ekt 441 microwave communications chapter 3: microwave network analysis (part 1)
TRANSCRIPT
EKT 441MICROWAVE COMMUNICATIONS
CHAPTER 3:
MICROWAVE NETWORK ANALYSIS (PART 1)
NETWORK ANALYSIS
Many times we are only interested in the voltage (V) and current (I) relationship at the terminals/ports of a complex circuit.
If mathematical relations can be derived for V and I, the circuit can be considered as a black box.
For a linear circuit, the I-V relationship is linear and can be written in the form of matrix equations.
A simple example of linear 2-port circuit is shown below. Each port is associated with 2 parameters, the V and I.
Port 1 Port 2
R
CV1
I1 I2
V2
Convention for positivepolarity current and voltage
+
-
NETWORK ANALYSIS
For this 2 port circuit we can easily derive the I-V relations.
We can choose V1 and V2 as the independent variables, the I-V
relation can be expressed in matrix equations.
21
11
2
221
VCjVI
CVjII
RR
C
I1I2
V2jCV2
R
V1
I1
V
2
211
1 VVIR
2 - Ports
I2
V2V1
I1
Port 1 Port 2
R
CV1
I1 I2
V2
2
1
2221
1211
2
1V
V
yy
yy
I
I
2
111
11
2
1V
V
CjI
I
RR
RR
Network parameters(Y-parameters)
NETWORK ANALYSIS
To determine the network parameters, the following relations can be used:
For example to measure y11, the following setup can be used:
0211
11
VV
Iy
0121
12
VV
Iy
0212
21
VV
Iy
0122
22
VV
Iy
This means we short circuit the port
2
1
2221
1211
2
1V
V
yy
yy
I
I
VYI or
2 - Ports
I2
V2 = 0V1
I1
Short circuit
NETWORK ANALYSIS
By choosing different combination of independent variables, different network parameters can be defined. This applies to all linear circuits no matter how complex.
Furthermore this concept can be generalized to more than 2 ports, called N - port networks.
2 - Ports
I2
V2V1
I1
2
1
2221
1211
2
1I
I
zz
zz
V
VV1 V2
I1 I2
2
1
2221
1211
2
1V
I
hh
hh
I
VLinear circuit, because allelements have linear I-V relation
ABCD MATRIX
Of particular interest in RF and microwave systems is ABCD parameters. ABCD parameters are the most useful for representing Tline and other linear microwave components in general.
221
221
2
2
1
1
DICVI
BIAVV
I
V
DC
BA
I
V
02
1
2
IV
VA
02
1
2
VI
VB
02
1
2
VI
ID
02
1
2
IV
IC
(4.1a)
(4.1b)
2 -Ports
I2
V2V1
I1
Take note of the direction of positive current!
Short circuit Port 2Open circuit Port 2
ABCD MATRIX
The ABCD matrix is useful for characterizing the overall response of 2-port networks that are cascaded to each other.
3
3
33
33
1
1
3
3
22
22
11
11
1
1
I
V
DC
BA
I
V
I
V
DC
BA
DC
BA
I
VI2’
V2V1
I1I2
V3
I3
11
11
DC
BA
22
22
DC
BA
Overall ABCD matrix
THE SCATTERING MATRIX
Usually we use Y, Z, H or ABCD parameters to describe a linear two port network.
These parameters require us to open or short a network to find the parameters.
At radio frequencies it is difficult to have a proper short or open circuit, there are parasitic inductance and capacitance in most instances.
Open/short condition leads to standing wave, can cause oscillation and destruction of device.
For non-TEM propagation mode, it is not possible to measure voltage and current. We can only measure power from E and H fields.
THE SCATTERING MATRIX
Hence a new set of parameters (S) is needed which Do not need open/short condition. Do not cause standing wave. Relates to incident and reflected power waves, instead of
voltage and current.
• As oppose to V and I, S-parameters relate the reflected and incident voltage waves.• S-parameters have the following advantages:1. Relates to familiar measurement such as reflection coefficient, gain, loss etc.2. Can cascade S-parameters of multiple devices to predict system performance (similar to ABCD parameters).3. Can compute Z, Y or H parameters from S-parameters if needed.
• As oppose to V and I, S-parameters relate the reflected and incident voltage waves.• S-parameters have the following advantages:1. Relates to familiar measurement such as reflection coefficient, gain, loss etc.2. Can cascade S-parameters of multiple devices to predict system performance (similar to ABCD parameters).3. Can compute Z, Y or H parameters from S-parameters if needed.
THE SCATTERING MATRIX Consider an n – port network:
Each port is considered to beconnected to a Tline withspecific Zc.
Linear n - portnetwork
T-line orwaveguide
Port 2
Port 1
Port n
Reference planefor local z-axis(z = 0)
Zc2
Zc1
Zcn
THE SCATTERING MATRIX There is a voltage and current on each port. This voltage (or current) can be decomposed into the incident (+) and reflected component (-).
V1+ V1
-
Linear n - portNetwork
Port 2
Port 1
Port n
z = 0
V1
I1
+z
Port 1
111 VVV
V1
V1+
+
- V1-
+
1111
111
VV
III
cZ
222
22
0 VVVV
eVeVzV zjzj
222
22
0 IIII
eIeIzI zjzj
THE SCATTERING MATRIX The port voltage and current can be normalized with respect to the impedance
connected to it. It is customary to define normalized voltage waves at each port as:
ciii
ci
ii
ZIa
Z
Va
(4.3a)
Normalizedincident waves
Normalizedreflected waves
ciii
ci
ii
ZIb
Z
Vb
(4.3b)
i = 1, 2, 3 … n
THE SCATTERING MATRIX Thus in general:
Vi+ and Vi
- are propagatingvoltage waves, which canbe the actual voltage for TEMmodes or the equivalent voltages for non-TEM modes.(for non-TEM, V is defined proportional to transverse Efield while I is defined propor-tional to transverse H field, see[1] for details).
Vi+ and Vi
- are propagatingvoltage waves, which canbe the actual voltage for TEMmodes or the equivalent voltages for non-TEM modes.(for non-TEM, V is defined proportional to transverse Efield while I is defined propor-tional to transverse H field, see[1] for details).
V2+
V2-
V1+ V1
-
Vn+
Vn-
Linear n - portNetwork
T-line orwaveguide
Port 2
Port 1
Port nZc1
Zc2
Zcn
THE SCATTERING MATRIX If the n – port network is linear (make sure you know what this means!), there is a linear relationship between the
normalized waves. For instance if we energize port 2:
V2+
V1-
Vn-
Port 2
Port 1
Port nZc1
Zc2
Zcn
V2-
Linear n - portNetwork
Constant thatdepends on the network construction
2222
VsV
22
VsVnn
2121
VsV
THE SCATTERING MATRIX Considering that we can send energy into all ports, this can be
generalized to:
Or written in Matrix equation:
Where sij is known as the generalized Scattering (S) parameter, or just S-parameters for short. From (4.3), each port i can have different characteristic impedance Zci
nnnnnnn
VsVsVsVsV 332211
(4.4a)
(4.4b)or
nn
VsVsVsVsV13132121111
nnVsVsVsVsV
23232221212
VSV
nnnnn
n
n
nV
V
V
sss
sss
sss
V
V
V
:
...
:::
...
...
:2
1
21
22221
11211
2
1
THE SCATTERING MATRIX
Consider the N-port network shown in figure 4.1.
Figure 4.1: An arbitrary N-port microwave network
THE SCATTERING MATRIX
Vn+ is the amplitude of the voltage wave incident on port n.
Vn- is the amplitude of the voltage wave reflected from port n.
The scattering matrix or [S] matrix, is defined in relation to these incident and reflected voltage wave as:
nNNN
N
n V
V
V
SS
S
SSS
V
V
V
.
.
.
....
......
......
......
.....
...
.
.
.2
1
1
21
11211
2
1
[4.1a]
THE SCATTERING MATRIX
VSVor [4.1b]
jkforVj
iij
k
V
VS
,0
A specific element of the [S] matrix can be determined as:
[4.2]
Sij is found by driving port j with an incident wave Vj+, and measuring
the reflected wave amplitude, Vi-, coming out of port i.
The incident waves on all ports except j-th port are set to zero (which means that all ports should be terminated in matched load to avoid reflections).Thus, Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, and Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.
THE SCATTERING MATRIX For 2-port networks, (4.4) reduces to:
Note that Vi+ = 0 implies that we terminate i th port with its
characteristic impedance. Thus zero reflection eliminates standing wave.
2
1
2
1
2221
1211
2
1
V
VS
V
V
ss
ss
V
V(4.5a)
(4.5b)01010202
2
1
12
2
2
22
1
2
21
1
1
11
VVVV
V
Vs
V
Vs
V
Vs
V
Vs
THE SCATTERING MATRIX
2 – Port Zc2
Zc2Zc1
Zc1Vs
V1+
02021
2
21
1
1
11
VV
V
Vs
V
Vs
V1-
V2-
V1-
2 – PortZc1
Zc2Zc1
Zc2Vs
V2-
V2+
01012
1
12
2
2
22
VV
V
Vs
V
Vs
Measurement of s11 and s21:
Measurement of s22 and s12:
THE SCATTERING MATRIX Input-output behavior of network is defined in terms of normalized
power waves S-parameters are measured based on properly terminated
transmission lines (and not open/short circuit conditions)
1* ][][ tss
THE SCATTERING MATRIX
THE SCATTERING MATRIX
THE SCATTERING MATRIXReciprocal and Lossless networks Impedance and admittance matrices are symmetric for reciprocal
networks A symmetric network happens when:
It is also purely imaginary for lossless network (no real power can be delivered to the network)
A matrix that satisfies the condition of (4.6b) is called a unitary matrix
tss ][][
1* ][][ tss
(4.6a)
(4.6b)
THE SCATTERING MATRIX Transpose of a Matrix (taken from Engineering
Maths 4th Ed by KA Stroud)
dc
baS
db
catS
Transpose of [S], written as [S]t
THE SCATTERING MATRIX Symmetrical Matrix (taken from Engineering
Maths 4th Ed by KA Stroud)
2212
2111
aa
aaS
It is symmetrical when aij = aji
When a [S] is symmetric, it is also reciprocal
If
THE SCATTERING MATRIXReciprocal and Lossless networks (cont) The matrix equation in (4.6b) can be re-written in;
OR
(4.7)
0
1
1
*
1
*
N
kkjki
N
kkiki
SS
SS For i = j
For i ≠ j
Used to determine reciprocality for a 2 port
network
1||||2111SS
THE SCATTERING MATRIX (Ex)
Find the S parameters of the 3 dB attenuator circuit shown in Figure 4.2.
Figure 4.2: A matched 3 dB attenuator with a 50 Ω characteristic impedance.
THE SCATTERING MATRIX (Ex)
From the following formula, S11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated with a matched load (Z0 =50 Ω);
The equation becomes;
jforkV
j
i
ijkV
VS
0
0220
)1(
0
)1(
0
)1(
0
1
1
11 Z
in
in
VV ZZ
ZZ
V
VS
On port 2
THE SCATTERING MATRIX (Ex)
To calculate Zin(1), we can use the following formula;
Thus S11 = 0. Because of the symmetry of the circuit, S22 = 0.
S21 can be found by applying an incident wave at port 1, V1
+, and measuring the outcome at port 2, V2-. This is
equivalent to the transmission coefficient from port 1 to port 2:
50)5056.8(8.141
)5056.8(8.14156.8
)1(
inZ
0
1
2
212
VV
VS
THE SCATTERING MATRIX (Ex)
From the fact that S11 = S22 = 0, we know that V1- = 0 when
port 2 is terminated in Z0 = 50 Ω, and that V2+ = 0. In this
case we have V1+ = V1 and V2
- = V2.
Where 41.44 = (141.8//58.56) is the combined resistance of 50 Ω and 8.56 Ω paralled with the 141.8 Ω resistor. Thus, S21 = S12 = 0.707
11227071.0
56.850
50
56.844.41
44.41VVVV
THE SCATTERING MATRIX (Ex)
A two port network is known to have the following scattering matrix:
a) Determine if the network is reciprocal and lossless.
b) If port 2 is terminated with a matched load, what is the return loss seen at port 1?
c) If port 2 is terminated with a short circuit, what is the return loss seen at port 1?
02.04585.0
4585.0015.0S
THE SCATTERING MATRIX (Ex)
Q: Determine if the network is reciprocal and lossless Since [S] is not symmetric, the network is not reciprocal. Taking the 1st
column, (i = 1) gives;
So the network is not lossless. Q: If port two is terminated with a matched load, what is the return loss
seen at port 1? When port 2 is terminated with a matched load, the reflection coefficient
seen at port 1 is Γ = S11 = 0.15. So the return loss is;
dBRL 5.16)15.0log(20||log20
1745.0)85.0()15.0(|||| 222
21
2
11 SS
Used to determine reciprocality for a 2 port
network
1||||2111SS
THE SCATTERING MATRIX (Ex)
Q: If port two is terminated with a short circuit, what is the return loss seen at port 1?
When port 2 is terminated with a short circuit, the reflection coefficient seen at port 1 can be found as follow
From the definition of the scattering matrix and the fact that V2+ = - V2
- (for a short circuit at port 2), we can write:
2121112121111
VsVsVsVsV
2221212221212VsVsVsVsV
THE SCATTERING MATRIX (Ex)
The second equation gives;
Dividing the first equation by V1+ and using the above result gives the
reflection coefficient seen as port 1 as;
1
22
21
2 1V
S
SV
22
2112
11
1
2
1211
1
1
1 S
SSS
V
VSS
V
V
452.02.01
)4585.0)(4585.0(15.0
00
THE SCATTERING MATRIX (Ex)
The return loss is;
Important points to note: Reflection coefficient looking into port n is not equal to Snn, unless all
other ports are matched Transmission coefficient from port m to port n is not equal to Snm,
unless all other ports are matched S parameters of a network are properties only of the network itself
(assuming the network is linear) It is defined under the condition that all ports are matched Changing the termination or excitation of a network does not change its
S parameters, but may change the reflection coefficient seen at a given port, or transmission coefficient between two ports
dBRL 9.6)452.0log(20||log20