Download EECS 420 – Electromagnetics II Lab

EECS 420 – Electromagnetics II Lab
Lab 3a – Time Domain Reflectometry
1. Objectives: At the completion of the lab, you should have each done the following:
a. Examine the reflections from discontinuities on transmission lines through the use
of the Agilent E5071C Network Analyzer.
b. Determine the type and values of unknown passive loads by measuring the
reflection coefficient in the time domain.
2. Pre-lab Homework: [Need not be turned-in, but should be included in the report as
theoretical calculations and plots. I strongly recommend that you do this homework
before coming to the lab.]
a. Read section 11-3-7 in Engineering Electromagnetics, K. R. Demarest.
b. Read the lab handout! You need the lab handout to do the pre-lab homework.
c. For each of the following loads that you will be measuring in lab, determine the
reflection coefficient at t  0 and t   for a step response. A capacitor looks
like a short circuit at t  0 (high frequencies pass easily through a capacitor) and
an open circuit at t   (DC signals don’t pass through a capacitor). An inductor
looks like an open circuit at t  0 (high frequencies are stopped by an inductor)
and a short circuit at t   (DC signals pass through an inductor). After you have
found the initial value ( t  0 ) of the reflection coefficient and the final value of
the reflection coefficient ( t   ), roughly sketch the reflection coefficient
versus time (just an exponential function starting at the initial value and
asymptotically approaching the final value). Also give the time constant for the
load assuming that that source impedance is 50 []. Remember the time
constant is RC and L/R depending on the storage element type and R is the
Thévenin resistance looking out of the two inputs of the storage element. Assume
that the characteristic impedance of the transmission line connecting the
source to the load is 50 []. Two of the loads have been done for an illustration.
Note that the rough sketch can be done by hand and does not need to be plotted
using Matlab or similar programs.
i.
Load is a 33 [] resistor.
1)  
RL  Z 0 33  50  17


 0.2048 (for both t  0 and t  
RL  Z 0 50  33 83
because the value RL is constant. RL will be different at t  0 and at
t   for loads that include an inductor or capacitor).
2) The sketch of the reflection coefficient is just a straight line of value
- 0.2048 .
3) No time constant is necessary for a non-reactive load.
1
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-7
x 10
Figure 3-1: Reflection coefficient of a 33 [] load
ii.
Load is a 0.39 [H] inductor.
1)  
RL  Z 0   50
R  Z 0 0  50

 1 at t  0 and   L

 1 at
RL  Z 0 50  
RL  Z 0 50  0
t  .
2) The graph is an exponentially decaying function with an initial value of 1
at t  0 and asymptotically approaching –1 at t   .
3) The

time
constant
looking
out
of
the
inductor
is
L 0.39 H 

 7.8 ns  where R is the source impedance of 50 [].
R
50 
Gamma [V/V]
1
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
time [s]
0.6
0.7
0.8
0.9
1
-7
x 10
Figure 3-2: Plot of reflection coefficient for a 0.39 [H] load
iii.
Load is a 130 [] resistor.
iv.
Load is a 220 [pF] capacitor.
v.
Load is a 0.39 [H] inductor in series with a 130 [] resistor.
vi.
Load is a 0.39 [H] inductor in parallel with a 130 [] resistor.
vii.
Load is a 220 [pF] capacitor in series with a 130 [] resistor.
viii.
Load is a 220 [pF] capacitor in parallel with a 130 [] resistor.
3. Background: Reflections on transmission lines are the causes of many problems in
transmission systems. Every connector, if it is not perfectly matched, can cause
reflections (good connectors have reflection coefficients of the order of 0.01 or less).
In order to determine the reflections caused by the connectors or the discontinuities in
the system, we must be able to measure them. This experiment deals with such
measurements.
a. The Agilent E5071C Network Analyzer has three frequency-to-time transform
modes: time-domain bandpass, time-domain low-pass step, and time-domain
low-pass impulse. This experiment uses the time-domain low-pass step mode,
which simulates the time domain response to a step input by Fourier transforming
the swept frequency-measurements into the time domain. As in a traditional time
domain reflectometry (TDR) measurement, the distance to the discontinuity in the
DUT, and the type of discontinuity (resistive, capacitive, inductive) can be
determined. The low-pass gives the best resolution for a given bandwidth in the
frequency domain.
If a mismatch exists in the system, part of the incident wave is reflected. The
reflected wave is readily identified since it is separated in time from the incident
wave. The information carried by the reflected wave is valuable since it reveals the
nature of the mismatch. The reflection coefficient in the frequency domain, then, is
expressed as
  
Z L    Z 0
Z L    Z 0
where Z L   is the impedance looking into a discontinuity.
An exact expression of the reflection coefficient in the time domain can be obtained
by taking an inverse Fourier transform of   . However, a simpler and more direct
analysis is possible.
As is shown in the text, it is possible to use a Thévenin
equivalent circuit of the transmission line as seen by the load to determine the
reflected response. This analysis shows that for single time-constant systems, the
reflected waveform is always an exponential transition between an initial value
( t  0 ) and a final value ( t   ). These initial and final values, along with the timeconstant, are functions of the load configuration, the characteristic impedance of the
transmission line, and the voltage level of the incoming step voltage.
For example, in the case of the series RL combination, at t = 0 the reflected voltage is
+Ei (assume the incident voltage is +Ei). This is because the inductor will not accept
a sudden change in current (zero current implies open circuit); it initially looks like an
infinite impedance, and  = 1 at t  0 . Then current in L builds up exponentially and
its impedance drops toward zero. When t goes to infinity, therefore,  is determined
only by the value of R since

R  Z0
R  Z0
when t   . The exponential transition of (t) has a time constant determined by the
effective resistance seen by the inductor. Since the source impedance is Z0, the
inductor sees Z0 in series with R, and

L
R  Z0
where  here is the time constant. The inductor will initially see the characteristic
impedance Z0 of the transmission line and then it will see the source impedance. In
this case they are equal and therefore no mismatch in impedance needs to be
accounted for.
Three other cases (resistor and inductor in parallel, resistor and capacitor in series,
and resistor and capacitor in parallel) can be analyzed in a similar way.
reflection coefficients of these four cases are found to be, for R-L in series,
    RR  ZZ0  1  RR  ZZ0  exp  t  
t     0   exp  t
0

0

where

L
R  Z0
for R-L in parallel,
    1   RR  ZZ0   1 exp  t  
t     0  exp  t


0
where

L
R 1  Z 01 1
for R-C in series,
    1   RR  ZZ0  1 exp  t  
t     0   exp  t
where
  R  Z 0 C
and for R-C in parallel,

0

The
    RR  ZZ0    1  RR  ZZ0  exp  t  
t     0  exp  t
0

0

where

  R 1  Z 01
1C
4. Equipment: You will need the following pieces of equipment:
a. Agilent E5071C Network Analyzer
b. One 2 [ft] transmission line cable
c. Calibration Standards
d. Three Circuit Boards. The values of components used for the circuit boards are
R1 = 33 [, R2 = 130 [, C = 220 [pF] or 180 [pF], and L = 0.39 [H]. The
resistor used in combination with C or L is R2. If the capacitor is 180 [pF] it will
be marked. If the capacitor is unmarked, it is 220 [pF]. The R39 on the inductor
stands for Radix point (generic term for decimal point that is not base-ten
specific) followed by 39 with units of [H].
Network
Analyzer
PORT 1
PORT 2
Cable
L
R2
R1
C
C
L
R2
R2
R2
Circuit Boards
Figure 3-3: Test setup for this lab
C
R2
L
5. Procedure:
1) Press [ON] to turn on the power of the Network Analyzer.
2) Set the stop frequency to 1.5 [GHz] (use the [STOP] button under stimulus
control to do this).
3) Set number of points to measure to 401 by pressing [SWEEP SETUP] [
POINTS] [401] [x1]
4) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [SET FREQ
LOWPASS]. (This sets the low-frequency limit so that the start frequency is an
exact sub-harmonic of the stop frequency.)
5) Perform an S11 1-port calibration.
6) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [TRANSFORM
ON/OFF].
7) Press
[SYSTEM]
[RETURN]
[ANALYSIS]
[TRANSFORM]
[TYPE]
[LOWPASS STEP].
8) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [START] [-5] [
G/n] to select a start time of -5 nanoseconds .
9) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [STOP] [60] [
G/n] to select a stop time of 60 nanoseconds .
10) Press [FORMAT] [REAL] to view the step response of the reflectance . The
step response should be zero for negative time because the unit step function is
zero for negative time.
11) Save your current settings to register [STATE01].
12) Connect an open termination to the cable and press [SCALE] [AUTO SCALE] to
center the display. An open has a reflectance of one, so the output should be one
for positive time.
13) Connect a short termination to the cable and press [SCALE] [AUTO SCALE].
Notice the polarity of the step response. A short has a reflectance of negative one,
so the output should be negative one for positive time.
14) Connect a matched load termination to the cable and press [SCALE] [AUTO
SCALE]. A matched load has a reflectance of zero, so the output should be zero
for all time.
15) Connect one of the circuits on the circuit boards to the cable (it is preferred to test
the resistors first).
16) Press [SCALE] [AUTO SCALE] to center the display if the trace of the signal is
not properly displayed.
17) If the transient time is too short, set stop time at a lower value. If the transient
time is too long, set stop time to a higher value.
18) Set marker one at the beginning of the initial response (will be slightly greater
than 0 [ns] because of the small delay introduced by the micro-strip line
connecting the SMA connector and the load). Set marker two at separate stable
point (after the initial response, before the transient response is finished, and away
from major ripples). Set marker three at steady state (latest time possible since
transient effects decrease as time increases). Read step (21) to make sure you
understand what values you will need.
19) Save your measurement screens.
20) Repeat steps 14) to 18) for all eight circuits.
21) From the data you recorded in step 18 and 19, find the values of the resistors,
capacitors, and inductors in each circuit. The only assumption that you may make
is that the components are in one of the eight different configurations discussed in
the lab handout. Compare your results with the given values. Your results should
be with in 50% of the nominal value. The following equation will be useful:
 
t      0  exp  t
For each circuit you should have measured the initial response, (0), and the
steady-state response (). You know the form that  will take based on the type
of step response that you measured (in the lab homework you might have
sketched each of the possible responses and you can match the measured response
to the corresponding sketch to find the component configuration). And you have
another independent point from markers two that can be used to solve for .
Finally, you need to determine the two unknowns (RL or RC) using the  value
and either (0) or () depending on the unknown load. Note that all of your
time values have to be shifted so that the initial response lines up with t  0 . The
time at which you obtain your initial response (0) is the time offset you should
subtract from all of your measurements. This compensates for the additional time
delay caused by the micro-strip line between the SMA connector and the load.
NOTE: YOU NEED TO SUBMIT ONLY ONE REPORT FOR LABS 3a AND 3b
TOGETHER AND IT IS DUE ONE WEEK AFTER YOU FINISH LAB 3b. THIS
REPORT CAN BE UPTO 8 PAGES LONG.