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50 EE290C High-Speed Electrical Interfaces
Spring
2004.

HOMEWORK 1.

Due: Thursday, February 12, 2003 at 5pm in 558 Cory

This is an individual assignment!


Problem 1: Getting to be familiar with transmission lines.

a) Using phasor notation, calculate the voltage and current waves on a transmission line
by solving the phase equation. Assume that R, L, C, G are all non-zero and independent
of frequency. From these solutions derive the formulas for the characteristic impedance
and the phase velocity of the transmission line.

b) Using the voltage and current equations from part a), derive the formulas for the
reflection and transmission coefficients at the interface of two transmission lines with
characteristic impedances Z
1
and Z
2
.

c) Calculate the voltage at the point A as a function of time, for t = 0 until t = 20ns for the
combination of transmission lines from Figure 1. Assume that the source produces a step
of 1V at time t = 0. Properties of transmission line segments are indicated in the figure.

3ns
.
50
A
100
150
1ns
150
4ns
150
50

Figure 1.

d) Using the circuit parameters derive the characteristic impedance and phase velocity of
a lossless transmission line that includes mutual inductance and capacitance, as shown in
Figure 2, and is excited in the odd mode. Repeat for the case of the even mode. .
.
C
self
C
self
C
mutual
L
self
L
self
L
mutual
Figure 2.

Hint: Find the equivalent inductance, capacitance seen by a conductor and substitute in
the formulas in (a):
Note: In the W-element models L
11
=L
self
, L
12
=L
mutual
, C
11
=C
self
+C
mutual
, |C
12
|=C
mutual
.

Problem 2:Time-Domain Reflectometry
Figure 3.

a) The setup from Figure 3 is used to obtain the odd- and even-mode TDRs for various
components:
DUT
50
1ns
.
50
.
50
vin1
vin2
tlin1
tlin2
nin1
nin2
Figure 3.


The simulation results for the various cases are contained in the following files from
Table 1.
Table 1.
Component tr0
File
10 FR4 channel
FR4_channel_even / FR4_channel_odd
10 Rogers channel
Rogers_channel_even /
Rogers_channel_odd
4 FR4 linecard trace
FR4_LC_even / FR4_LC_odd

Assuming that the components are lossless, find the W-element models.

b) The DUT in problem 2(a) is the cascade of a 3 FR4 linecard trace and a 6 FR4
channel terminated at R=50
. Calculate and draw the odd- and even-mode transient
waveforms at the node tlin1 until time t=20ns.

c) The DUT in problem 2(a) is now the cascade of a FR4 linecard trace of length 5, a
backplane via of length 10cm and a nelco channel of length 7. All components are
assumed to be lossless. The odd and even mode TDR results are contained in the files
TDR_Pr2c_odd.tr0 and TDR_Pr2c_even.tr0. Using these simulations, determine the W-
element models for each of the components.

d) The following setup is used to measure the S-parameters of the DUT. The AC sources
produce sinusoids of equal/opposite phases in the case of even/odd mode operation. The
amplitude of the source signals is 2V. The D.U.T. is a 10 FR4 channel same to the one
used in Problem 2(a), only now the channel has losses. The AC simulation results for the
even/odd modes can be found in the files Sparam_FR4_even.ac0 and
Sparam_FR4_odd.ac0, respectively. Using these, find the complete W-element model for
the channel.

50
1ns
50
50
vin1
vin2
tlin1
tlin2
nin1
nin2
50
50
D.U.T.
nout1
nout2
Figure 4.

Note 1: As in many practical channels, you can assume that R
o,12
=0 and G
o,11
=G
o,12
=0.
This means that you have to find the parameters R
o,11
, R
s,11
, R
s,12
, G
d,11
, G
d,12
. Note 2: The resistance per unit length at a frequency f is given by:
f
R
R
f
R
even
odd
s
o
even
odd
×
+
=
/
,
/
)
(

The conductance per unit length at a frequency f is given by:
f
G
G
f
G
even
odd
s
o
even
odd
×
+
=
/
,
/
)
(

where:
12
,
11
,
,
s
s
odd
s
R
R
R =
,
12
,
11
,
,
s
s
even
s
R
R
R
+
=
,
12
,
11
,
,
d
d
odd
d
G
G
G
+
=
and
12
,
11
,
,
d
d
even
d
G
G
G =
.