EEE 194RF
Assignment #6
Due 14 March 2001
Assignment #6
4.5
Employ the conversion table for the different parameter representations of the two-port
network and find the h-matrix representation for a Darlington Pair shown in Figure 4-7
under the assumption that the transistors are specified by the same h-matrices derived in
Example 4-2.
As a first step replace Q1 connected to R by a cascade of two networks. To find a matrix
representation of this connection we have to convert the h-parameter matrix for the
transistor into an ABCD form and multiply it by the ABCD representation of the resistor
A B 
 A B   A B
 A B  1 R 
=
=
C D 



 


Q1+ R  C D Q1  C D R C D Q1  0 1 
The result of this operation has to be converted back to h-matrix form and added to the hparameter matrix of Q2. The result is:
 R∆ h + h11
 Rh + 1
[ h] = [ h ]Q 2 +  h22

21
 Rh + 1

22
h12 
Rh22 + 1 

h22 
Rh22 + 1 Q1
where ∆h is the determinant of the h-parameter matrix for Q1.
4.9
Find the ABCD parameters for a generic three element pi-network, as depicted in Figure
4-2.
One way to solve this problem is to directly evaluate each parameter by using the
definition for the ABCD matrix. Alternately, we can use out knowledge of the ABCD
representations for the individual circuit components as given in Table 4-1. With
reference to Figure 4-2 we can write
 A B   1 0  1 Z B   1
C D  =  −1   0 1   −1

  Z A 1 
 ZC
Z

1+ B

0
ZC
=
1  Z A + Z B + Z C

Z AZ C


ZB 

ZB 
1+
Z A 
EEE 194RF
4.10
Assignment #6
Due 14 March 2001
Compute the ABCD parameters for an RF transformer with turn ratio N=N1 / N2 , where
N1 and N2 are the number of turns in the primary and secondary, respectively.
Since the voltages in the transformer are related as v 1 / N1 = v 2 / N2 , we can use the
definition of the ABCD matrix and write
v
N
A= 1
= 1 =N
v2 i =0 N 2
2
Because of v 1 / N1 = v 2 / N2 for the ideal transformer the condition v 2 = 0 immediately
leads to v 1 = 0. Therefore
v
B= 1
=0.
−i2 v =0
2
Furthermore since there are no losses in an ideal transformer, the input power is equal to
the output power
P1 = P2 ⇒ v 1 i1 = -v 2 i2
Here we took in to account that the direction of the current flow in the secondary winding
of the transformer is opposite to the current direction at port 2. Taking into account the
relation between the voltages at ports 1 and 2, we can write
v
N
i
i1 = − 2 i2 = − 2 i2 = − 2
v1
N1
N
Using the relations between the input and output currents we can calculate the remaining
two parameters as
i
i
1
C= 1
= 0, D = 1
= .
v2 i = 0
−i2 v =0 N
2
2
4.12
TBD
4.14
Find the forward gain of the circuit in Example 4-8.
From Equation 4.47 we observe that the power at port 1 is computed as
1 2
2
P1 = a1 1 −  Γin 
2
Following an identical derivation we can obtain the power delivered to the load as
1
1 2
2
2
2
P1 = a L 1 − Γ L  = b2 1 −  ΓL 
2
2
Our objective is to find the power gain which is equal to the ratio of the power delivered
to the load over the input power:
(
)
(
)
b2
P
G= L =
P1 a
L
2
(
)
(1 − Γ  )
(1 − Γ  )
2
L
2
2
L
With reference to Figure 4-23(b) we can write b2 = S21 a1 + S22 ΓLb2 or b2 =
S 21
a1
1 − S22Γ L
EEE 194RF
Assignment #6
Due 14 March 2001
Substitution yields the gain of the circuit
G=
(1 − Γ )
Γ (1 − Γ )
S21
1 − S 22
2
2
L
2
L
where Γin = S11 +
4.15
2
in
S12 S 21
ΓL .
1 − S 22 ΓL
Given that the input of an amplifier has a VSWR = 2 and the output is given by VSWR =
3, find the magnitudes of the input and output reflection coefficients. What does your
result mean in terms of S11 and S22 ?
VSWRin − 1 1
=
VSWRin + 1 3
VSWRout − 1 1
Similarly the output VSWR is Γout =
=
VSWRout + 1 2
Since the input VSWR = 2, Γin =
The output reflection coefficient can be represented as
S S
Γin = S11 + 12 21 Γ L
1 − S 22 ΓL
Since the system is symmetric we can write an almost identical expression for the output
reflection coefficient of the amplifier
S S
Γout = S22 + 12 21 Γ S
1 − S1Γ S
In the general case, where all S-Parameters and source and load reflection coefficients are
nonzero, we cannot estimate the values of the S11 and S22 parameters of the amplifier
solely based on the values of the input and output reflection coefficients. However, if the
load and source reflection coefficients are equal to zero, i.e. load and source impedances
are matched to the characteristic impedance of the system, then
S11 = Γin and S 22 = Γout
EEE 194RF
Assignment #6
Due 14 March 2001