DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
EEG416
ACTIVE NETWORKS; ANALYSIS, SYNTHESIS AND DESIGN
TUTORIAL EXERCISES
Q1a.
Define sensitivity. State an expression for multi-element variation sensitivity of the
parameters of a biquad filter.
b. Show that the transfer function of the circuit of fig. 1 admits a biquad function
representation.
i. Define all its biquad parameters.
ii.
Define the sensitivity for the biquad
parameters in terms of component variations.
C1
R1
R2
V1
V2
fig. 1
C2
A voice signal of frequency (0.3 – 3.4)kHz occupies a baseband telephone channel
(0 – 4)kHz. The channel is multiplexed with others such that the transition gap between
channels is 4kHz. If a maximum attenuation of 1dB is allowed in each telephone channel,
and a minimum attenuation of 20dB required elsewhere. Design an appropriate filter
using Butterworth approximation to process the signal.
b. Realize the filter as an active network using positive feedback topology.
Q2a.
Q3a.
Show that the circuit of fig. 2 is a generalized biquad circuit whose transfer function
admits the expression

R10
R s
R 
s
1

 10  10  s 2 

R 7 R1R 4C 1C 2 R8R1C 1 R 9 
R 2C 1 R 3R 4C 2C 1 
VO

s
1
VI
s2 

R 2C 1 R 3R 4C 1C 2
Show that the circuit can be used to realize low pass and band pass functions amongst
others. Obtain expressions for its low pass filter and band pass filter outputs.
b. Use the circuit to synthesize the delay equalizer whose transfer function is given as
s 2  10s  2500
T ( s)   2
s  10s  2500
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R3
R2
C1
C2
R6
R10
R4
R1
R5
R7
R8
R9
VO
VI
fig. 2
Q4ai. Explain the need for frequency transformation in the synthesis of filter characteristics.
ii.
State the expressions required to transform a bandpass filter characteristic to a
normalized realizable filter function. Define all variables therein. Use appropriate
characteristic diagrams to demonstrate the process.
b. Using Butterworth approximation method, obtain the gain function for a filter described
by the characteristic of fig. 3.
A()
dB
1 = 200rad/s
2 = 5000rad/s
20
3 = 100rad/s
4 = 10000rad/s
1

3
1
2
fig. 3
4
Q5a. Verify that the following functions are realizable. State at least three conditions to justify
your assertion.
s 2  4s  3
s 2  4s  3
Z ( s) 
Z ( s)  3
i.
ii.
s 2  2s
s  6 s 2  8s
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b.
Realise the functions in (a) with either Foster method or Cauer method.
Q6a.
Define gain sensitivity and state the general expression for
i.
gain variation
ii.
pole frequency variation.
For the circuit of fig. 4, obtain an expression for the gain.
If the values of the resistors vary by 10%, estimate the relative change in the value of
1.
pole frequency
2.
pole Q
C1
R2
bi.
ii.
R1
C2
Vi
Vo
fig. 4
Q7ai. Show that the circuit of fig. 5 realizes a bandpass filter. Hence, obtain an expression for
the filter’s transfer function.
ii.
Define all the biquad parameters for the circuit.
b.
Use the circuit of (a) to realize the transfer function
H ( s) 
400 s
s  400 s  1.024  107
2
C2
RF
C1
R1
Vi
r2
Vo
fig. 5
r1
Q8a. State two characteristic properties of filters realized using each of the following
approximation functions.
i.
Butterworth ii.
Chebyshev
iii.
Cauer
b.
Use Chebyshev approximation to obtain the gain function of the filter having the
characteristic of fig. 6.
c.
Propose a realization of the circuit with a Salen and Key circuit.
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A()
dB
15
1
2000
5000

rad/s
fig. 6
Q9.a. Define sensitivity.
b.
Starting from basic definitions, derive an expression for the sensitivity of the gain
function of a biquad filter in terms of its components.
c.
State the factors affecting the variation in gain.
Q10a. A filter is required to meet the specifications:
fP = 10kHz, APmax = 1dB,
fS = 20kHz, ASmin = 30dB and dc gain = 0dB
Obtain the filter transfer function using Butterworth approximation.
b.
Realize the filter using op-amp biquad in negative feedback topology.
State two advantages of the positive topology over negative topology.
Q11a. Analyze the circuit of fig. 7 and obtain its transfer function.
b. Use the obtained transfer function to evaluate the sensitivity of it’s pole frequency and
pole quality to the variation of all its resistive elements.
c. If the values of the resistances increase by 10% due to increase in temperature, calculate
the percentage change in the values of the pole parameters.
C2
R2
C1
R1
Vi
VO
fig. 7
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Q12. The transfer function of a band pass filter designed with Chebyshev polynomial has the
expression
Ks 2
H ( s)  2
s  3s  81 s 2  4s  64
a. What is the order of its equivalent normalized low pass filter.
b. How many biquads are required to realize this function. Define all the parameters of the
biquad sections.
c. Synthesize the function using the circuit of fig. 8. Calculate the gain constant.



R2
C2
C1
R1
VI
VO
R
R(k-1)
Q13.
fig. 8
For the function
s2
H ( s)  2
s  200s  640,000
a. Evaluate ωp and Qp
b. Synthesize the function with practical element values and state the topology used.
c. If the values of resistances vary by ±10% and capacitors by ±5%, evaluate the variation
in the gain of the network.
Q14.i. Show that the circuit of fig. 9 will realize a band pass filter.
ii.
Hence, realize the function
150s
H (s)  2
s  5s  625
iii.
Assuming C1 = C2 = 1F and R1 = R2 = R3 = R obtain the values of all the elements in
the circuit.
iv.
Draw a well labeled realized circuit
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R2
R1
C2
+
Vi
Vo
C1
Q15.
Fig. 9
RB
R3
RA
Given the function
15s 2
H ( s)  2
s  4s  144
Synthesize the network with Sallen and Key circuit using C1 = C2 = 1F, and R1 = R2 = R
obtain the values of all other components. Draw the complete realized circuit
diagram.
Q16.a. Show that for each of the following network functions, the magnitude functions
|H(jw)|2 are constant.
s 2  as  b
s a
H 2 (s )  H 2
H 1 (s )  H
s b
s  as  b
b.
Obtain expressions for the phase function of the networks.
Q17 Determine the transfer function for a low pass filter having a maximally flat gain
magnitude characteristic, which is 1dB down at 2 rad/s and 30dB down at 6rad/s.
Q18 Find a low pass maximally flat magnitude network function which is down 3dB at 1kHz
and 20dB at all frequencies greater than 2.5kHz.
Q19. Show that the transfer function H(s) for the twin-T network of fig. 1 assumes the
expression
C
C
R
Fig. 1
Vi
2C
R
½R
V
o
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Q20.
Determine whether the following functions are positive real.
Hint: For the functions with factorized denominator, perform partial fraction expansion.
(i)
(iii)
(ii)
(iv)
Q21.
A low pass filter has the specifications;
APmax = 1dB, Asmin = 35dB, fP = 1000Hz and fs = 3500Hz.
(a) Find the Butterworth approximation function of the filter
(b) Determine its loss at 900Hz
(c) Determine the pole Q of the gain function.
Q22.
Estimate the order of (i) Butterworth, (ii) Chebyshev approximation function needed to
realize the filter described in fig. 2
Loss dB
55
Fig. 2
0.2
ω
1000
1500
2000
3000
rad/s
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