##### 13.1 INTRODUCTION

We saw in Chapter 12 that voltage sweep generators are used where smaller deflections of the electron beam are required, such as in CROs with small screens mainly because of the requirement of supply voltages, as the deviation from linearity is negligible only when the supply voltage is significantly higher than the sweep amplitude. This condition implies that supply voltages could be prohibitively high if larger sweep amplitude is required to achieve larger deflections of the electron beam, as in the case of television and radar receivers. In such instances, voltage sweep generators will not meet the requirements and hence, the need for current sweep generators. The basic principle employed in current sweep generators is electromagnetic deflection. In these sweep generators, when a voltage is applied to a coil of inductance *L*, the current in the inductance increases linearly with time. Usually a coil or a set of coils—called the yoke—is mounted externally onto the gun structure of the tube and the current in the yoke produces a magnetic field that deflects the electron beam. However, in practice, the current sweep so generated may not necessarily be linear. In order to linearize the current sweep, the driving waveform needs to be adjusted. It will be shown that a trapezoidal driving waveform generates a linear current sweep. The application of a current sweep in a television receiver is considered in the chapter.

#### 13.1.1 A Simple Current Sweep Generator

A simple current sweep generator is shown in Fig. 13.1(a). The circuit basically consists of a transistor used as a switch driven by a trigger signal. The current in the inductor rises exponentially during the period the switch is closed (the transistor is ON). At the end of the trigger signal, the device switches into the OFF state and the current in the inductor decays. If at a time *t* = 0, a voltage *V* is applied to a coil of inductance *L* in which the current is initially zero, then the inductor current *i*_{L} will increase linearly with time, *i*_{L} = (*V/L*)*t* (in a capacitor, *V* = (*I/C*)*t*). A time-base circuit using this principle is shown in Fig. 13.1(a). The waveforms of this sweep circuit are shown in Fig. 13.1(b).

The trigger signal *V*_{B} operates between the two levels. The lower level (negative going period) keeps the transistor in the cut-off state, while the upper level (positive going period) drives the transistor into saturation.

As long as the input is negative, *Q*_{1} is biased OFF and the inductor current is zero. At *t* = 0+, as the input goes positive, *Q*_{1} is driven into saturation. The current *i*_{L} increases linearly with time {*i*_{L} = (*V _{CC}/L*)

*t*}. The inductor current attains a maximum value of

*I*

_{L}, (see Fig. 13.2).

During the sweep period, *Q*_{1} is ON and the voltage at its collector is *V*_{CE(sat)} which is small. The diode *D* does not conduct since it is reverse-biased. The sweep terminates at *t* = *T*_{s} when the trigger signal drives the transistor to cut-off. The current in the inductor at *T*_{s} is the peak current *I*_{L}. A spike of amplitude *I _{L}R_{d}* appears across the inductance

*L*. The net result is that the voltage at the collector of the transistor is (

*V*

_{CC}+

*I*

_{L}

*R*

_{d}). The diode

*D*is now forward-biased. The inductor current then continues to flow through the diode

*D*and the resistance

*R*

_{d}till it decays to zero eventually. This decay is exponential with a time constant

*τ*=

*L/R*

_{d}where,

*R*

_{d}is the sum of the damping resistance and the diode forward resistance:

At *t* = 0−, *V*_{CE} = *V*_{CC}. When the transistor is ON, *V*_{CE} = *V*_{CE(sat)}, which is very small. At *t* = *T*_{s}, *Q*_{1} is turned OFF. The net voltage at the collector of the transistor is (*V*_{CC} + *I*_{L}*R*_{d}), and must be limited to make sure that it would not exceed the break down voltage of the collector base junction. The peak current *I*_{L} is chosen on deflection requirements, and as a result a spike of magnitude *I*_{L}*R*_{d} is generated. There is an upper limit to the size of *R*_{d} to limit the magnitude of this spike to a safe value. The spike decays with the same time constant as the inductor current. Thus, we see that the spike duration depends on *L*, whereas the spike amplitude does not depend on *L* but instead depends on *R*_{d}. So far, the resistance of the inductor *R*_{L} and the collector saturation resistance of the transistor, *R*_{CS} are neglected. Taking *R*_{L} and *R*_{CS} into account, *i*_{L} is written as:

**FIGURE 13.1(a)** A current sweep generator

**FIGURE 13.1(b)** The waveforms of a current sweep circuit

Expanding the exponential into series

The peak-to-peak excursion of the sweep is *V*_{CC}/(*R*_{L} + *R*_{CS}) and the sweep amplitude is *I*_{L}.

The slope error is, therefore, given as:

From Eq. (13.1) it is evident that the sweep is non-linear. The current sweep is linear if the slope error is small. Therefore, from Eq. (13.2), to ensure linearity the voltage (*R*_{L} + *R*_{CS})*I*_{L} must be small when compared with the supply voltage, *V*_{CC}. To understand the procedure to plot the waveforms let us consider an example.

##### EXAMPLE

*Example 13.1:* In the circuit shown in Fig. 13.2(a) the resistance of the coil is 10 Ω, collector saturation resistance of the transistor is 5 Ω and the diode forward resistance is zero.

(a) For a 400 µs sweep, draw waveforms of *i*_{L} and *v*_{CE} indicating the voltage, current levels and time constants.

(b) Calculate the retrace time (time during which the inductor current falls to 10 per cent of its maximum value) if (i) *R*_{d} = 150 Ω and (ii) *R*_{d} = 1000 Ω.

(c) Calculate the slope error of the sweep.

**FIGURE 13.2(a)** A current sweep generator

(a)

At

*t* = *T*_{s}, *i*_{L} = *I*_{L}

The waveforms of *i*_{L} and *v*_{CE} are shown in Fig. 13.2(b).

The amplitude of spike = *I*_{L}*R*_{d} = 0.0582 × 150 = 8.73 V

(b)

Let the current fall to 10 per cent of the maximum value at *t* = *T*_{r} (retrace time).

We have

At

Therefore,

*e ^{Tr/τ}* = 10

*T*_{r1} = *τ* ln10 = 2.30*τ* = 2.3 × 0.66 × 10^{−3} = 1.53 ms

(ii) Let

*R*_{d} = 1000 Ω

Then

*T*_{r2} = *τ* ln 10 = 0.1 × 10^{−3} × 2.30 = 0.23 ms

(c)

**FIGURE 13.2(b)** The waveforms of *i*_{L} and *v*_{CE}

#### 13.1.2 Linearity Correction through Adjustment of the Driving Waveform

The current sweep shown in Fig. 13.1(a) with pulse as the driving waveform may not necessarily generate a linear output. One simple method to produce a linear current sweep is by adjusting the driving waveform. The non-linearity encountered in this circuit, as seen from Eq. (13.1), results from the fact that as the inductor (yoke) current increases, the current in the internal resistance of the source *R*_{s} also increases. Consequently the voltage across the inductor decreases, as shown in Fig. 13.3(a). So, *v*_{L} = *V*_{s} − *i*_{L}*R*_{s} as *v*_{s} is constant at *V*_{s}.

If we can compensate for the voltage developed across the resistance *R*_{s} then the current sweep tends to be linear. This can be achieved as shown in Fig. 13.3(b). Let *R*_{s} be the internal resistance of the source *v*_{s}.

**FIGURE 13.3(a)** The inductor voltage decreases as the inductor current increases

**FIGURE 13.3(b)** The driving waveform for generating a linear current sweep

The total circuit resistance is (*R*_{s} + *R*_{L}). Now, if we want the inductor current to vary linearly, i.e., *i*_{L} = *Kt* (where *K* is the constant of proportionality) then the source voltage *v*_{s} is,

As,

Therefore,

This waveform consists of a step followed by a ramp (*R*_{s} +*R*_{L})*Kt*. Such a waveform is called a trapezoidal waveform. We can thus see that if the driving signal is trapezoidal as given by Eq. (13.3) then the current sweep is linear (i.e., *i*_{L} = *Kt*). The Norton representation of the driving source, using Eq. (13.3) is:

The waveform of this current source is also a step followed by a ramp, as shown in Fig. 13.4. Thus, a trapezoidal driving waveform generates a linear current sweep. At the end of the sweep, the current once again will return to zero exponentially with a time constant *τ* = *L*/(*R*_{s} + *R*_{L}). Generally, *R*_{s} >> *R*_{L}, hence, *τ* ≈ *L/R*_{s}.

The question now is, should *R*_{s} be small or should it be large? The resistance *R*_{s} is chosen based on two conflicting requirements. If *R*_{s} is small, the current will decay slowly and a long period will have to elapse before another sweep is possible, i.e., the fly-back time becomes unacceptably long. This could be construed as a disadvantage. However, the advantage is that the peak voltage developed across the inductor (= *I*_{L}*R*_{s}) may not be unduly large. As a result, the voltage at the collector of the transistor when the sweep terminates (= *V*_{CC} + *I*_{L}*R*_{s}), which reverse-biases the base-collector diode of the transistor may not be large enough to damage the device.

Alternately, if *R*_{s}; is large, the current will decay rapidly. This means that the retrace time is negligible, which enables us to initiate the next sweep immediately after the sweep duration. On this count, *R*_{s} is required to be large. However, a large peak voltage will appear across the inductor and the voltage that now reverse-biases the base-emitter diode of the transistor may be excessively large and can damage the device.

**FIGURE 13.4** The trapezoidal current source and the wave form

Generally, a compromise has to be struck such that the spike amplitude is not appreciably large and also the inductor current decays in a smaller time interval. Normally, to achieve this, a damping resistance *R*_{d} is connected across the yoke to limit the peak voltage. Let *R* be the parallel combination of *R*_{s} and *R*_{d}. Then the retrace time constant is *τ*_{r} = *L/R*. The trapezoidal waveform required to improve the linearity of the current sweep is generated using the circuit shown in Fig. 13.5 from the figure, we have:

**FIGURE 13.5** The generation of a trapezoidal waveform

Generally, *R*_{2} *R*_{1}:

Dividing by *R*_{2}:

Generally *R*_{2} 1. Therefore,

Expanding the exponential as a series and limiting to the first few terms of the expansion, we have:

If

Thus, the first term is a step and the second term represents a ramp. Therefore, *v*_{o} is a step followed by ramp (trapezoidal). A practical linear transistor current sweep is shown in Fig. 13.6.

Here, *Q*_{1} acts as switch, it is ON when the input is zero and is OFF when the input goes negative. *R*_{1}, *R*_{2} and *C*_{1} generate the trapezoidal driving waveform. *Q*_{2} and *Q*_{3} combination is a Darlington pair and *R*_{E} stabilizes *i*_{L}. The Darlington emitter follower provides a large input resistance and thus eliminates the loading on the driving source by its input. The current *i*_{L} varies linearly with time.

**FIGURE 13.6** A practical linear current sweep generator