u3. power transfer theorems and image parameters...
TRANSCRIPT
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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
Curso 2016-2017
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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
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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
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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
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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
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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
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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
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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
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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
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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
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11
Transmission loss
ZL
Eg
Zg
P1 P2
circuit
D
C
B
A
V1 V2
I1 I2
Circuit Analysis / Fundamental theorems / Transmission units
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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?
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13
Transmission units • Transmission gain:
• Insertion gain:
Circuit Analysis / Fundamental theorems / Transmission units
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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
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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.
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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
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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
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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
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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.