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TUTORIAL
EC4105
DISCRETE & INTEGRATED ANALOG CIRCUITS
Module 1:
RC Filters : RC loss pass and high pass filters and their response to sinusoidal, step,
pulse, square wave and ramp inputs.
1.
Refer to figure 1. Explain why it is called a high pass RC circuit. Show that its transfer
function is given as
Vo (s)
-------Vi (s)
RCS
= A(s) = ----------- and
1+RCS
1
| A(s) | = ---------------------------- where fo = 1 / 2RC
[1+(f )o/ f)2] ½
C
Vs
R
V0
Fig. 1
And  = arctan (fo / f).
Where  is the angle by which output voltage leads the input voltage and f o is the cut-off
frequency. At this frequency fo, the magnitude of capacitive reactance is equal to the
resistance and the gain is 0.707.
2.
A step voltage is applied to the circuit in figure 1, at time t = 0. (a) Determine the output
vo (t); (b) Show that the steady state is achieved (final value) if t > 5 for most of the
engineering applications. (c) Show that the voltage across capacitor cannot change
instantaneously, provided that the current remains finite.
3.
A voltage pulse of amplitude V and duration t p is applied to the circuit in figure 1. Derive
vo(f) for all t. Sketch the output waveform for the cases RC >> t p and RC << tp.
4.
A voltage square wave is applied to the circuit in figure 1. The average value of the
square wave input is Vd.c.
(a) Write the equations to determine the output voltage. Specialize these equations to the
case if the input were a symmetrical square wave.
(b) Sketch the output waveform vo(t) for the cases
(i)
RC>>T (ii) RC<<T and (iii) RCT where T=T1+T2, is the time period of the
square wave.
5.
A ramp (or sweep) voltage is applied at t = 0 to the circuit shown in figure 1. The slope of
ramp is . Determine the output voltage vo (t) and sketch it to an arbitrary scale for the
cases RC>>T and RC<<T. Also obtain the expression for the transmission error.
6.
Under what conditions the circuit shown in figure 1 will act as a differentiator? What will
be the output if (i) square wave (ii) ramp signal and (iii) a sinusoidal signal is applied to its
input terminals?
7.
A symmetrical square wave of peak amplitude V and frequency f is applied to a high
pass RC circuit. Show that percentage tilt is given by:
1 – e-1/2fRC
P = ------------------ x 200 percent and if tilt is small than
1 + e-1/2fRC
This equation reduces to
T
P = ------------ x 100 percent.
2RC
8.
A 10 Hz symmetrical square wave whose peak to amplitude is 2V impressed upon a high
pass circuit whose lower 3-dB frequency is 5Hz. Calculate and sketch the waveform. In
particular what is peak output amplitude?
9.
A 10 Hz square wave is fed to an amplifier. Calculate and plot the output under the
following conditions; the lower 3 dB frequency is (i) 0.3 Hz (ii) 3.0 Hz (c) 30 Hz.
10.
(a) A square wave whose peak-to-peak value is 1V extends  0. 5V with respect to
ground. The duration of the positive section is 0.1 sec and of the negative section is 0.2
sec. If this waveform is impressed upon an RC differentiating circuit whose time constant
is 0.2 sec., what are the steady state maximum and minimum values of the output
waveform?
(b) Prove that the area under the positive section equals that under the negative section
of the output waveform. What is the physical significance of this result?
(c) Write the equations to determine the output voltage.
11.
Refer to figure 2. Explain why is it called a low pass RC circuit, show that its transfer
function is given as
Vo (s)
RCS
-------- = A(s) = ----------- and
R
Vi (s)
1+RCS,
1
| A(s) | = ---------------------------- and  = tan-1 (f / f2)
[1+(f / f2)2] ½
1
Where f2 = ---------------- , the upper 3 dB frequency.
2RC
C
v0
Fig. 2
12.
A step voltage is applied to the circuit in figure 2 at time t = 0. (a) Obtain the expression
for vo(t), (b) Obtain expression for rise time in terms of time constant RC and cut-off
frequency f2.
13.
A voltage pulse of amplitude V and duration t p is applied to the circuit in figure 2. Derive
vo (t) for all t. Sketch the waveform for the cases (i) RC >>T and (ii) RC<<T. State the
rule of thumb for preserving the shape of the pulse.
14.
A voltage square wave is applied to the circuit in figure 2, whose average values V d.c. (a)
Write the equations to determine the output voltage vo(t). Simplify these equations for the
symmetrical square wave with zero average value and obtain the solution. (b) For voltage
square wave input of part (a), sketch the output waveform vo(t) for the following cases (i)
RC << T (ii) RC  T and (iii) RC >> T where T = T1+T2 is the time period of the square
wave.
15.
At t=0, a ramp voltage is applied to the circuit shown in figure 2, whose slope is .
Determine the output waveform vo(t) and sketch it for the cases RC<<T and RC>>T
where T is the total ramp time of interest. Also obtain the expression for the transmission
error.
16.
Under what conditions the circuit shown in figure 2 will act as an integrator? Discuss its
limitations and advantages over the differentiator of problem No. (6). If input to this circuit
were (i) a ramp voltage (ii) a square wave and (iii) a sinusoidal signal; then determine the
output waveform vo(t).
17.
An ideal 1 sec. pulse is fed to an amplifier. Calculate and plot the output waveform
under the following conditions; the upper 3dB frequency is (i) 10 MHz (ii) 1 MHz, (iii) 0.1
MHz.
18.
A pulse is applied to a low pass RC circuit. Prove by direct integration that the area under
the pulse is the same as the area under the output waveform across the capacitor. Give
physical interpretation of this result.
19.
A symmetrical to square wave whose peak-to-peak amplitude is 2V and whose average
value is zero is applied to an RC integrating circuit. The time constant equals the half of
the square wave. Calculate and sketch the waveform. In particular, find the peak-to-peak
amplitude of the output amplitude.
20.
A symmetrical square wave whose average value is zero and has peak-to-peak
amplitude of 20V and a period of 2  sec. This waveform is applied to a low pass circuit
whose upper 3-dB frequency is 1/2 MHz. Calculate and sketch the steady state output
waveform In particular, what is peak to peak output amplitude.