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Jawaharlal nehru technological university anantapur college of engineering
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UNIT – I
LINEAR WAVESHAPING
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High pass, low pass RC circuits, their response for sinusoidal, step, pulse, square and ramp
inputs. RC network as differentiator and integrator, attenuators, its applications in CRO probe,
RL and RLC circuits and their response for step input, Ringing circuit.
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A linear network is a network made up of linear elements only. A linear network can be
described by linear differential equations. The principle of superposition and the principle of
homogeneity hold good for linear networks. In pulse circuitry, there are a number of waveforms,
which appear very frequently. The most important of these are sinusoidal, step, pulse, square
wave, ramp, and exponential waveforms. The response of RC, RL, and RLC circuits to these
signals is described in this chapter. Out of these signals, the sinusoidal signal has a unique
characteristic that it preserves its shape when it is transmitted through a linear network, i.e. under
steady state, the output will be a precise reproduction of the input sinusoidal signal. There will
only be a change in the amplitude of the signal and there may be a phase shift between the input
and the output waveforms. The influence of the circuit on the signal may then be completely
specified by the ratio of the output to the input amplitude and by the phase angle between the
output and the input. No other periodic waveform preserves its shape precisely when transmitted
through a linear network, and in many cases the output signal may bear very little resemblance to
the input signal.
The process whereby the form of a non-sinusoidal signal is altered by transmission through a
linear network is called linear wave shaping.
THE LOW-PASS RC CIRCUIT
Figure 1.1 shows a low-pass RC circuit. A low-pass circuit is a circuit, which transmits
only low-frequency signals and attenuates or stops high-frequency signals.
At zero frequency, the reactance of the capacitor is infinity (i.e. the capacitor acts as an
open circuit) so the entire input appears at the output, i.e. the input is transmitted to the output
with zero attenuation. So the output is the same as the input, i.e. the gain is unity. As the
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frequency increases the capacitive reactance (Xc = H2nfC) decreases and so the output decreases.
At very high frequencies the capacitor virtually acts as a short-circuit and the output falls to zero.
Sinusoidal Input
The Laplace transformed low-pass RC circuit is shown in Figure 1.2(a). The gain versus
frequency curve of a low-pass circuit excited by a sinusoidal input is shown in Figure 1.2(b).
This curve is obtained by keeping the amplitude of the input sinusoidal signal constant and
varying its frequency and noting the output at each frequency. At low frequencies the output is
equal to the input and hence the gain is unity. As the frequency increases, the output decreases
and hence the gain decreases. The frequency at which the gain is l/√2 (= 0.707) of its maximum
value is called the cut-off frequency. For a low-pass circuit, there is no lower cut-off frequency.
It is zero itself. The upper cut-off frequency is the frequency (in the high-frequency range) at
which the gain is 1/√2 . i-e- 70.7%, of its maximum value. The bandwidth of the low-pass circuit
is equal to the upper cut-off frequency f2 itself.
For the network shown in Figure 1.2(a), the magnitude of the steady-state gain A is given by
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Step-Voltage Input
A step signal is one which maintains the value zero for all times t < 0, and maintains the
value V for all times t > 0. The transition between the two voltage levels takes place at t = 0 and
is accomplished in an arbitrarily small time interval. Thus, in Figure 1.3(a), vi = 0 immediately
before t = 0 (to be referred to as time t = 0-) and vi = V, immediately after t= 0 (to be referred to
as time t = 0+). In the low-pass RC circuit shown in Figure 1.1, if the capacitor is initially
uncharged, when a step input is applied, since the voltage across the capacitor cannot change
instantaneously, the output will be zero at t = 0, and then, as the capacitor charges, the output
voltage rises exponentially towards the steady-state value V with a time constant RC as shown in
Figure 1.3(b).
Let V’ be the initial voltage across the capacitor. Writine KVL around the IOOD in Fieure 1.1.
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Expression for rise time
When a step signal is applied, the rise time tr is defined as the time taken by the output
voltage waveform to rise from 10% to 90% of its final value: It gives an indication of how fast
the circuit can respond to a discontinuity in voltage. Assuming that the capacitor in Figure 1.1 is
initially uncharged, the output voltage shown in Figure 1.3(b) at any instant of time is given by
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