u3. power transfer theorems and image parameters...
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1
U3. Power transfer theorems
and image parameters Circuit Analysis, GIT
Philip Siegmann
Departamento de Teoría de la Señal y Comunicaciones
philip.siegmann@uah.es
Curso 2016-2017
Index
• Introduction: Typical circuit configuration
• Recall
– Equivalent impedance
– Equivalent Thevenin and Norton
• Maximum power supply. Impedance Maching
• Power transfer. Everitt Theorem.
• Transmission Units
• Image Parameters
• Cascaded association using image param.
2
Typical circuit configuration
ZL
Eg
Zg
Two-port
passive
circuit
D
C
B
A
Energy supply
(real voltage or
current source)
Load impedance
(receiver)
Equivalent impedance, Zeq
Equivalent Thevenin, (ETh, ZTh)
Eg
Zg
B
A
Zeq
ETh
ZTh
D
C
ZL
3
Equivalent circuits:
Circuit Analysis / Fundamental theorems / Typical circuit configuration
4 4
Recall: Equivalent Impedance
• Any passive circuit (i.e. without independent
sources) between two terminals (A,B) can be
simplified with a equivalent impedance, Zeq:
B
A
Zeq
Circuit Analysis / Fundamental theorems / Equivalent impedance
The equivalent impedance can be obtained by:
• Parallel and/or serial association of the impedances within the passive
circuit.
• Applying a test generator Eg:
Passive
circuit
B
A
Zeq
g
g
eqI
EZ Eg
Ig
Passive
circuit
B
A
Zeq
5 5
Recall: Equivalent Thevenin &
Norton • Any linear circuit with independent sources (i.e. active
circuit) between two terminals (A,B) can be simplified by a real voltage source (Equiv.Thevenin) or real current source (Equiv. Norton):
Active
circuit
B
A
ETh
B
A
ZTh
B
A
ZN IN
Equivalent Thevenin Equivalent Norton
Circuit Analysis / Fundamental theorems / Equivalent Thevenin & Norton
6 6
Recall: Equivalent Thevenin
• ETh is the voltage between A and B in open circuit:
• Zth is the equivalent impedance seen trough A-B
Active
circuit
B
A
o.c.in ThBAAB EVVV
Deactivated
(i.e. passive)
circuit
B
A
circuit in the sources dindependen
theannullingby , )(g
g
ThI
EZ
Circuit Analysis / Fundamental theorems / Equivalent Thevenin & Norton
____
(*) see slide 4
ETh
B
A
ZTh
ZTh
Eg
Ig
7 7
Recall: Equivalent Norton
• IN is the current between A and B in short circuit:
• ZN (as ZTh) is the equivalent impedance seen trough A-B
• Equivalencies between the real Thevenin and Norton
sources:
Active
circuit
B
A
s.c.in NAB II
B
A
ZN IN
NTh
NNTh
ZZ
IZE
B
A
ZN IN ETh
B
A
ZTh
Circuit Analysis / Fundamental theorems / Equivalent Thevenin & Norton
8
Maximum power supply
• When does a real generator (Eg, Zg) supply the maximum
power to a load impedance ZL?
Circuit Analysis / Fundamental theorems / Maximum power supply
Eg
Zg ZL
Ig
2
2
2
j2]Re[
2
1
ybxa
xEZIP
g
LgZL
yxZ
baZ
L
g
j
j fixed
variable
PZL
a
EP
g
ZL8
2
maxand *
maxmatching Impedance
gLZLZL ZZ
by
axPP
9
Everitt theorem
• When there is maximum power supply at the input
gate (P1) of a non-dissipative circuit (i.e. made of L
and C only) then there is also maximum power supply
at the output (P2) of the circuit and vice versa:
Circuit Analysis / Fundamental theorems / Everitt theorem
ZL
Eg
Zg
P1 P2
No
dissipative
circuit
D
C
B
A
D-Cat matching Impedance B-Aat matching Impedance
max22max11
PPPP
10
Transmission units
• For measuring the signal transmission trough a circuit
the following magnitudes are usually used:
– Transmission loss:
– Insertion loss:
Circuit Analysis / Fundamental theorems / Transmission units
ZL
Eg
Zg
P1 P2
circuit V1 V2
I1 I2
signaloutput circuits
signalimput circuits
signaloutput circuits
circuit without signal supplied
Zg
P20
V20
I20
ZL Eg
11
Transmission loss
ZL
Eg
Zg
P1 P2
circuit
D
C
B
A
V1 V2
I1 I2
Circuit Analysis / Fundamental theorems / Transmission units
12
Insertion loss
ZL
Eg
Zg
P2
circuit V2
I2
Zg
P20
V20
I20
ZL Eg
Circuit Analysis / Fundamental theorems / Transmission units
Are the different insertion loss the same if in dB and Np?
13
Transmission units • Transmission gain:
• Insertion gain:
Circuit Analysis / Fundamental theorems / Transmission units
14
Image parameters of a bilateral Q
Z02
Z01 Z02
Z01
Image impedances Z01 y Z02
R-L-C
Propagation constant
I1 I2
V2 V1 ZL 02
22
112exp
ZZL
IV
IV
R-L-C
Circuit Analysis / Two-port networks / Image parameters
15
Image parameters for a bilateral and
symmetrical Q
Characteristic impedance (Z0 = Z01=Z02) and
propagation constant () can be obtained as follows:
tanh1
tanh1ln
2
1,tanh,
..
......0
CO
CSCSCO
Z
Z
A
BCZZ
C
BZ
being ZO.C. y ZS.C. impedances seen trough Q when the ports of
the opposite side are respectively open- and short-circuit:
ZO.C.
o.c.
ZO.C
o.c.
ZS.C. ZS.C.
Circuit Analysis / Two-port networks / Image parameters
s.c.
16
Image parameters for a bilateral and
symmetrical Q
coshsinh1
sinhcosh
0
0
Z
Z
DC
BA
Propagation constant
000
00
2
1
2
1
2
1
2
1
2
1
ln2
1lnln]Np[
loss)insertion ( :,difference phase:,j
,exp
ZZZZZZ
ZZZZ
LLL
LL
P
P
I
I
V
V
I
I
V
V
loss ontransmissi
“T” as function of Z0 and
I1 I2
V2 V1 Z0
R-L-C &
symmetrical
Circuit Analysis / Two-port networks / Image parameters
2
110
2
110
2
110
log10
log20
log20]dB[
P
P
I
I
V
V
17
Equivalent Thevenin for a bilateral and
symmetrical Q when Zg=Z0
Equivalent Thevenin:
)exp(
cosh·sinh1
0
0
g
g
g
g
th E
ZZ
E
ACZ
EE
Eg
Zg=Z0
A
B
, Z0
Eth
Zth=Z0
A
B
Circuit Analysis / Two-port networks / Image parameters
18
Summarizing, for a bilateral and symmetric
quadripole
Equivalent circuits:
Eth
Zout
ZL V2
Eg
Zg
Zin V1
0
0
e then if
ZZ
EEZZ
out
gth
g
A
B
I1 I2
V2 V1
Eg
Zg
ZL
Zin Zout
A
B
C
D
C
D
, Z0
2
1
2
1
0in
0 exp
Z
then if
I
I
V
V
Z
ZZL
Circuit Analysis / Two-port networks / Image parameters
19
Cascaded association using image
parameters
N bilateral and symmetric quadripoles (all with the same Z0)
connected in cascade:
N
NNNN
eII
eVeeVeVV
N
NNN
...
0
...
021
21
211 ...
0V1
0
Z
N
Z
0
0ZZL
2
0
Z1V 2V NV
0I1I 2I NI
Circuit Analysis / Two-port networks / Cascaded Association using image param.
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