Tutorial 4
Tutorial 4:
1.
A two port network is known to have the following scattering matrix:

0.85  40  

0.200  
S    0.150 

0.8540
i).
Determine if the network is reciprocal and lossless.
ii).
If the port two is terminated with a matched load, what is the return loss
seen at port 1?
iii).
If the port two is terminated with a short circuit, what is the return loss
seen at port 1?
(Answers: (i) since [S] is not symmetric, it is not reciprocal. It is also not lossless
(but this needs proof). (ii) RL = 16.5 dB; (iii) RL = 10.66 dB)
2. A lossless T-Junction power divider has source impedance 50 Ω. Find the output
characteristic impedances so that the input power is divided in a 2:1 ratio.
Compute the reflection coefficients seen looking into the output ports.
(Answers: Z  150 ; Z  75 ;   0.67 ;   0.33 )
1
3.
2
1
2
Design a maximally flat low-pass filter with a cutoff frequency of 2 GHz,
impedance of 50 Ω, and at least 15 dB insertion loss at 3 GHz.
(Answers: Designed using ladder diagram with parallel C element first;
C  0.983 pF , L  6.437nH ; C  3.183 pF ; L  6.437nH ; C  0.984 pF )
1
4.
3
2
5
4
Design an equal ripple low-pass filter (3.0 dB ripple) with a cutoff frequency at 4
GHz, impedance of 50 Ω, and at least 20 dB insertion loss at 5.2 GHz.
(Answers: For IL = 20 dB, N = 5. Designed using ladder diagram with parallel C
first; C  2.77 pF ; L  1.515nH ; C  3.611 pF ; L  1.515nH ; C  2.77 pF )
1
2
3
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4
5
Tutorial 4
5.
Design a three-section bandstop lumped element filter with a 0.5 dB equal-ripple
response, a bandwidth of 10% centered at 3 GHz, and impedance of 50Ω. What is
the resulting attenuation at 3.1 GHz?
(Answers:   1.52 ; attenuation at 3.1 GHz = 11 dB; L  16.615nH ;
C  0.169 pF ; L  0.291nH ; C  9.674 pF ; L'3  L1' ; C3'  C1' ; R  50 )
'
1
'
1
6.
'
'
2
2
L
Design a three-section band-pass lumped element filter having a maximally flat
group delay response. The bandwidth should be 12% with a center frequency of 5
GHz. The impedance is 70Ω. What is the resulting attenuation at 5.5 GHz?

 1  0.591 ; attenuation at 5.5 GHz is 13 dB; Using
c
shunt cap as the first element; L  0.267nH ; C  3.7889 pF ; L  37.1313nH ;
C  0.0273 pF ; L'3  L1' ; C3'  C1' ; R  70 )
(Answers:   1.591 ;
7.
'
'
'
1
1
2
'
'
2
L
Why the receiver designs are required to have a preamplifier with low noise
figure and higher gain?
(Answers: first stage of a receiver front end has the dominant effect on the noise
performance; maximum gain at the first stage will decrease the noise effect for
second and others stage; FT  F1 
8.
F2  1 F3  1

 .. )
G1
G1G2
The single stage low noise amplifier is designed by using the AT-36133 FET
transistor from Avago Technologies which have the following S-parameter at
frequency of 2.4 GHz (characteristic impedance, Z0 = 50Ω):
S11  0.8264.9 
S 21  1.28  35.5
S12  0.134  36 
S 22  0.5776.2 
i)
Determine the stability of the transistor at 2.4 GHz.
ii)
Calculate the transducer power gain in dB, if the source impedance
is 40Ω and the load impedance is 20Ω.
EKT 441 MICROWAVE COMMUNICATIONS
Tutorial 4
iii)
Calculate the transducer power gain in dB for unilateral case.
(Answers: (i)    0.4181  j 0.4561  0.6187132.51 ; K  1.124  1; (ii)

  0.111 ;   0.4286 ; in  0.3409  j 0.6752  0.756463.21 ;
S
L
out  0.1316  j 0.5712  0.586277.03 ; GT  1.037  0.158dB ;
GTU  1.0346  0.1477dB )
Full Solution:
i)
  S11S 22  S 21S12


 

  0.8264.9  0.5776.2   1.28  35.5  0.134  36 
   0.4181  j 0.4561  0.6187132.51
 1
1  S11  S 22  
2
K
2
2 S12 S 21
1  0.82  0.57  0.6187
2
K
2
2
2
2 1.280.134
K  1.124  1
The transistor is stable at frequency 2.4 GHz.
ii)
 Z 0 40  50


 Z 0 40  50
 Z 0 20  50


 Z 0 20  50
S S 
in  S11  12 21 L
1  S 22 L
ZS
ZS
Z
L  L
ZL
S 
 10
 0.111
90
 30
 0.4286
70
 0.134  36 1.28 35.5  0 .4286
1  0.5776.2   0.4286


in  0.8264.9  


in  0.3409  j 0.6752  0.756463.21
2
GT 

S 21 1  S
2
1   
2
L
1  S in 1  S 22 L
2

1.28 1  0.111 1  0.4286
2
GT 

2
2
2

1   0.1110.756463.21 1  0.5776.2 0.4286
GT  1.037  0.158dB
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2
2

Tutorial 4
9.
The S parameter for the HP HFET-102 GaAs FET at 2 GHz with a bias voltage
Vgs =0 are given as follows (Z0 = 50 Ω):
0.894  60.6 
0.0262.4  
S   


0.781  27.6  
 3.122123.6
Determine the stability of this transistor by using K – Δ test, and plot the stability
circles on a Smith chart.
(Answers:   0.696 ; K  0.607 ; device is potentially unstable;
C  1.36147 ; R  0.50 ; C  1.13268 ; R  0.199 )


L
S
L
S
Full Solution:
  S11S 22  S12 S 21  0.696  83  0.696
1  S11  S 22  
2
K
2
2 S12 S 21
2
 0.607
Thus we have   0.696 < 1 and K >1, so the unconditionally stability criteria is
not satisfied, and the device is potentially unstable.
CL
S

RL 
CS 
RS 
22
 S11 
S 22  
2
2
 1.36147 
2
 0.50
S12 S 21
S 22  
2
S
11


 S 22
S11  
2
2
S12 S 21
S11  
2
2
 1.13268
 0.199
EKT 441 MICROWAVE COMMUNICATIONS
Tutorial 4
10.
Design an amplifier for maximum gain at 4.0 GHz using single stub matching
sections. Calculate and plot the input return loss and the gain from 3 to 5 GHz.
The GaAs FET has the following S parameters (Z0 = 50 Ω):
ƒ (GHz)
3.0
4.0
S11
0.80∟-890
0.72∟-1160
S21
2.86∟990
2.60∟760
S12
0.03∟560
0.03∟570
S22
0.76∟-410
0.73∟-540
5.0
0.66∟-1420
2.39∟540
0.03∟620
0.72∟-680
(Answers:   0.488 ; K  1.195 ; device is unconditionally stable at 4 GHz.
    0.872123 ;   

S


in
L
out
 0.87661 ; G  4.17  6.20dB ;

S
G  6.76  8.3dB ; G  1.67  2.22dB ; G
0
T max
L
 16.7dB )
Full Solution:
  S11S 22  S12 S 21  0.488  162   0.488
1  S11  S 22  
2
K
2
2 S12 S 21
2
 1.195
Since   0.488 < 1 and K > 1, the transistor is unconditionally stable at 4 GHz.

S  in 

L  out 
GS 
B1  B12  4 C1
2C1
B2  B2 2  4 C 2
2C 2
1
1  S
2
2
 0.872123
2
 0.87661
 4.17  6.20dB
2
2
G0  S 21  2.6  6.76  8.3dB
GL 
1  L
2
1  S 22 L
2
 1.67  2.22dB
GT max  6.2  8.3  2.22  16.7 dB
EKT 441 MICROWAVE COMMUNICATIONS