*value combinations for the fourth element. (b) If the circuit is known to be a DC circuit that has been in the present state for a long time, identify whether the fourth element is a passive element or active element. (c) Can the nature of fourth element be identified if the circuit is known to be a circuit with time-varying voltages and currents? - The voltage that appears across two terminals of a circuit is
*v*(*t*) = [10 + 5 sin 100*πt*] V for all*t.*Current delivered to the circuit (as per passive sign convention) is seen to beAre all the elements in the circuit passive elements?

- A circuit is known to contain only linear resistances. Two terminals are brought out from this circuit and a voltage source with
*v*(_{s}*t*) = [10 + 10sin*t*] is connected across the terminals. If the current into the circuit is*i*(*t*) = [2 + Asin(t –*θ*)] A find*A*and*θ.*