## 44

## Open-Loop and Closed-Loop System

##### OBJECTIVES

**In this chapter you will learn about:**

- The difference between open-loop and closed-loop systems
- How feedback signals are compared with input signals to give the error signal
- To have an appreciation of transients and the concept of damping
- To explain the setting time for a signal
- To understand open-loop gain and closed-loop gain and the relations between the two
- To understand the importance of integration

**Feedback control system**

##### 44.1 INTRODUCTION

All the elements necessary to accomplish the control objective are described by the term control system. Control means methods to force parameters of the system to have specific values. The basic strategy which a control system operates is logical and natural. In fact, the same strategy is employed in living organisms to maintain temperature, fluid flow rate, and a host of other biological functions. This is a natural process control. The technology of artificial control was first developed using a human as an integral part of the control action. The term automatic control came into use after we learned how to use mechanics, electronics and computers to replace the human functions.

We often use feedback to improve the performance of a system. Feedback systems consider the output of the system and use this information to modify the input signal to achieve the desired result. Such techniques are very widely used and form the basis of most forms of automatic control systems. The use of feedback often simplifies the system design and reduces the importance of linearity and accuracy in many of the key components.

##### 44.2 CONTROL SYSTEM

A control system is one that performs some function, checks its success, and takes further action until the objective is attained, and if in the meantime the objective varies, then the system will respond to the change.

In process control, the basic objective is to regulate the value of some quantity. The term regulate indicates maintaining that quantity at some desired value regardless of external values. The desired value is called the reference value or set point.

As an example, in Figure 44.1 liquid is flowing into a tank at some rate, *Q _{in},* and out of the tank at some rate

*Q*

_{out}_{}. The liquid in the tank has some height or level

*h*. It is known that the output flow rate varies as the square root of the height,

*Q*√

_{out}= K*h.*The higher the level, the faster the liquid flows out. If the output flow rate is not exactly equal to the input flow rate, the level will drop if

*Q*

_{out}>

*Q*or rise if

_{in}*Q*

_{out}

*< Q*

_{in}.

**Figure 44.1 **The Objective is to Regulate the Level of Liquid in the Tank* h* to the Value* H*

This process has a property called self-regulation. This means that for some input flow rate, the height will rise until it reaches a height for which the output flow rate matches the input flow rate. A self-regulating system does not provide regulation of a variable to any particular reference value. If we want to maintain the level at some particular value, *H* (as shown in Figure 44.1), regardless of the input flow rate, then something more than self-regulation is needed.

Figure 44.2 illustrates a modification of the tank system to allow artificial regulation of the level by a human. To regulate the level so that it maintains the level *H*, it will be necessary to employ a sensor. This has been provided via a sight tube *S,* as shown in Figure 44.2. The actual liquid level or height is called the controlled variable. In addition, a value has been added so that the output flow rate can be changed by the human. The output flow rate is called the manipulated variable or controlling variable.

**Figure 44.2 **A Human can Regulate the Level Using a Sight Tube* S* to Compare the Level *h*, to the Objective *H* and Adjust Values to Change the Level

The human measures the height in the sight tube and compares the value with the set point. By a succession of incremental opening and closing of the valve, the human can bring the level to the set point value, *H*, and maintain it, thereby continuously monitoring the sight tube and adjusting the value. The height is thus regulated.

To provide automatic control, the system is modified as shown in Figure 44.3 so that machines, electronics, or computers replace manual intervention. An instrument called sensor is added that can measure the level and convert it to a proportional signal, *s*; this signal is provided electronically as input to a machine, circuit or computer called the controller. The controller performs the function of the human in evaluating the measurement and providing an output signal, that is, to change the valve setting via an actuator connected to the valve by a mechanical linkage. When automatic control is applied to systems similar to the one in Figure 44.3, which are designed to regulate the value of some variable to a set point, it is called process control.

**Figure 44.3 **An Automatic Level Control System Replaces the Human with a Control and Uses a Sensor to Measure the Level

##### 44.3 SERVOMECHANISMS

Yet another commonly used type of control system that has a slightly different objective from process control is the servomechanism. In this case, the objective is to force some parameter to vary in a specific manner. This may be called a tracking control system. Instead of regulating a variable value to a set point, the servomechanism forces the controlled variable value to follow the variation of the reference value.

For example, in an industrial robot arm, similar to the one shown in Figure 44.4, servomechanisms force the robot arm to follow in a path from point *A* to point *B*. This is carried out by controlling the speed of motors driving the arm and the angles of the arm parts.

**Figure 44.4 **Servomechanism-type Control System is Used to Move a Robot from Point *A* to Point* B* in a Controlled Manner

The principle of servomechanisms is similar to that for process control systems; however, the dynamic differences between regulation and tracking result in differences in the design and operation of the control system.

* Note:* Control systems depend on the quantities that we can measure. The five basic quantities are as follows:

- Displacement,
- Force,
- Pressure,
- Temperature, and
- Velocity.

There are other more complete quantities, such as (1) light intensity, (2) chemical compositions, (3) rate of flow, and (4) conductivity.

##### 44.4 OPEN-LOOP AND CLOSED-LOOP SYSTEMS

An ammeter is an example of a useful open-loop device. To obtain the desired accuracy, the magnet is carefully formed and patiently aged the moving coil is exactly designed and the mechanism is precisely fabricated. After the instrument is calibrated, it is assumed that the output deflection is an accurate indication of the input current.

There are two approaches to controlling the temperature of a room. The first is to use heaters that have a control that varies the heat output. The control is accordingly set in the heater to produce a certain amount of heat output to obtain the desired temperature. An approximate setting will gradually be learned by experience. If the setting is too low, the setting will not reach the desired temperature, whereas if the setting is too high the temperature will rise above the desired value. If an appropriate setting is chosen, the temperature of the room should stabilize at the right temperature, but will become too hot or too cold if the external factors, such as the outside temperature or the level of ventilation, are changed. Such a system, as shown in Figure 44.5, is called an open-loop system.

**Figure 44.5** An Open-loop System

An alternative approach is to use a heater equipped with a thermostat. The user then sets the thermostat to the temperature required and it then increases or decreases the heat output to achieve and then maintain the temperature of the room even if external factors, such as the outside temperature or the level of ventilation change. Such a system, as shown in Figure 44.6, is called a closed-loop system.

**Figure 44.6** A Closed-loop System

The user is simply the person using the system, the goal is the desired result and the output of the system is the achieved result. The forward path is the part of the system that converts the input to the output. Closed-loop system also has a feedback path through which the output is fed back for comparison with the goal. The difference between the output and the goal is represented by an error signal, which is used as an input to the forward path.

In open-loop systems, the goal is not an input to the system but simply to guide the user in his input to the system. In closed-loop systems, the goal is the primary input to the system. In both cases, the forward path may have secondary inputs, for example, electrical energy. Such inputs are simply assumed to be present.

##### 44.5 AUTOMATION

Many automatic control systems are used daily, but most of the sophisticated control systems are found in industrial applications. These applications are often referred to as automation. The following are some of the many advantages of automation:

- Consistent production,
- Release of production operators for some useful work, or
- Improved conditions for the operators.

If the work is more complex, other advantages would include:

- Improved accuracy of manufacture, or
- Economic use of an expensive plant.

##### 44.6 COMPONENTS OF A CONTROL SYSTEM

The basic control system, (as shown in Figure 44.7), has the following four significant features:

**Figure 44.7** Basic Control System

- Input signal: This sets the intended condition. It is sometimes called the reference signal or even the set signal.
- Output signal: This represents a measurement of the outcome achieved and hence provides the feedback signal. It is sometimes called the reset signal.
- Comparator: This is the part of the system in which the input and output signals are compared. The differences are the error that is fed to controller as shown in Figure 44.7.
- Controller: This is the device that causes the required activity to happen. This path is like that in the amplifier in that the controllers control the supply of energy from a separate power supply.

Many control systems need to operate not only on the error but also on the rate of reaction necessary. This introduces a further complication. If the feedback circuit detects a change of error, it may be that it will not properly reflect this because it does not respond directly. We can describe this indirect response in terms of the transfer function.

##### 44.7 TRANSFER FUNCTION

A feedback circuit comprising a resistor *R* and a capacitor *C* is shown in Figure 44.8(a). Let us assume the introduction of a sudden error as shown in Figure 44.8(b); this type of signal is described as a step change. The output signal will take the form shown in Figure 44.8(c).

**Figure 44.8** Signal Transfer in an RC Network

We can predict such a reaction; therefore, we should be able to specify it by the same function. In fact, we can specify the reaction by means of a transfer function. The transfer function is the ratio of the output signal to the input signal, making due allowance for the (time) element.

A step change is one that rises from zero to a finite value instantaneously. We could also have predicted the outcome in response to, say, a sinusoidal input. The output would be sinusoidal, of a smaller peak value and would lag the input by up to a quarter of a period.

For a step change, the time variation can be considered in two parts: (1) the transient period and (2) the steady-state period. On the basis of the *RC* circuit shown in Figure 44.8, we can illustrate these periods as shown in Figure 44.9.

**Figure 44.9** Response Periods

It is not just the *RC* network that can provide the responses shown in Figure 44.9, an alternative would be the sudden application of an electromagnetic field to a rotating coil. Switching on the current to create the field as shown in Figure 44.10 does not immediately create the field, instead it builds up exponentially. It follows that the e.m.f. induced in the rotating coil also builds up; thus, the transfer function is of the same form as that for the *RC* circuit.

**Figure 44.10** A Basic Regulator System

Proceeding a step forward, the circuit details can be replaced by a block diagram. The significant factor is the transfer function, which relates the output to the input; the manner in which the transfer is affected is of little importance.

Similar to amplifiers, we assume that the system will not saturate and that the relation is derived from linear components. This is not always the case, especially in devices with ferromagnetic cores. This involves hysteresis, which has to be minimized to obtain an almost linear performance.

* Note:* More complex signals require the application of Laplace transform.

##### 44.8 REGULATORS AND SERVOMECHANISMS

Automatic control systems usually require some outcome that is not electrical. Most outcomes are mechanical in nature and these can be divided into the following two groups:

- Regulators, and
- Servomechanism [also known as remote position controllers (rpcs)].

The difference between the two groups is the form in which the error appears. For steady-state conditions regulators require an error signal, whereas servomechanisms require zero error. A system incorporating servomechanism is known as a servo system.

In either case, the mechanism will have inertia to be overcome, if motion is required. This infers that the mass of the mechanism has to be accelerated to produce movement and this will probably be followed by deceleration so that the mechanism stops at the required position. This is especially the case in servomechanisms, but regulators also have to respond to the demand for movement.

To induce motion, we have to introduce a motor in the system to drive the mechanism we refer to as load. Having added these components, a regulator system takes the form shown in Figure 44.10.

**Example 44.1**

The regulator is a speed controller. The load is driven at a constant speed set by an input of 80 V. Steady-state condition is assumed, i.e., the load is driven at constant speed. For the feedback transducer, the transfer function is 40 mV per r/min load; for the amplifier the transfer function is 20 volt per volt for the motor load the transfer function is 100 r/min per volt. Explore the effect of the error.

**Solution:**

The error signal fed into the amplifier is the difference between the input signal and the output signal supplied by the feedback transducer. If the steady-state speed is *N _{r},* then the feedback signal is (40 × 10

^{–3}×

*N*

_{r}). Given that the input voltage is 80 V, the error signal is

*N*

_{r})

The amplifier transfer function is 20 V per volt, hence the amplifier output voltage is 20 (80 – (0.04 *N _{r}*

_{}) = 1600 – 0.8

*N*.

_{r}The motor speed is given by the transfer function being applied to the amplifier output; hence:

*N*= 100 (1600 – 0.8

_{r }*N*), 81

_{r}*N*= 160 000

_{r }*N*= 1975.3 r/min

_{r }

The desired load speed to produce a feedback 80 V is 2000 r/min. The error voltage is 0.988 V. The regulator requires an error of 24.7 r/min to ensure steady speed. To attain a steady speed of 2000 r/min, we require offsetting the input signal by the error voltage; thus, an input signal of 80.988 V would cause the load to be driven at a constant speed of 2000 r/min.

The difference between the feedback and the input signal is the error not only under steady-state conditions but also during the transient period. Once the system has settled down, the difference between the desired speed and the actual speed developed under steady-state conditions is called the accuracy, usually expressed as a percentage of the desired speed, which in the situation just investigated is 1.2–35 per cent.

In the regulator, to obtain a different speed, it would be necessary to adjust the input signal voltage. This can be performed simply by deriving it from a potentiometer, as illustrated in Figure 44.11. Yet another potentiometer can be used for positioning the load and the two potentiometers aligned for zero error. With zero error, there will be no drive to the motor from the amplifier. If there is an error signal, it will be amplified to drive the motor and produce the desired movements.

**Figure 44.11** Simple Servomechanism or RPC

Once the load is aligned with the position determined by the input signal, no further motion will occur unless the input signal is varied. Should it be varied, then further action will be required to reposition the load.

In the *RPC* or servomechanism, the steady-state condition occurs when the error is zero, which is distinctly different from the regulator that requires an error.

##### 44.9 TRANSIENT PERIODS

There are two transient conditions. First is the situation that arises when the system is switched on. If we consider the constant speed drive shown in Figure 44.10 at the time of switching on the motor will be at rest. Therefore, it has to accelerate towards the operational speed required. In practice, it accelerates and still continues gaining speed even after it reaches the operational speed; the result is that it accelerates beyond the desired speed and subsequently has to slow down. This response is shown in Figure 44.12.

**Figure 44.12** Transient Response of a Simple Regulator

It is quite likely that the deceleration will produce a further overshoot and the speed may fluctuate several times before a steady state is reached.

In the *RPC*, a similar situation can arise. Here, the load has an inertia that carries it beyond the desired position. As soon as it has passed the desired position, the error reverses and the motor is thrown into reverse. This helps stop the load, but starts it often in the opposite direction. Again an overshoot can take place, but each time it is smaller until the load stops in the desired position, there being no error signal to drive the motor further.

The other transient condition occurs if there is a change in condition after the steady state in the condition has been achieved. Typically, the constant speed regulator might be disturbed if the torque of the load were to change, say it were increased, the motor would slow down, creating an error signal. This would increase the speed back to the original value, but this would give rise to a transient variation in speed for a short period of time.

Regulators quite often have to chase a continuously varying objective. For example, the load always varies in a generator as consumers switch on and off the loads.

Servo systems can also experience similar problems if the input signal is continuously varying, thus causing the load to be continuously chasing the desired moving target. It can also be difficult if the load experiences a force trying to displace it, thus causing it to move from the desired position and hence an error is detected. This causes the motor to produce a force seeking to return the load to the desired position. However, a point of stability is achieved when the displacement force is equal but opposite of the driving torque. This is not at the desired position but one at which the error causes the motor to offset the displacement force.

##### 44.10 DAMPING

In the transient response, the output position moves towards the required objective until it is reached and then the objective is passed, i.e., there is overshoot. The output then approaches the objective from the other side, again resulting in overshoot. This happens two, three, or more times. Such a response, as illustrated in Figure 44.13, is said to be undamped.

**Figure 44.13** Undamped System Response to Step Change

We can reduce the overshoot by introducing a second form of feedback. This depends on the velocity of approach of the load position to the objective. For instance, at the point where the error is zero, there will be a feedback signal due to the overshoot of the position characteristic. The feedback, therefore, has to oppose the error positional signal so that at the given instance it will cause the load to be braking. On introducing sufficient velocity feedback, the characteristic would take the form shown in Figure 44.14.

**Figure 44.14 **Response with Critical Damping

In this response, sufficient velocity feedback has been introduced to ensure that no overshoot takes place. The velocity feedback ceases when the steady-state condition has been reached, the velocity being zero.

This introduction of velocity of feedback is known as damping. When sufficient velocity feedback ensures that there is no overshoot, it is known as critical damping. A slight reduction in velocity feedback would result in overshoot and the system would be under-damped. An increase in velocity feedback would not result in overshoot, but the transient would be even longer. Such a system is said to be over-damped.

System responses that are under-damped, over-damped, and critically damped are illustrated in Figure 44.15.

**Figure 44.15 **System Responses: Under-damped, Critical, and Over-damped

Care has to be undertaken when introducing velocity damping; if too much is introduced, a system can become unstable, i.e., instead of approaching a steady-state condition the overshooting exceeds the initial error and progressively the errors become larger as the load swings backward and forward as shown in Figure 44.16.

**Figure 44.16** The Response of an Unstable System

A control system should not only be stable but also respond as quickly as possible. Critically damped or over-damped systems are slow to move the load to the desired objective. The under-damped system arrives at the objective more quickly, but it produces overshoot.

Most systems can afford a degree of overshoot and therefore most control systems are designed with under-damping. This necessitates two design parameters: the tolerance limits and the settling time.

The relationship between these two parameters is shown in Figure 44.17. The tolerance limits define the extent to which we can expect variation between the load position and the objective position. This is usually expressed as a percentage of a step function giving rise to the change. Values of 5 or 2 per cent are common.

**Figure 44.17** Settling Time and Tolerance Limits

The settling time is the time taken from the start of the step change until the output finally remains within tolerance.

The response characteristics are usually sinusoidal in form except that the peaks are being decremented. The decrement is the ratio of one peak to the next. As an example, let us assume the first peak of the sinusoid to be 100 V and let there be a decrement factor of 4. This means that the next peak that occurs half a cycle later will be 100/4 volts, i.e., 25 V. Applying the same approach, it follows that the next peak is 25/4 = 6.25 V, etc. This is illustrated in Figure 44.18.

**Figure 44.18 **A Decremented Response (Decrement Equal to 4)

A common decrement is 10 so that the first overshoot is limited to 10 per cent of the step change. If the tolerance is also 10 per cent, this provides a very quick response by the system yet remaining within a reasonable proximity of the objective. Furthermore, the return motion will be limited to a 1 per cent overshoot. Similarly, decrements of 5 will take just over two half-cycles before the setting time is achieved for a tolerance limit of 1 per cent.

##### 44.11 THE BASIC FEEDBACK CONTROL SYSTEM

One approach to design is to synthesize a possible system, analyze its performance, and compare its performance with the specifications. The performance in terms of accuracy and stability may be deduced from the response of the system to various inputs. In predicting the response, it is convenient to assume that the system is linear and the input is an impulse, a step, or a sinusoid. The basic control system with negative feedback is shown in Figure 44.19, called the standard diagram. Equations (44.1–44.5) describe the performance of this closed-loop system.

**Figure 44.19 **The Standard Diagram

Equations:

*C*=

*G∈*(44.1)

*∈*=

*R*–

*C*(44.2)

Therefore,

*C*=

*G*(

*R*–

*C*) = (

*GR*–

*GC*) (44.3)

Or

*C*+

*GC*=

*GR C*(1 +

*G*) =

*GR*(44.4)

And

The input to the system is *R*, which stands for the reference variable; the output is *C,* the controlled variable; and *∊* is the error. It is the error times the loop gain *G* that gives the controlled variable output *C*. Equations (44.1–44.5) are quite specific in describing the operation of this loop. Table 44.1 shows the general performance of Eq. (44.5), i.e., this table defines the closed-loop gain. *C/R* of the standard diagram in Figure 44.19 for different values of open-loop gain *G*. For the magnitude of the open-loop gain *G* >>1, the denominator of Eq. (44.5) becomes essentially *G*; therefore, the closed-loop gain is 1; this indicates that when open-loop gain is high, the closed-loop gain is unity. For an open-loop gain *G* << 1, the denominator becomes essentially one; therefore, the closed-loop gain is effectively *G*. This is shown in Figure 44.20.

**Figure 44.20** Closed-Loop Gain and Open-Loop Gain as a Function of Frequency

**Table 44.1 **Closed-Loop Gain as a Function of Open-Loop Gain

Figure 44.20 is a diagram of an assumed gain function that slopes from the upper left to the lower right. When the open-loop gain function *G* > 1, the closed-loop gain *C/R* is equal to unity. When the open-loop gain function *G* < 1, then the closed-loop gain *C/R* is essentially the same as the open-loop gain.

The third case in Table 44.1 occurs when the open-loop gain is approximately unity. For the open-loop gain equal to unity, the closed-loop gain is an unknown value ranging from ½ to ∞; if the open-loop phase is zero, then the denominator has a value of 2 and the closed-loop gain is ½; if the open-loop phase is 180° then the denominator is zero and the gain is – ∞. Stability is of concern when open-loop gain is close to unity, because as long as the gain is much greater than one the closed-loop gain is close to unity; as long as the gain is much less than one, the closed-loop gain is roughly equal to the gain itself. The only region where instability can possibly occur is where the gain is approximately unity. Recognizing this fact makes the servo stability problem relatively easy to handle.

Another aspect of the standard diagram, Figure 44.19, is its error gain characteristics. Equations (44.6–44.9) describe this error-gain performance.

Therefore,

or

The development of these equations is straight forward. Equation (44.6) states that the error is equal to the controlled variable divided by the gain. Equation (44.7) shows that the controlled variable is equal to the reference variable twices *G*/(1 + *G*) from Eq. (44.5). Combining Eqs. (44.6) and (44.7), Eq. (44.8) is obtained. Equation (44.9) restates this in error gain form.

To summarize the characteristics of Eq. (44.9), for the open-loop gain greater than one, the error gain is essentially 1/*G*; for the loop gain less than one the denominator becomes essentially unity, and so the error gain is unity. For the third case where the gain is approximately unity, the error gain can range from ½ to α. Figure 44.21 demonstrates the characteristics of this error gain expression and of those tabulated in Table 44.2.

**Figure 44.21** Error Gain (∈/*R*) and Open-loop Gain *G* Versus Frequency (*ω*)

**Table 44.2 **Error Gain as a Functional Open-loop Gain

To obtain good error performance (small errors), the open-loop gain of a servo should be large. This is indicated in Table 44.2, where it is shown that for *G* greater than one, the error gain is equal to 1/*G*; therefore, the errors decline as the open-loop gain increases. As the open-loop gain is increased in a servo, a point is reached where oscillations occur. In general, the performance of a servo is improved by increasing gain, but only to the point where instability becomes intolerable.

**Example 44.2**

For an open-loop gain of 99, calculate (1) the closed-loop gain *G* of a fully feedback system (standard diagram) and (2) the error gain for this system.

* Note:* (1) An amplifier with an open-loop gain of 99 with a 1-V input would provide a 0.99-V output.

(2) In the above amplifier, the error between the input and output would be 0.01.V.

**Example 44.3**

Calculate the closed-loop gain (*G*) and error gain (*ϵ/R*) for each of the following open-loop gains:

**Solution:**

#### 44.11.1 The Frequency-response Approach

To describe the performance of servos simply and directly, the frequency–response point of view is generally used. Figure 44.22 demonstrates the frequency–response approach.

**Figure 44.22** The Frequency-response Approach

An input signal *A* sin *ω*t is fed into a gain block [described as *G* ( *jω*)] and the output of this block is *B*(* jω*)sin (*ωt* + *θ*). In the frequency–response approach, *G*( *jω*) may be defined as the relation between the output signal and the input signal (also called the transfer function). The magnitude of *G*(*jω*) may be described as *B*(* jω*)|*A*| and the phased *G*( *jω*) is *θ*. Therefore *G*(*jω*) may be described as where *θ* is a function of frequency.

Figure 44.23 demonstrates the above relationship in phasor representation. The input is *A* sin *ω*t (along the real axis) and the output is the phasor *B* sin (*ωt* + *θ*) which is counter-clockwise from *A* sin *ω*t by angle *θ* from the real axis.

**Figure 44.23 ** The Phasor Relationship Between Input and Output Sine Waves

#### 44.11.2 Frequency–response Testing

The input to an amplifier or servo is a sinusoidal signal. The output of the amplifier or servo is a sine wave of the same frequency, having an amplitude and phase different from the amplitude and phase of the input. The relationship between output and input amplitude and phase defines the gain. It is assumed that the system is linear, and thus no harmonics appear at the output. This linearity assumption is satisfactory for practical servos.

Frequency–response testing is accomplished as suggested in Figure 44.24(a). Figure 44.24(b) shows a servo whose input has a sine wave angle variation. This angle command coming into the servo varies sinusoidally with twice and the output of the servo therefore also varies sinusoidally with time; the relative amplitude and phase of the two sine waves (output and input) define the gain of the servo.

**Figure 44.24 ** Frequency–responses Tests

##### 44.12 THE INTEGRATOR

Integrators are very important, because every servo behaves as if it contained an integrator inside the loop. The best first-order approximation of the open-loop gain of a servo is an integrator. Figure 44.25 shows a block diagram of an integrator that has a sine wave input. The input to the integrator is *A* sin *ωt* and the output is *B* sin (*ωt* + *θ*). *A* is arbitrary. *B* is related in amplitude to *A* and *θ* is the phase angle due to the integration, where

**Figure 44.25 ** Frequency–response Test of an Integrator

Equation (44.10) develops the relationship between the output and the input and defines the integration process. Using Eq. (44.10), the gain of this integrator is expressed in Eq. (44.11):

Equation (44.11) shows that an integrator has the property of causing a 90° phase lag and causing a change in gain inversely proportional to frequency. This is shown in Figure 44.26, where the frequency is plotted as the abscissa and the magnitude of gain is plotted as the ordinate; the lower part of the diagram gives the phase angle (–90°) plotted against frequency.

**Figure 44.26** Frequency–response Plot of an Integrator

Figure 44.26 illustrates the logarithmic plot (Bode plot) of the gain function for an integrator as expressed in Eq. (44.12).

*G*= 1/

*jω*(44.12)

Equation (44.13) demonstrates why the bode plot of an integrator is a straight line.

In Figure 44.27, the abscissa is log *ω* (the logarithm of the angular frequency in radianes per second) and the ordinate is the logarithm of the magnitude of the gain function. The Bode plot, therefore, represents the logarithm of the gain function plotted against the logarithm of the frequency. The gain function as expressed in Eq. (44.12) shows that the gain magnitude changes according to the reciprocal of frequency. By taking the logarithm of this expression, Eq. (44.13) can be written. Equation (44.14) expresses and summarizes the relationship that the logarithm of the magnitude of the gain is equal to –log *ω*.

*|G| = –*log

*ω*(44.14)

The plot of Eq. (44.14) is a straight line with a slope of –1, as shown in Figure 44.27. The phase plot is shown in the lower part of Figure 44.27. It can be seen that the phase is a constant –90°.

**Figure 44.27 **Logarithm Plot (Bode Plot) of an Integrator

Figure 44.28 illustrates asymptotic plots of some of the typical mathematical functions encountered in servo design and analysis. These are all straight lines having various slopes.

**Figure 44.28 **Bode Gain Plots of Several Functions

#### 44.12.1 Integration in the Time Domain

A clear understanding of the integration process from several points of view is fundamental to obtaining a physical visualization of the operation of servos. Figure 44.29 shows an input signal consisting of two step functions of constant magnitude in sequence. The integrals of these input step function signals are ramps, because the time integral of a constant value is a time-increasing value. Yet another way of considering it is that the total area under any constant function increases linearly with time.

**Figure 44.29** Time Integral of an Input Function

All servo motors and servo drives exhibit the integration characteristic, because a constant command signal input to a motor or drive system results in a constant velocity output. Of course, the output velocity changes as a function of load torques and other disturbances, but this does not alter the fact that a motor drive acts as an integrator. From a first-order-approximation standpoint, it is valid to consider a servo drive device as simple integration.

Figure 44.30 illustrates the torque–speed characteristics of an electric motor for a servo drive. This may be a two-phase a.c. servo motor or a hydraulic servo device system. For load torques less than *T*_{1}, the velocity of the motor is essentially unaffected by the load torque and it is proportional to the input voltage. As the torque required from the motor increases from *T*_{1} to *T*_{st} (the stall torque), the velocity falls off markedly. If it is assumed that no torque greater than *T*_{1} is demanded of the motor, then the torque–speed characteristic of the drive is essentially flat as shown in the Equation in Figure 44.30. The motor velocity is directly proportional to the motor voltage.

**Figure 44.30 **Torque–speed Curve of an Electric Motor

The motor shown in Figure 44.30 may be considered as a position integrator as shown in Figure 44.31 and by Eqs. (44.15–40.17). The potentiometer supplies the voltage *e _{m}* to the motor through the power amplifier; the output angle of the motor as a function of the input angle to the potentiometer is now determined as

**Figure 44.31** An Open-loop Motor Control (Integrator)

*dθ*/

_{m}*dt*=

*kθ*(44.16)

_{m}Equation (44.15) gives the motor velocity in terms of the motor voltage and in terms of the potentiometer angular setting. It is assumed that the potentiometer in Figure 44.30 has an angular travel of 2*π* rad (360°). The motor displacement angle *θ _{m}* is equal to a constant

*k*times the time integral of

*θ*

*m*. The motor acts as an integrator; this integral characteristic is a very important characteristic.

**Example 44.4**

Find the integral of the time function shown in Figure 44.32.

**Figure 44.32 **Voltage Regulator

**Solution:**

Integral = area

To be useful a control system must be able to cope with changing conditions such as a sudden change in load or a disturbance at some point in the system. In evaluating a system, an approach is to determine the dynamic behaviour described by the complete response to a step function. An alternative approach is to consider the frequency response for steady-state sinusoidal inputs. In either approach, we rely on the transfer function.

The purpose of a voltage-regulating system is to hold the controlled voltage within specified limits in spite of changes in load or other operating conditions. The output voltage *V _{c}* of the unregulated, separately excited, constant-speed, d.c. generator in Figure 44.33(a) varies with changes in load current because of the armature resistance voltage drop. The variation can be reduced by closing the loop through a human operator with an eye on the voltmeter and a hand on the rheostat (Figure 44.33(b)). The controller notes the error between the voltmeter pointer position and the reference and takes necessary action to reduce the error to zero.

**Figure 44.33 **Operation of a Manual Voltage Regulator

The voltage regulator in Figure 44.34 represents an improvement. Here, the output voltage is compared with a reference, and the difference is amplified and used to decrease the discrepancy. The operation is: A sudden increase in load current, reduces output voltage υ_{c} and greatly increases the error voltage υ_{c} = *V*_{+} – *V*_{c}. The error voltage is then amplified and applied to the field circuit tending to increase υ_{f} and restore the output voltage.

For linear operation, the governing equations are as follows:

When each of these relations is represented by an element, it produces the block diagram shown in Figure 44.34(b). In this representation, is introduced as disturbance.

**Figure 44.34 **Wiring and Block Diagrams of Voltage Regulation

* Note:* In anticipation of a rapid response, the variables are shown as instantaneous values.\

##### 44.13 STEADY-STATE OPERATION OF A COMPOUND GENERATOR

A constant-speed compound generator is shown in Figure 44.35. The effect of a load change is fed back by means of series winding. If operation is on the linear portion of the magnetization curve and changes are slow, the governing equation is

*V*=

_{c}*E*–

*I*=

_{A}R_{A}*k*(

*N*+

_{F}I_{F}*N*) –

_{ser}I_{A}*I*

_{A}R_{A}*V*–

_{np}*V*=

*kN*–

_{F}I_{F}*I*(

_{A}*R*–

_{A}*kN*) (44.19)

_{ser}

where, *N* is the number of turns.

The corresponding block diagram is shown in Figure 44.35(b). For flat compounding, the term *R _{A}-kN_{ser}* is made equal to zero, Δ

*V*= 0; and

*V*=

_{c}*V*

_{nl}. There are two inadequacies in this system; first is the feedback loop.

**Figure 44.35 **Steady-state Operation of a Compound Generator (a) Circuit Method (b) Block Diagram

Steady-state operation of a compound generator contains an element *R _{A}* that changes with temperature (and therefore with

*I*) and factor

_{A}*k*that changes with magnetic flux (and therefore with

*I*

_{A}_{})

*.*The second inadequacy is there is no amplification in the loop and the total error is incorporated in the output.

##### 44.14 AUTOMATIC FREQUENCY CONTROL

Power-generating stations should maintain their voltage and frequency to an inter-connected system and a power grid to provide continuity of service and improved stability in the event of overload and/or outage.

As the system load fluctuates throughout the day, it is necessary for the prime movers of all the alternators on the system to speed up or slow down. Consequently, some method of automatic control is required to bring the system frequency to the basic system frequency (say 50 Hz). As both the alternators and their prime movers have a relatively high mass, and as fossil-fuelled prime movers have a high thermal inertia, load fluctuations result in a relatively slow response to an automatic control system.

Figure 44.36 illustrates a typical closed-loop system for correction of frequency. The bus voltage output frequency (*f*_{o}) is sensed by a frequency compartor, which also receives an input from a frequency reference standard (*f _{r}*) In the event of a frequency difference, an error signal (Δ

*f*) is transmitted to a power amplifier that drives a servo motor (

*M*)

*.*The servo motor either opens or closes a steam valve, which in turn either increases or decreases the steam turbine input. If the output frequency of the bus is too high, the error signal reduces the steam turbine input and the speed of the alternator. If the output frequency of the bus is too low, the error signal increases the steam turbine input and the speed of the alternator automatically.

**Figure 44.36 **Automatic Frequency Control of Power System Frequency

Because of the relatively slow system response, the generating control system is designed to correct all paralleled alternators simultaneously, after the frequency error has accumulated to a gain (or loss) of up to 100 Hz. In a 24-hour period, the alternators should generate a total of 5,184,000 Hz to maintain an average 24-hour frequency of 60 Hz. The system shown in Figure 44.37 may be used to control one or motor alternators simultaneously and so maintain a relatively constant frequency to the utilization portion shown in Figure 44.37.

**Figure 44.37** Line Diagram of a Typical Utilization System

##### SUMMARY

- Control systems can be either open looped or closed looped.
- An open-loop system does not recognize the output in the belief that the input will be achieved.
- A closed-loop system feeds back information of the output to ensure that the intended input is achieved.
- The device that produces the feedback signal is called a transducer.
- The error is produced by the comparator and supplied to the controller.
- Control systems can be divided into regulators and RPCs.
- The controller causes a motor to cause the desired output movement.
- Regulators control speed.
- RPCs control position.
- A change of input signal gives rise to a transient change before a new steady-state condition being achieved.
- The steady-state condition is achieved with an error that is not zero but is insufficient to cause further change.
- To reduce the transient period, damping is introduced.
- Damping can be critical damping, over-damping or under-damping.
- Excessive damping can make the system unstable.
- The settling time is the time taken for the transient to reduce within given tolerance limits.

##### MULTIPLE CHOICE QUESTIONS (MCQ)

- The most commonly used input signal(s) in control system is (are)
- Ramp or velocity function
- Step function
- Accelerating function
- All of the above

- The system described by the Eq.
*Y = Ax = a + bx, a >*0,*b >*0 is- Linear
- Non-linear
- Dynamic
- Time varying

- Linear systems obey
- Reciprocity principle
- Principle of super-position
- Principle of maximum power transfer
- All of the above

- If the initial conditions for a system are in erentlyzero, what does it signify?
- The system stores energy
- The system does not store energy
- The system is working with zero reference input
- None of the above

- In a control system, the use of feedback
- Increases the reliability
- Eliminates the chances of instability
- Reduces the effects of disturbance and noise signals in the forward path
- Increases the influence of component parameters on system performance

- Which one of the following effects is not caused by negative feedback?
- Reduction in gain
- Increase in bandwidth
- Increase in distortion
- Reduction in output impedance

- Damping in a control system is a function of
- Gain
- 1/(gain)
- √gain
- 1/
*V*gain

- In the time domain specification, the time delay is the time required for the response to reach
- 75 per cent of the final value
- 50 per cent of the final value
- 33 per cent of the final value
- 25 per cent of the final value

- How can a steady-state error in a system be reduced
- By increasing the system gain
- By decreasing the system gain
- By decreasing the static error constant
- By increasing the input

##### ANSWERS (MCQ)

- (b)
- (a)
- (d)
- (c)
- (c)
- (c)
- (d)
- (b)
- (b)

##### CONVENTIONAL QUESTIONS (CQ)

- Find the closed-loop gain
*G*’(*jω*) and the error gain ϵ(*jω*)/*R*(*jω*) for the following values of*G*(*jω*):- 100∠0°
- 10∠0°
- 1∠0°
- 0.5∠0°
- 100∠–90°
- 10∠–90°
- 1∠–90°
- 0.5∠–90°
- 1∠–135°
- 1∠–175°
- 1∠–180°

- Find the time integrals of the following functions and state the units of the answers:
- 7 V from 0 to 35
- 8
*t*V, where,*t*= time from 0 to 2.5 - 4
*e*^{–5t}V from 0 to 0.2 S

- Explain the concept of a closed-loop system. Describe the manner in which it differs from an open-loop system.
- Describe the essential components of a closed-loop system with which you are familiar. What are the advantages and disadvantages associated with the use of feedback in such a control system?
- Describe the settling time in a servo system.
- Explain the terms under-damped, over-damped and critically damped.
- Explain the manner in which a transducer can produce a signal proportional to angular displacement.
- Why don’t power inputs appear in a control system block diagram?
- What information on system performance is provided by step response?
- Explain in system terminology how body temperature is regulated?
- Define transform, transform function and step response.

##### ANSWERS (CQ)

1. (a) 0.99∠0° 0.0099∠0° (b) 0.91 ∠0°, 0.091∠0° (c) 0.5 ∠0°, 0.5 ∠0° (d) 0.33∠0°, 0.67∠0° (e) 1∠–.57°, 0.01∠+89.43° (f) I∠+84.3° (g) 0.707∠–45°, 0.707∠+45° (h) 0.45∠–63.4°, 0.9∠+26.6° (i) 1.3∠–67.5°, 1.3∠+67.5 (j) 11.5∠+87.5° (k) –∞, +∞ (2.(a)) 21 V.s, (b) 16 V.s (c) 0.506 V.s.