u3. power transfer theorems and image parameters...

1

U3. Power transfer theorems

and image parameters Circuit Analysis, GIT

Philip Siegmann

Departamento de Teoría de la Señal y Comunicaciones

[email protected]

Curso 2016-2017

mailto:[email protected]

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

http://www.google.es/url?q=http://www.uarm.edu.pe/rree_convenios&sa=U&ei=Stk6U6ijD-6Y0AXS0oCwCA&ved=0CC4Q9QEwAA&usg=AFQjCNFM3rqv0benLHnTuxI3c9bp6XoefQ

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


____

(*) 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



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



12

Insertion loss

ZL

Eg

Zg

P2

circuit V2

I2

Zg

P20

V20

I20

ZL Eg


Are the different insertion loss the same if in dB and Np?


13

Transmission units • Transmission gain:

• Insertion gain:



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.


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


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


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


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.

u3. power transfer theorems and image parameters...

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