KHON KAEN UNIVERSITY
DEPARTMENT OF ELECTRICAL ENGINEERING
CM - 09 : A LUMPED-ELEMENT MODEL OF TRANSMISSION LINES
OBJECTIVES:
1. To study and investigate a lumped-element model of transmission lines.
2. To be able to calculate characteristic impedance, propagation constant, and
phase velocity from lumped-element model parameters of transmission lines.
3. To determine and measure resistive load reflection coefficients.
INTRODUCTION
Introductory experiments in a transmission line course are generally carried out
at MHz or GHz frequencies. The objectives in such experiments often include plotting
the standing wave pattern, finding the wavelength, maxima, minima, the VSWR
(Voltage Standing Wave Ratio), and the propagation constant. More involved
experiments might look at the effect of load changes on the VSWR and ask for
matching a line by reactive stubs connected near a mismatched load. The reason for
operation at high frequencies is that the line segments under study are physically limited
to lab table dimensions; therefore, to be able to examine a few half-wave lengths of the
standing wave necessitates short wavelengths. Since the components and the
instrumentation at these frequencies are costly, such setups are often out of reach of
some engineering programs. It turns out that a good substitute for actual lines for the
basic experiments is the transmission line model based on the lumped-element
equivalence circuit. With an adequate number of lumped elements and appropriate
component values, these models can provide the required multiple half-waves along
their “lengths” at much lower frequencies rather than at the GHz ranges. Such models
can be built using off-the shelf RLC components, and the measurement equipment for
the lower range frequency, such as oscilloscope, is readily available in any electrical or
electronic laboratory.
Figure 1 shows a lumped-element model of a lossless transmission line
connected to a voltage source vS with source resistance RS at the left end and at the right
end terminated with a load resistance RL.
Lz
Lz
Lz
Lz
RS
vS (t )
C z
z
C z
z
C z
z
C z
z
Figure 1 A lumped-element model lossless transmission line circuit.
In this experiment, we will investigate transient responses of this circuit with
various values of load resistances. Important transmission line properties such as
characteristic impedance, propagation constant, and phase velocity will be calculated
from lumped-element model parameters of the tested transmission line.
RL
2
THEORY
To construct or set up a lumped-element model, the values of the section
components (R, L G and C) and the total number of sections for the line must be chosen
such that the line covers at least a few half-wavelengths, with enough data points per
wavelength so that an accurate graph of the standing wave can be plotted. In this
experiment, standing wave pattern will not be investigated so the number of section is
chosen to be 10 which should be adequate for transient responses study. The lumpedelement model of a transmission line is made up of 10 concatenated sections, each
section has element parameters as shown in Figure 2.
z  z
z
R
L
C
G
z
Figure 2 A lumped-element model parameters for one section of a transmission line.
The line constants R, L, C, and G, are called the distributed parameters and for
real lines are given in “per unit length” units. In the case of the lumped-element model,
these line constants are “per section”. Consequently, the results that normally have
unit-length in their units must be expressed in “sections”. For example, wavelength is
measured in number of sections rather than in meters, the attenuation constant in Nepers
per sections, etc.
By applying the circuit laws to a section and solving the resulting differential
equations, one is able to find the line voltage and current as a function of time and
distance, and also a number of other useful relations that characterize the properties of
the wave propagation along the line. These are summarized below:
Case (1) Lossy transmission line
(1)
For example a 50  lossy transmission line has a series resistance R = 5  and
56H inductor. The 22 nF shunt capacitor is assumed to have no loss and so G = 0 
So the parameters for each section are: R = 5 , L = 56H, G = 0 and C = 22 nF.
Case (2) Lossless transmission line
This is an ideal case which is hardly achieved in most practical transmission
lines. Real transmission lines always have losses both in their conductors and dielectric.
3
Note that in our experiment we do not have a lossless transmission line model
since each real inductor has a nonzero series resistance.
Case (3) Distortionless transmission line
In lossy transmission lines, the attenuation constant  is frequency dependent so
that wave amplitudes of different frequencies will be attenuated differently. This results
in waveform distortion. If we can make the attenuation constant  to be frequency
independent and the phase constant  to be linearly dependent on frequency, we get a
distortionless transmission line. If we select
R G

L C
(2)
From (1) we get the propagation constant
j L 
jC 

  RG 1 
1 

R 
G 

jC 

 RG 1 
    j
G 

(3)
And so the attenuation constant and phase constant become
  RG  0
   LC
(4)
For this example case, we put a 500  across each capacitor and so the
parameters for each section are: R = 5 , L = 56H, G = 1/500 and C = 22 nF.
Note that the ratio R /L is approximately G/C.
The phase velocity is the same as case (2) but the characteristic impedance will
be
Z0 
R  1  j L / R 


G  1  jC / G 
R
L


 R0  jX 0  R0
G
C
(5)
4
The voltage reflection coefficient Γ from a load terminator with impedance Z is given
by
(6)
Note that these lumped element model can be used up to a certain frequency
called “Cutoff frequency, C ”. Beyond this frequency, measurement results will not be
valid. The cutoff frequency can be estimated from the values of inductance and
capacitance of each section.
(7)
At low frequency (C ) the phase velocity is approximated as
u p  LC 
1
(8)
LC
Assume that the signal generator in Figure 3 is used to inject a single voltage
step at time t = 0; the voltage step is from an initial value of 0 V to a final value of +1
V at the terminals of the signal generator. Assume that the potentiometer VRL has been
set to the value L / C , which is very nearly equal to the characteristic impedance Z0
for frequencies below the cutoff frequency.
VR1
RS
vS (t )
Function Generator
Lz
Lz
C z
C z
Transmission Line, Z0
Figure 3 The transmission line is terminated at each end with adjustable resistances,
VR1 on the left (source) end and VRL on the right (load) end.
The signal generator acts as an ideal voltage source in series with a source
resistance, RS. If the source resistance is smaller than the characteristic impedance Z0
then a series variable resistor or potentiometer VR1 must be added so that RS + VR1 = Z0.
Applying steady sinusoidal signals of various frequencies can excite the many
resonances of the line; applying a low-frequency square wave introduces a series of
VRL
5
nearly independent voltage step inputs whose propagation down the line and reflections
from the terminators may be studied.
Since RS + VR1 and Z0 form a voltage divider (with RS + VR1 ≅ Z0 ), the voltage
step applied to the transmission line is nominally from 0 V to +1/2 V, and it is this
smaller step which introduces waves at the left end of the line which then propagate
toward the termination ZR.
1.00
750.00m
Output
Input
S5
S10
500.00m
250.00m
0.00
0.00
10.00u
20.00u
30.00u
Time (s)
40.00u
50.00u
Figure 4 Step response of a lossless transmission line.
Input =  1V 100 Hz Square wave with 1 s rise time
Figure 4 shows a computer simulation result of the step response at the input end
of a 50 lossless transmission line (R = 0 , L = 56H, G = 0  and C = 22 nF), at
the mid end (S5), and at the load end (S10) down the line. The ripples or ringing in the
response waveforms are caused by the varying characteristic impedance with
frequency, Z0(ω).
(9)
The response is less sharp at positions further down the line because the phase
velocity is slower for the higher-frequency components of the step. The time axis is in
units of the low-frequency propagation time delay/section.
Note that until the signal has had time to propagate down the line to the righthand termination and back again, the transmission line behaves as though it extends to
infinity, because there can be no reflected wave to modify the response. Now consider
the effect of an open or of a shorted termination at the right-hand end. The open
termination has Γ = +1, so the reflected step has the same sign as the incoming step.
The shorted termination has Γ = −1, so it inverts the incoming step. The resulting
waveforms are shown in Figure 5 and Figure 6.
6
2.00
1.50
Output
S10
S5
1.00
Input
500.00m
0.00
20.00u
10.00u
0.00
50.00u
40.00u
30.00u
Time (s)
Figure 5 Reflection from an open circuit load
1.00
Input
s5
Output
500.00m
S10
0.00
-500.00m
-1.00
0.00
10.00u
20.00u
30.00u
Time (s)
40.00u
50.00u
Figure 6 Reflection from a shorted circuit load
Both plots show the waveform at the input, mid, and load ends of the
transmission line. Figure6 shows the response with the right-hand end shorted (Γ =
−1 ), so the reflection is inverted. Figure 5 shows the response with the right-hand end
open (Γ = +1 ), so the reflection reinforces the original step. The length of the line is 10
sections, so the reflection arrives 20 time units after the initial stimulus. The source end
of the line is properly terminated, so no further reflection takes place at that end. The
final, equilibrium voltage on the line ( t → ∞ ) is 0 for the shorted case and 1 for the
open case (signal generator output = 0 V to + 1 V).
For a lossy transmission line (R = 5 , L = 56H, G = 0  and C = 22 nF) the
simulation responses are as shown in Figure 7.
7
1.00
750.00m
Output
Input
S5
500.00m
S10
250.00m
0.00
0.00
10.00u
20.00u
30.00u
Time (s)
40.00u
50.00u
Figure 7 Step response of a lossy transmission line.
PRELIMINARY REPORT
Given lumped parameters R = 2 , L = 68H, G = 0  and C = 22 nF, calculate
 Characteristic impedance
 Propagation constant
 Phase velocity
APPARATUS
1. Function Generator and Oscilloscope
2. Digital Multimeter and LCR Meter
3. Transmission Line model Board
4. 1 k( VR1) and 5 k( VRL) potentiometers
PROCEDURE
Measure the inductance (L) and capacitance (C) of each section using an LCR meter.
Measure the DC resistance of each inductor using a digital multimeter. Find the average
values of L, C, and R. Recalculate the values of characteristic impedance, propagation
constant, and phase velocity using the measured values of L, C, and R.
1. The load reflections of a step input
1.1 Calculate the cutoff frequency C according to equation (7). Set the function
generator to obtain a 4 Vp-p amplitude square wave at repetitive frequency less than
1/100 of the cutoff frequency C /2 (Hz).
1.2 Connect a function generator to the transmission line as shown in Figure 3.
With the value of characteristic impedance Z0 computed from the measured values of L,
C, and R and the specification of the function generator, determine the value of VR1 so
that RS + VR1 = Z0. Since RS + VR1 and Z0 form a voltage divider (with RS + VR1 ≅ Z0 ),
the voltage step applied to the transmission line must be +1 V at the left end of the line
which then propagate toward the termination VRL. Connect section 5 output to section 6
input. Set VRL to its maximum value (VRL >> Z0).
8
1.3 Set VRL to its maximum value. Use oscilloscope to measure responses at the
input of the transmission line and at 9 positions along each section down the line.
Record waveforms at the input of the transmission line (Source end), the connection of
section 5 and section 6 (Mid line), and at the load resistor VRL (Load end). Set the
oscilloscope time/div and horizontal position so that the step input can be viewed
clearly on the oscilloscope display. See Figure 4 for example.
1.4 Set VRL to its minimum value (VRL << Z0). Repeat 1.3.
2. Characteristic impedance
2.1 Determine the characteristic impedance Z0 (in the low-frequency limit) by
adjusting the variable resistor VRL until the load reflection disappear. Disconnect the
load resistor and measure the value of VRL using a digital multimeter set as an
Ohmmeter.
2.2 Set VRL to VRL = 2Z0. Use oscilloscope to measure responses and record
waveforms at the input of the transmission line (Source end), the connection of section
5 and section 6 (Mid line), and at the load resistor VRL (Load end).
2.3 Set VRL to VRL = 0.5Z0. Use oscilloscope to measure responses and record
waveforms at the input of the transmission line (Source end), the connection of section
5 and section 6 (Mid line), and at the load resistor VRL (Load end).
3. The propagation constant
3.1 Set VRL to VRL = Z0. Use oscilloscope to measure the value of attenuation per
section, and show its relation to attenuation constant, .
3.2 Use oscilloscope to measure time delay of each section, what can be
concluded from this measurement (i.e. what is the relation of time delay/section to
phase constant, ). You may get more accurate result by measuring time delay at the
mid line and load end and calculate the average delay per section.
FINAL REPORT
Write a report on how this experiment is performed. Include the followings:
1. Discuss the waveforms obtained in each procedure.
2. Discuss the results when the load is changed to be: shorted circuit (VRL << Z0),
matched (VRL = Z0), open circuit (VRL >> Z0), VRL = 2Z0 , and VRL = 0.5Z0.
3. Discuss the measured values of attenuation and phase constants.
QUESTIONS
1. From the measurement results, determine the time delay and attenuation per
section of this transmission line model.
2. In this experiment we cannot implement a lossless transmission line model.
Explain the reason(s) why.
3. From the experiment, do you notice the ringing voltage at the source end of the
transmission line model ? Explain why. Find the relation of the ringing
frequency to the circuit element values of the transmission line model.
4. Determine the round trip time of this transmission line.
5. Do you think that this transmission line model can be used to explain a real
transmission line ? If not explain why.
Boonying Charoen
September 2014