# Experiment 4 Measurement of Transfer Functions and Impedance

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```					                                                                       ECEN 2260 Experiment 4

Experiment 4
Measurement of Transfer Functions and Impedance
ECEN 2260

1.       Introduction
In this experiment, you will measure the magnitudes and phases of transfer functions and
impedances. You will measure how the magnitude and phase depend on frequency, and
will plot the results on semi-log axes to produce a Bode plot. Your measurements will be
performed on our now familiar R–L–C circuits, as well as on several other simple
filter circuits. Some of the material below is excerpted from the supplementary lecture
notes on frequency response and Bode plots written by Prof. Erickson.
Network under test
The impedance between two                            i(t)
terminals of a network is measured as                              +
illustrated in Fig. 1. A sinusoidal                              v(t)            Z
source of angular frequency ω is                                   –
applied to the terminals, and any
Fig. 1. Measurement of an impedance Z(s).
other independent sources within the
network (including initial conditions)                               Network under test
are set to zero. The relative
+
magnitudes and phases of the
vin(t) +                          H                vout(t)
–
terminal sinusoidal voltage and
–
current are measured. The magnitude
and phase of the impedance at angular        Fig. 2. Measurement of a transfer function H(s).
frequency ω is then given by

V
Z(jω) =
I
∠ Z (jω) = ∠ V – ∠ I                                                           (1)
where V     and I are phasors representing the sinusoids v(t) and i(t), respectively. To
measure the dependence of the impedance on frequency, the process is repeated with the
sinusoidal source set at various frequencies.
The transfer function of a network containing an input signal vin(t) and an output
signal vout(t) is defined in the Laplace domain as the ratio of the output to the input:

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ECEN 2260 Experiment 4

V out(s)
H (s) =
V in(s)                                                 (2)
The transfer function describes the function of a linear ac signal processing network. It
finds wide application in areas such as filter networks, control systems, audio systems,
and communications systems.
The transfer function of a linear network can be measured as in Fig. 2. A
sinusoidal source of angular frequency ω is applied to the input terminals, and any other
independent sources within the network (including initial conditions) are set to zero. The
relative magnitudes and phases of the input sinusoid vin(t) and output sinusoid vout(t) are
measured. The magnitude and phase of the transfer function at angular frequency ω is
then given by

V out
H (jω) =
V in
∠ H (jω) = ∠ V out – ∠ Vin                                                     (3)
where V in and V out are phasors representing the sinusoids vin(t) and vout(t), respectively.
To measure the dependence of the transfer function on frequency, the process is repeated
with the sinusoidal source set at various frequencies.

2.     The decibel (dB)
It is traditional in electrical engineering to express magnitudes of complex-
valued quantities in decibels. The magnitude of a dimensionless quantity G can be
expressed in decibels as follows:
G        = 20 log 10 G
dB
(4)
Decibel values of some simple magnitudes are listed in Table 1. Care must be used when
the magnitude is not dimensionless. Since it is not proper to take the logarithm of a
quantity having dimensions, the                Table 1. Expressing magnitudes in decibels
magnitude must first be normalized. For
Actual magnitude        Magnitude in dB
example, to express the magnitude of an
1/2                  – 6dB
impedance Z in decibels, we should
1                    0 dB
normalize by dividing by a base
2                    6 dB
impedance Rbase:
5 = 10/2            20 dB – 6 dB = 14
Z
Z        = 20 log 10                                             dB
dB                  R base
10                      20dB          (5)
3
1000 = 10             3 ⋅ 20dB = 60 dB

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ECEN 2260 Experiment 4

The value of Rbase is arbitrary, but we need to tell others what value we have used. So if
|| Z || is 5Ω, and we choose Rbase = 10Ω, then we can say that || Z ||dB = 20 log10(5Ω/10Ω)
= – 6dB with respect to 10Ω. A common choice is Rbase = 1Ω; decibel impedances
expressed with Rbase = 1Ω are said to be expressed in dBΩ. So 5Ω is equivalent to
14dBΩ. Other quantities having units, such as voltages or currents, can be expressed in a
similar way. For example, the FCC and some international agencies regulate currents at
the input port of device plugged into the utility power system; they express the current
magnitude in dBµA, or dB using a base current of 1µA. 60dBµA is equivalent to
1000µA, or 1mA.

3.      The Bode plot of transfer functions and impedances
A Bode plot is a plot of the magnitude and phase of a transfer function or other
complex-valued quantity, versus frequency. Magnitude in decibels, and phase in degrees,
are plotted vs. frequency, using semi-logarithmic axes. The magnitude plot is effectively
a log-log plot, since the magnitude is expressed in decibels and the frequency axis is
logarithmic. The use of a log-log plot is necessary because the range of frequencies and
magnitudes typically includes many orders of magnitude.
An example of the Bode plot of an impedance Z is given in Fig. 3. The
magnitude
|| Z(jω) || is found using phasor or Laplace-domain analysis. This magnitude is then
expressed in dBΩ using Eq. (5)
with Rbase = 1Ω. The procedure
is repeated at frequencies f
varying over the range 10 Hz
to 10 kHz (recall that ω = 2πf),
and the results are plotted on
the semilog axes shown. Phase,
in degrees, is also plotted on
the same axes.
As an example of the
above procedure, consider the
R–L circuit example of Fig. 4,
with the values R = 100 Ω and
L = 20 mH. Use of phasor
analysis to find the impedance
Fig. 3. Bode plot of an impedance Z.

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ECEN 2260 Experiment 4

jωLR
Z=
R + jωL                                     (6)
Z
R              L
The magnitude is

Z =           ωLR
2          2
R + ωL
(7)   Fig. 4. R–L circuit example.
This quantity is expressed in dBΩ as follows:

Z          = 20 log 10       ωLR / 1Ω
dB Ω                       2         2
R + ωL
(8)
with ω = 2πf. This equation is plotted in Fig. 5, for f varying from 100 Hz to 10 kHz.
The phase of Z is given by

∠Z = 90° – tan– 1 ωL
R                                                                          (9)
This equation is also plotted in Fig. 5.

50 dBΩ                                                                     150˚

|| Z ||                         120˚
40 dBΩ

30 dBΩ                                                                         90˚
|| Z ||                                                                                        ∠Z
20 dBΩ                                                                         60˚

∠Z
10 dBΩ                                                                         30˚

0 dBΩ                                                                           0˚
100 Hz                         1 kHz                              10 kHz
f
Fig. 5. Bode plot, R-L impedance example.

4.    Measurement of transfer functions
The measured Bode plot of a transfer function or impedance can be generated
automatically by a network analyzer or frequency response analyzer. These devices
contain a sinusoidal source of controllable magnitude and frequency, for excitation of
the circuit input. They also contain two inputs channels, for measurement of the circuit
input and output waveforms. The network analyzer determines the magnitude and phase
of its two input channels, and evaluates the transfer function as in Eq. (3). It sweeps the

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ECEN 2260 Experiment 4

excitation frequency over a given range, and the plots the measured magnitude and
phase on semilog axes.
A signal generator and oscilloscope can be used to obtain manual measurements
of magnitude and phase, as in Fig. 2. The signal generator produces the sinusoid vin(t).
The oscilloscope is connected to measure vin(t) and vout(t). The peak-to-peak magnitudes
of vin(t) and vout(t) are measured, perhaps using the cursor or waveform parameter
measurement features of the oscilloscope (caution: the built-in peak-to-peak voltage
measurement feature of most oscilloscopes is easily corrupted by noise). The ratio of
these magnitudes is equal to the transfer function magnitude || H(jω ) || of Eq. (3).
It is more difficult to measure phase than magnitude, and some care and
diligence is required to obtain accurate results. To measure the phase using an
oscilloscope, the oscilloscope is triggered by vin(t), and the time per division is adjusted
so that one period of vin(t) nearly fills the entire width of the screen (from the left edge of
the graticule to the right edge). The trigger and horizontal postion controls are adjusted
so that a positive-going zero crossing of vin(t) occurs at the center of the screen. The
positive-going zero crossing of vout(t) is then found, and the horizontal distance between
the positive-going zero crossings of vout and vin is measured. The phase shift between vout
and vin is then given by
∠ H (jω) = ∠ V out – ∠ Vin
time between zero crossings
= 360°
waveform period
(10)
The phase is positive when the positive-going zero crossing of vout(t) occurs before the
positive-going zero crossing of vin(t). The phase can also be measured using the period
and zero crossing waveform measurement functions of the oscilloscope.

5.      Measurement of impedances
The impedance measurement of Fig. 1 can be performed using the transfer
function procedure of Section 4, with v(t) and i(t) substituted for vout(t) and vin(t),
respectively. The only difficulty is in measuring the current waveform i(t) using the
oscilloscope. Current probes can be obtained for this purpose; unfortunately there are
none of these in our lab. An alternative way to measure a current waveform is to put the
current through a resistor R, and measure the voltage across the resistor using the
oscilloscope. Since the measured voltage is equal to i(t)R, the magnitude and phase of
the current sinusoid can be deduced.

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ECEN 2260 Experiment 4

Network under test
Non-idealities can corrupt                       R i(t)
the above procedure. In particular,                 +           +
wire-wound        resistors   contain             v1(t)        v2(t)                 Z
significant series inductance, and                  –           –
should be avoided here. Carbon
Fig. 6. Measurement of impedance, without using a
composition and metal film resistors            current probe.
are acceptable.
Figure 6 illustrates the measurement procedure. The impedance Z is given by

V2
Z(j ω) =
I                                                           (11)
But the current i(t) is described by the phasor

V1 – V2
I=
R                                                              (12)
Substitution of Eq. (12) into Eq. (11) yields

V2
Z(jω) = R
V1 – V2
(13)
The voltages v1(t) and v2(t) are connected to channels 1 and 2 of the oscilloscope, and
the oscilloscope is triggered by v2(t). The add and invert functions of the oscilloscope are
employed, to display v1(t) – v2(t). The magnitude and phase of the impedance are then
given by

V2
Z(j ω) = R
V1 – V2
∠ Z (jω) = ∠ V2 – ∠ V1 – V2
(14)
To obtain an accurate measurement, it is necessary to choose the value of R such that the
voltage drop across R [i.e., v1(t) – v2(t)] is similar in magnitude to v2(t). At different
frequencies, it may be necessary to use different values of R.

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ECEN 2260 Experiment 4

R
Laboratory Procedure.                                                             1 kΩ
+
1. Measure Bode plot of R-C low-pass filter.
+          C
Connect the R-C network illustrated in Fig. 7. Apply a sinusoid       v1
–        0.1 µF            v2

to the circuit using the function generator, and set the frequency
–
to the corner frequency of the R-C circuit. Measure the input
Fig. 7 R-C test circuit.
and output voltage waveforms using the two channels of the
oscilloscope.

2. Measure Bode plot of R-C band-pass filter.
C1
Construct the band-pass filter                      R1               1 µF
1 kΩ
circuit illustrated in Fig. 8. Use the
+
Low-Q approximation discussed in
class to compute the approximate        v1 +
C2    R2
v2
–                           0.1 µF 1 kΩ
corner frequencies of this filter.
Measure the magnitude and phase                                                                  –
Fig. 8 Bandpass filter circuit.
of the transfer function of this
filter, over the frequency range extending one decade above and below the filter
passband.

3. Measure Bode plot of RLC circuit transfer function.
L
Obtain an R–L–C network board from
+                                                  +
your TA. A schematic of this board is
given in Fig. 9; this is the board from vin(t)                  C             R            vout(t)
1 µF         470 Ω
Exp. 3.
Measure the transfer function             –                                                  –

V out                    Fig. 9. Schematic of R–L–C network board.
H (jω) =
Vin
Take measurements of magnitude and phase over the frequency range 100 Hz to 10 kHz.
Take enough points to obtain reasonably smooth curves.

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ECEN 2260 Experiment 4

4. Measure Inductor impedance and deduce value of L.
Measure the impedance (using equation 14) of the inductor. Take measurements of
magnitude and phase over the frequency range 100 Hz to 10 kHz. Take enough points to
obtain reasonably smooth curves. Deduce the value of the inductance.

5. Measure input impedance of RLC circuit.
Measure the input impedance of the R–L–C network (with the output terminals open-
circuited). Take measurements of magnitude and phase over the frequency range 100 Hz
to 10 kHz. Take enough points to obtain reasonably smooth curves.

6. Measure output impedance of RLC circuit.
Measure the output impedance of the R–L–C network (with the input terminals short-
circuited). Take measurements of magnitude and phase over the frequency range 100 Hz
to 10 kHz. Take enough points to obtain reasonably smooth curves.

It is important that you check whether your measured data makes sense before you leave
the lab.

Post-lab write-up.
a. Attach copies of all of your Bode plots.
b. For each measurement (i.e., parts 6-11), construct asymptotes for the theoretical
magnitude and phase. Overlay these theoretical asymptotes on your data, and compare.
Computer simulations will not be accepted for this part.
c. Attach your analysis of the corner frequencies of the bandpass filter of Fig. 8.
Construct the magnitude and phase Bode plots of the transfer function of this filter, and
compare with your measurements. Computer simulations will not be accepted for this
part.

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ECEN 2260 Experiment 4

Experiment 4
Measurement of Transfer Functions and Impedances
Lab Score Sheet
Instructions: Ask your TA to initial each section as you complete it in the lab. Turn in
this score sheet with your write-up.

Names of lab-group members:

____________________________                   _________________________________

Points         Initials

Laboratory procedure

Part 1. (10 points) Measure Bode plot of R-C low-
pass filter

Part 2. (10 points) Measure Bode plot of R-C
bandpass filter

Part 3. (20 points) Measure Bode plot of R-L-C
circuit transfer function

Part 4. (10 points) Measure inductor impedance
and deduce value of L.

Part 5. (15 points) Measure input impedance of R-
L-C circuit

Part 6. (15 points) Measure output impedance of
R-L-C circuit

Post-lab writeup (20 points)

Total score (100 points possible)

9

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