Formula Sheet:
Two port networks
V1   Z11 Z12   I1 
V    Z
 
 2   21 Z 22   I 2 
 b1   S11 S12   a1 
b    S
 
 2   21 S 22  a2 
 I1  Y11 Y12  V1 
 I   Y Y  V 
 2   21 22   2 
V1   A B  V2 
 I   C D   I  (I2=-I2 above)
 2 
 1 
Mason’s Gain Rule
T s  
P 
k
k
k


P11  P2  2  P3 3  

∆ = 1 – (sum of all different loop gains) + (sum of products of all pairs of loop gains, for
non-touching loops) – (sum of products of all triples of loop gains, for non-touching loops) +
…
Pk = kth path from input to output.
∆k = The quantity ∆, but with all loops touching the kth path, Pk, removed.
Noise
T0=290 K
k=1.380  10-23 J/oK
Te 
Pn  kTB
Te 
No
kB
T1  YT2
Y 1
F
Si N i
So N o
vo  a0  a1vi  a2vi2  a3vi3  
DR f 
N g  No
No
 10 log
No  kGBT0  Te 
Tg  To
To
Te  F  1T0

L  1 L  S
1  G21
Te 
T
2
G21
L 1  S


2
T
Fn  1
T
T
Ten
F2  1 F3  1


Tcas  Te1  e 2  e3   
G1
G1G2
G1G2 Gn1
G1 G1G2
G1G2 Gn1
Nonlinearity
VIP 
ENR (dB)  10 log
T
1  G21 T
F  1 e  1
T0
G21 T0
T
F  1  ( L  1)
T0
Fcas  F1 
N o  GkTe B
4a1
3a3
P1
P21 2
P2 12  N o
 1
1
P3  
 
 G2 P3 P3
1
P 
  3 
 No 
23
Gain Equations
PL
power delivered to the load

Pavs power available from the source
P
power avaliable from the network
GA  AVN 
Pavs
power avaliable from the source
P
power deli vered to the load
GP  L 
Pin
power input to the n etwork
GT 


S 1  S 1  L
P
GT  L  21
2
2
Pavs
1  S in 1  S 22L
2
2
2

Figure of Merit
1
G
1
 T 
2
1  U  GTU 1  U 2
S11 S 21 S12 S 22
U
2
2
1  S11 1  S 22



Stability Circles
S12S 21
RL 
S 22  
2
S

CL
2
*
 S11
22
S 22  
2

*
RS 
2
S12S 21
S11  
2
CS
2
S

11
*
 S 22
S11  
2

*
2
Stability Parameters

1  S11
2
2
K
*
S 22  S11
 S12 S 21
B1  1  S11  S 22  
2
1  S11  S 22  
2
2
CS 
1  1  g s  S11
2
RS 
1  S
GS
gs 

GS max 1  S11S
2
  S11S 22  S12 S 21

1  g s 1  S11
1  1  g s  S11
1  S 
2
2
2
2 S12 S 21
Constant gain circles
*
g s S11
2
11
2
2
C
L

*
g L S 22
1  1  g L  S 22
1  L
GL
gL 

GL max 1  S 22 L
2
2

1  g L 1  S 22
RL 
1  1  g L  S 22
2

2
1  S 
2
2
22
Noise Figure Circles
2
F  Fmin
CF 
s  opt
s  opt
2
4R
R
N

 N Ys  Yopt F  Fmin  N
2
Zo 1   2 1   2
Gs
1  s
s
opt
opt
N 1
RF 


N N  1  opt
N 1
2


2
N
2
F  Fmin
1  opt
4 RN Z o
Power Amplifiers
Pout  Pin  1  Pout  1 
 1  
 1  
PDC
 G  PDC  G 
Distributed amplifiers
 2 Ri C gs2 Z g
Zd
g 
 j Lg C g  C gs l g   d 
 j Ld C d  C ds l d 
2l g
2 Rdsl d
 PAE  PAE 


g m2 Z d Z g e  N g l g  e  N d ld
G
2
 l
4
e g g  e  d ld
Oscillators

1  C1  C2 


L3  C1C2 
0 

2
N opt 
C2 g m

C1 Gi
ln  g l g  d ld 
 g l g   d ld
1
C3 L1  L2 
0 
ZT I , s   Zin I , s   Z L s 
L1 g m

L2 Gi
Reactive Diode Multipliers



nPnm
0


n 0 n   n1  m2

mPnm
0
1  m2
  n
&
n   n  0
Reactive Diode Multipliers

m P
2
m
m 0
0
Transistor Multipliers
I 0  I max
cos

T
Pn 

2
T
I n  I max
2Vt  Vg max  Vg min
Vg max  Vg min
1 2
I n RL
2
Gc 
4 cosn T 
for n>0
T 1  2n T 2
2
Pin 
Vg Ri
2 Ri  j 0C g s
Pn
Pavail
RL 
Hybrids
90 Hybrid
180 Hybrid
0
j
S    1 
2 1

0
0
1
S    j 
2 1

0
j
0
0
1
1
0
0
1
1
0
0
j
1
0
0
1
0
1 
j

0
0
 1
1

0
2
Vg  Vg max  Vgg
Vd max  Vd min
2I n
R 1  g m Gi L3
 2

Gi
o C1C2 C1