Introduction to Frequency Domain Analysis (3 Classes) Many thanks to Steve Hall, Intel for the use of his slides Reference Reading: Posar Ch 4.5 http://cp.literature.agilent.com/litweb/pdf/59521087.pdf. Outline. Motivation: Why Use Frequency Domain Analysis 2Port Network Analysis Theory
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Domain Analysis (3 Classes)
Many thanks to Steve Hall, Intel for the use of his slides
Reference Reading: Posar Ch 4.5
http://cp.literature.agilent.com/litweb/pdf/59521087.pdf
Here are the key concepts that you should retain from this class
I
I
I
I
1
1
2
2
port
2
2


port
+
+
+
+
Port 2
V
V
V
V
Port 1
1
1
2
2
Network
Network




Case 1: Inject current I1 into port 1 and measure the open circuit voltage at port 2 and calculate the resultant impedance from port 1 to port 2
Case 2: Inject current I1 into port 1 and measure the voltage at port 1
and calculate the resultant input impedance
Where:
If the impedance matrix is known, the response of the system can be predicted for any input
Or
Open Circuit Voltage measured at Port i
Current Injected at Port j
Zii the impedance looking into port i
Zij the impedance between port i and j
Calculate the impedance matrix for the following circuit:
R2
R1
R3
Port 2
Port 1
Step 1: Calculate the input impedance
R2
R1
+

R3
I1
V1
Step 2: Calculate the impedance across the network
R1
R2
+

R3
I1
V2
Step 3: Calculate the Impedance matrix
Assume: R1 = R2 = 30 ohms
R3=150 ohms
0.1nH
Port 1
Tline
0.1nH
Port 2
0.3pF
0.3pF
Zo=50 ohms, length=5 in
Question:
Assume F=5 GHz
0.1nH
Port 1
Tline
0.1nH
Port 2
0.3pF
0.3pF
Zo=50 ohms, length=5 in
Answer:
Without Probe Capacitance
0.1nH
Tline
Zo = 50
Port 1
Port 2
Port 2
Z21 = 50 ohms
With Probe Capacitance @ 5 GHz
Zo = 50
Port 2
Port 1
106 ohms
106 ohms
Z21 = 63 ohms
Advantages:
Disadvantages:
Note: The Admittance Matrix is very similar, however, it is characterized
with short circuit currents instead of open circuit voltages
a2
a1
2
2


port
port
Port 1
Port 2
R
R
Network
Network
b2
b1
ai represents the square root of the power wave injected into port i
bj represents the power wave coming out of port j
Where:
Sii = the ratio of the reflected power to the injected power at port i
Sij= the ratio of the power measured at port j to the power injected at port i
R=50
Zo
R=Zo
Z=l
Z=0
S11 is measure of the power returned to the source,
and is called the “Return Loss”
Zo
RL
Z=0
Z=l
50 ohms
S11 of a transmission line
will exhibit periodic effects
due to the standing waves
R=50
Zo
R=50
L=5 inches
0.45
0.4
0.35
0.3
S11, Magnitude
0.25
0.2
0.15
0.1
0.05
0
1.0
1.5
2.0
2.5
3..0
3.5
4.0
4.5
5.0
Frequency, GHz
0.45
1.76GHz
2.94GHz
0.4
Peak=0.384
0.35
0.3
S11, Magnitude
0.25
0.2
0.15
0.1
0.05
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Frequency, GHz
Note: The phase of the reflection should be either +1 or 1 at 1.76 GHz because it is aligned with the incident
Er=1.0 and Zo=75 ohms
a2=0
a1
2
2


port
port
V1
Zo
Zo
V2
Network
Network
b2
b1
S21 is measure of the power transmitted from
port 1 to port 2, and is called the “Insertion Loss”
Question:
Answer:
When the reflections are minimal, alpha can be estimated
Advantages:
Disadvantages:
s112 s122
s212 s222
s113 s123
s213 s223
s111 s121
s211 s221
3 cascaded s parameter blocks
a11
a21
b12
b22
a13
a13
b11
b21
a12
a22
b13
b13
I2
I1
2
2


port
port
V1
V2
Network
Network
I3
I2
I1
V3
V1
V2
This is the best way to cascade elements in the frequency domain.
It is accurate, intuitive and simplistic.
Z
Assignment 6:
Convert these to sparameters
Port 1
Port 2
Y
Port 1
Port 2
Z1
Z2
Port 1
Port 2
Z3
Y3
Y1
Y2
Port 2
Port 1
Port 1
Port 2
Port 1
Port 2
L2
L1
CVIA
Z1
Z2
Port 1
Port 2
Z3
Port 1
Port 2
2

port
50
50
Network X
Network
Port 1
Port 2
2

port
50
50
Network Y
Network
2

port
2

port
50
Network Y
50
Network X
Network
Network
Port 2
Port 1
Advantages:
Disadvantages:
The wave functions (a,b) used to define sparameters for a twoport network are shown below. The incident waves is a1, a2 on port 1 and port 2 respectively. The reflected waves b1 and b2 are on port 1 and port 2. We will use a’s and b’s in the sparameter follow slides
a1
s21
b2
s22
s11
GS
GL
s12
b1
a2
“In a signal flow graph, each port is represented by two nodes. Node an represents the wave coming into the device from another device at port n, and node bn represents the wave leaving the device at port n. The complex scattering coefficients are then represented as multipliers (gains) on branches connecting the nodes within the network and in adjacent networks.”*
Example
Measurement equipment strives to be match i.e. reflection coefficient is 0
See: http://cp.literature.agilent.com/litweb/pdf/59521087.pdf
It includes the effect of nonperfect loads.
We will show how the voltage transfer functions for a 2 port network is given by the following equation.
Notice it is not S21
Voltage Transfer functionLet’s see if we can get this results another way
a11
a21
b12
b22
a13
a13
b11
b21
a12
a22
b13
b13
s112 s122
s212 s222
s113 s123
s213 s223
s111 s121
s211 s221
a11
a21
b12
b22
a13
a13
s213
s211
1
s212
1
A11
B21
A12
B22
A13
B23
s221
s112
s222
s113
s223
b11
b21
a12
a22
b13
b13
s123
s121
1
s122
1
B11
A21
B12
A22
B13
A23
s112 s122
s212 s222
s113 s123
s213 s223
s111 s121
s211 s221
B
A
C
s213
s211
1
s212
1
A11
B21
A12
B22
A13
B23
s221
s112
s222
s113
s223
s123
s121
1
s122
1
B11
A21
B12
A22
B13
A23
B
A
C
Mason’s Rule
B
C
B
A
A
Assumption is that these are ~ 0
=
=
i.e. For a budget just add up the db’s and jitter
a
1
a
2
=
a
a
a
3
4
3
b
a
b
3
4
4
a
b
S
S
S
S
a
2
1
11
12
13
14
1
b
S
4

port
2
S
S
S
b
21
b
2
22
23
24
1
b
S
S
S
S
3
33
34
31
32
b
S
S
4
S
S
43
41
44
42
a
dm1
a
dm2
=
a
cm1
a
cm2
b
DS
DS
DCS
DCS
dm1
11
12
11
12
b
DS
dm2
DS
DCS
DCS
21
22
21
22
b
CS
CS
CDS
CDS
cm1
11
12
11
12
b
CS
CDS
cm2
CS
CDS
21
21
22
22
Matrix assumes differential and common mode stimulus
Common mode conversion Matrix:
Differential Stimulus, Common mode response. i.e., DCS21 = differential signal [(D+)(D)] inserted at port 1 and common mode signal [(D+)+(D)] measured at port 2
Differential Matrix:
Differential Stimulus, differential response
i.e., DS21 = differential signal [(D+)(D)]
inserted at port 1 and diff signal measured at port 2
b
a
DS
DS
DCS
DCS
dm1
dm1
11
12
11
12
b
a
DS
dm2
dm2
DS
DCS
DCS
21
22
21
22
=
b
a
CS
CS
CDS
CDS
cm1
cm1
11
12
11
12
b
a
CS
CDS
cm2
cm2
CS
CDS
21
21
22
22
differential mode conversion Matrix:
Common mode Stimulus, differential mode response. i.e., DCS21 = common mode signal [(D+)+(D)] inserted at port 1 and differential mode signal [(D+)(D)] measured at port 2
Common mode Matrix:
Common mode stimulus, common mode
Response. i.e., CS21 = Com. mode signal
[(D+)+(D)] inserted at port 1 and Com. mode
signal measured at port 2
Example Calculation, Differential Return Loss
The stimulus is equal, but opposite, therefore:
2
1
2
4


port
port
Network
Network
4
3
Assume a symmetrical network and substitute
Other conversions that are useful for a differential bus are shown
Differential Insertion Loss:
Differential to Common Mode Conversion (ACCM):
Similar techniques can be used for all multimode Parameters
Advantages:
Disadvantages:
X
Direction of
propagation
Z
For typical PCBs at 10 GHz with 5 mil traces (W=0.005”)
Y
Amperes Law:
A magnetic Field will be induced by an electric current
or a time varying electric field
Faradays Law:
An electric field will be generated by a time varying magnetic flux
This differential equation is easily solvable for Ex:
Note the positive exponent
is because the wave is
traveling in the opposite direction
Portion of wave traveling
In the +z direction
Portion of wave traveling
In the z direction
= permittivity in Farads/meter (8.85 pF/m for free space)
(determines the speed of light in a material)
= permeability in Henries/meter (1.256 uH/m for free space
and nonmagnetic materials)
Since inductance is proportional to & capacitance is proportional
to , then is analogous to in a transmission line, which
is the propagation delay
Incident sinusoid
RL
Reflected sinusoid
z=l
z=0
If a sinusoid is injected onto a transmission line, the resulting voltage
is a function of time and distance from the load (z). It is the sum of the
incident and reflected values
Note: is added to
specifically represent
the time varying
Sinusoid, which was implied
in the previous derivation
Voltage wave reflecting
off the Load and traveling
towards the source
Voltage wave traveling
towards the load
= Complex propagation constant – includes all the transmission line parameters (R, L C and G)
(For the loss free case)
(lossy case)
= Attenuation Constant (attenuation of the signal due to transmission line losses)
(For good conductors)
= Phase Constant (related to the propagation delay across the transmission line)
(For good conductors and good dielectrics)
Subsequently:
Zo
Zl
(Assume a line length of l (z=l))
Z=l
Z=0
This is the reflection coefficient looking into a tline of length l
Interaction: transmission line and a load
Zin
RL
Z=l
Z=0
Note: is
dependent on
and
This is the input impedance looking into a tline of length l
To demonstrate, lets assume a loss free transmission line
Remember, the input reflection takes the form
The frequency where the values of the real & imaginary
reflections are zero can be calculated based on the line length
Term 1
Term 2
Term 1=0
Term 2 =
Term 2=0
Term 1 =
Note that when the imaginary portion is zero, it means the phase
of the incident & reflected waveforms at the input are aligned. Also notice that value of “8” and “4” in the terms.
Calculate:
(assume a very low loss line)
Er_eff=1.0
RL
Zo=75
Z=l
Z=0
Coeff
Reflection
Imaginary
Example #1: Solution01

2.5E
01

2.0E
Real
01

1.5E

01
1.0E

5.0E
02
0.0E+00


02
5.0E
01

1.0E

01

1.5E


01

2.0E

2.5E

01
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
Frequency
3.0E+09
Step 1: Determine the periodicity zero crossings or peaks & use the relationships on page 15 to calculate the electrical length
Important Concepts demonstrated