ECEN Lab 2: Second Order Circuits Solution

    • Objectives

The purpose of the lab is to investigate the frequency response of second order circuits and further practice circuit design and analysis techniques in the frequency domain.

    • Introduction

It is useful to format a transfer function as a multiplication of known functions so that its frequency response can be easily sketched without the need for complex tools. Since the y-axis of Bode plots are in dB-scale, Bode plot of the overall transfer function can be simply obtained by graphically adding Bode plots of individual first or second order sections on the same frequency axis.

In case the derivation of H(s) already yields multiplication of known functions, then it is best to keep them in these formats. However, if the derivation yields multiplication of transfer functions with high order numerator or denominator polynomials, then you may need to decompose them into smaller known sections. Table 1 shows first order functions and their magnitude and phase responses.


Table 1: First Order Functions

H(s)
Bode Magnitude
Bode Phase
H(s)
Bode Magnitude
Bode Phase



ωo
0
ωo
ω



90


1





+20 DB/DEC




0dB
ω

1+ s
















s

−90


0DB
ω

0
ω

1+
−20 dB/dec


!o


ω


ω
!o






O


O







ωO



















ω
o



s
0DB


ω
90





s +!o


−20 dB/dec

0



ω







































































s +!o

+20 DB/DEC



























0




ω


s
0dB




ω
−90











ω








ω



















O









o










































0

ωo
ω







+20 DB/DEC

90
































!o

0dB
ωo
ω








s

0DB




ω


















































































ωO








s

−20 dB/dec

−90





!o




0



ω




















ω
O




































20log|K|

0




ω





0dB




ω
















































1








0



ω






























K





























0dB


ω
−180







K


−20log|K|
































































































In case any section of the transfer function has a second order denominator with complex poles, then that section cannot be decomposed into first order functions. Table 2 shows the second order functions with complex poles and their magnitude and phase responses. Remember that second order polynomials with real poles can always be expressed as a multiplication of two first-order polynomials, so that a combination of first order functions in Table 1 can still be used.










c    Department of Electrical and Computer Engineering, Texas A&M University

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Table 2: Second Order Functions




H(s)
Bode Magnitude

Bode Phase






0dB
ω
0
ωo
ω



!2










ωo






0






2

!0

2




s

+
Q
s + !0
−40 dB/dec
−180
























s2
0DB
ω
180







ωO




2

!0

2

0

ω
s

+
Q
s + !0
+40 DB/DEC










ωO














!0 s

ωO







0DB
ω
90
ωo




Q



0

ω










s2 +
!0 s + !02

−90





Q












ωO





s
2+!2
0DB
ω










ωO





0


0

ω



!0 s + !02




s2 +








Q






s2

!0 s + !02
ωO

ωO




Q

0DB
ω
0

ω










s2 +
!0 s + !02







Q









Calculations

    1. Derive the transfer functions for the circuits shown in Figs. 1(a), 1(b) and 1(c):



HLP (s) =
VLP (s)

HHP (s) =
VHP (s)
HBP (s) =
VBP (s)



Vi


Vi

Vi
R1
R2


C3
C4

C5
R6


VLP



VHP

VI
C1
C2
VI

R3
R4
VI
R5

(a)



(b)


(c)


Figure 1: Second order (a) lowpass filter (b) highpass filter (c) bandpass filter











































VBP
C6


Express the transfer functions as
H(s) =

P(s)
(1)

Q(s)





where P(s) and Q(s) are polynomials of s with coefficients in terms of resistors and capacitors. Remember that negative powers of s are not allowed in polynomials, so all the terms in P(s) and Q(s) must contain sn where n 0.








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2. Find the resistor and capacitor values such that the transfer functions can be formatted as follows:

HLP (s) =
1

1




,
f1 = 4kHz, f2 = 8kHz
(2)



















1 +
s
1 +

s












2 f2




2 f1








HHP (s) =
s



s
,

f3 = 4kHz, f4 = 8kHz
(3)













s + 2 f3



s + 2 f4




HBP (s) =

s
1


,
f5 = 4kHz, f6 = 8kHz
(4)

















s + 2 f5
1 +

s








2 f6























When calculating component values, you can make reasonable approximations. For example, 1 + x    1 if

x   1, which typically requires x < 0.1. Similarly, 1+x   x if x   1, which requires x > 10.

    3. Sketch the magnitude and phase Bode plots for HLP (s), HHP (s), and HBP (s).

    4. Calculate the output voltages VLP (t), VHP (t), and VHP (t) for Vi (t) = 0.5 sin(2 6000t).

Simulations

For all simulations, provide screenshots showing the schematics and the plots with the simulated values prop-erly labeled.

Draw the schematics for the circuits in Fig. 1 with the calculated component values. Perform the following simula-tions for each circuit:

    1. Obtain the magnitude and phase Bode plots of the transfer function using AC simulation, and measure the 3-dB frequencies and passband gains. Also measure the magnitude and phase of the transfer function at 6kHz.

    2. Apply the input Vi (t) = 0.5 sin(2 6000t) and obtain the time-domain waveforms for the input and the output voltage using transient simulation. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

Measurements

For all measurements, provide screenshots showing the plots with the measured values properly labeled.

Build the circuits in Fig. 1 with the simulated component values. Perform the following measurements for each circuit:

    1. Obtain the magnitude and phase Bode plots of the transfer function using the network analyzer, and measure the 3-dB frequency and passband gain. Also measure the magnitude and phase of the transfer function at 6kHz.

    2. Apply the input Vi (t) = 0.5 sin(2 6000t) and obtain the time-domain waveforms for the input and the out-put voltage using the scope. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

Report

    1. Include calculations, schematics, simulation plots, and measurement plots.

    2. Prepare a table showing calculated, simulated and measured results.

    3. Compare the results and comment on the differences.

    4. The same transfer functions can be obtained using different combinations of resistors and capacitors. Explain your reasoning for your selection. What are the trade-offs if you change your selection of components to realize the same transfer functions?



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Demonstration

    1. Build the circuits in Fig. 1(a), (b) and (c) on your breadboard and bring it to your lab session.

    2. Your name and UIN must be written on the side of your breadboard.

    3. Submit your report to your TA at the beginning of your lab session.

    4. For the lowpass filter in Fig. 1(a):

Show the frequency response using the network analyzer. Measure and verify -40dB/dec slope at the stopband.

    5. For the highpass filter in Fig. 1(b):

Show the frequency response using the network analyzer. Measure and verify +40dB/dec slope at the stopband.

    6. For the bandpass filter in Fig. 1(c):

Show the frequency response using the network analyzer.

– Measure the low and high 3-dB frequencies.

– Measure the magnitude and phase at a passband frequency fx determined by your TA. Show the time-domain input and output waveforms using the scope at the frequency fx .

– Measure the gain.

– Measure phase difference between the input and the output.








































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