Maximum Power Transfer

+−

+

−LV LR

LISR

SV SL

L

SVR

IR

=+

( )2 2

21 1 12 2 2

LL L L L S

SL

L

RPav V I R I VR R

= = =+

2

8 L

SVR

SR LR

( ) ( )( )

22

4

21 02

S SL LS S

S

LLL

L L

R R RdPav RR R

R RV R

R+ − +

= = ⇒ =∂ +

For fixed and ,s sV R∴maximum average power transfer to load LRoccurs when

Maximum Power Transfer

+−

+

−LV

LIS S SZ R Xj= +

SV

SL

L

SVZ

IZ

=+

2 21 1 1cos cos2 2 2L L L L L L L L LPav V I Z Z I Z R I= = =

, 0L L L LZ R jX R= + >

L L LV Z I=

( ) ( )2 2

2 2 21 12 2S S

S S S

L L

L L L

V VR RZ XR XZ R

= =+ + + +

To maximize , necessary to setL Sav LP X X−=

( )2

212L S

Lv

Sa

L

VR

RPR

⇒ =+

( ) ( )( )

22

4

21 02

S SL LS S

S

LLL

L L

R R RdPav RR R

R RV R

R+ − +

= = ⇒ =∂ +

2

m ax8L

Sa v

SVP

R=

LRSR0

SX−

LX

LavP

Theorem : Maximum Power TheoremOptimum Load Impedance

opt sLZ Z=

+−

+

−LV

LIS S SZ R Xj= +

SV

SL

L

SVZ

IZ

=+

( )1 cos2S S S LSPav ZV I Z= +

, 0L L L LZ R jX R= + >

opt SLZ Z=Max. Power Theorem : Conjugate-match condition

( )21 Re2 S S LI Z Z= +

( )21 Re2 SL LZI Z= +

( )2

21 Re2

SLS

S L

ZZ

VZ

Z= +

+

( )( ) ( )1 cos2 S S S SL LZ I I ZZ Z= + +i

( ) ( )2 2

2 21 1Re 22 2 2

S SS S S S

SS S

V VZ Z R

RZ ZPav∴ = + =

+

2

24 L

S

Sav

VR

P= =

Efficiency 1 or 50%2 2

S

L L

L

av av

av av

P PP P

= = =

L SZ Z=

Comments :1. Under conjugate-match condition, 50% of the power delivered by the

source is lost as heat dissipation in RS. Power company neverconjugates their loads!

2. Max. Power Theorem is used extensively in communication circuits to extract maximum power from preceding stages.

∴

Network Functions

j UU U e=

( )( )( )

Y jH jU j

ωωω

LinearElements

No independent Sources

u y

( )( ) cosu t U t Uω= + ( )( ) cosy t Y t Yω= +

j YY Y e=

Definition

is called a Network Function.

+−iV

( )( )( )

o

i

V jH jV j

ωωω

N+

−oV

0oI =

iI

( )( )( )

o

i

V jH jI j

ωωω

N+

−oV

0oI =

+−iV

( )( )( )

o

i

I jH jV j

ωωω

N oI

iI

( )( )( )

o

i

I jH jI j

ωωω

N oI

Typical Application of Max. Power Theorem

+−

1600SZ = Ω

SV

( ) ( )22 22

Re1 1 1610 50 10 0.52 2 1600 16

Lo S

S L

ZP V WZ Z

− = = ≈ = + +

16LZ = Ω

HI-FIAmplifier

Loudspeaker

Input impedance = 16 Ω⇓

10V=

Let average power oP =

delivered to loudspeaker

For maximum power transfer, make 1600 .LZ = Ω

( )21 160010 252 1600 1600oP W = = +

For maximum power transfer, make 1600 .LZ = Ω

( )21 160010 252 1600 1600oP W = = +

Use a transformer :

:1n16 Ω ≡ ( )2 16 1600LZ n= =

2 100 10n n⇒ = ⇒ =

10:1HI-FI

AmplifierLoudspeaker25W 25W

non-energtransformer is 25 Watts of ric powe⇒

is delivered to loudspeaker.

FREQUENCY RESPONSE

C L R

( )( ) ( )

( )( ) ( )2 22 2

11

1 1 1 1

C LRZ j jC CR L R L

ω ωωω ωω ω

− −= +

+ − + −

01 1 Resonant frequencyC L LC

ω ω ωω= ⇒ = ←

0 ω

R( )R jω

2R

2R

−

00ω

0ω

ω

2R

R0ω =

ω = ∞0

0ω ω=( )R jω

( )X jω

( )Z jω

Resistance function ( )R jω→ ( )X jωReactance function ←

( )X jω

FREQUENCY RESPONSE

C L R

( )( ) ( ) ( ) ( )

2 2

2 22 2

11

1 1 1 1

C LRZ jC CR L R L

ω ωωω ωω ω

− = + + − + −

( )( )1

1tan 1

C LZ jR

ω ωω − − − =

01 1 Resonant frequencyC L LC

ω ω ωω= ⇒ = ←

0 ω

R

( )Z jω

2π

2π

−

00ω

0ω

ω

( )Z jω

2R

R0ω =

ω = ∞0

0ω ω=( )R jω

( )X jω

( )Z jω

( )Z jω

FREQUENCY RESPONSE

C L R

( ) ( )1 1Y j j CR Lω ω ω= + −

01 1 Resonant frequencyC L LC

ω ω ωω= ⇒ = ←

0 ω

1R

( )G jω

Cω

0

0ωω

( )B jω

ω = −∞

0

0ω ω=

( )G jω

( )B jω

( )Y jω

1R

ω = ∞

( )G jω ( )B jω Susceptance function ←

Conductance function ↑

1Lω

−

FREQUENCY RESPONSE

C L R

( ) ( ) ( )221 1Y j CR Lω ω ω= + −

( ) 11

tan 1C LY j

R

ω ωω − − =

01 1 Resonant frequencyC L LC

ω ω ωω= ⇒ = ←

0 ω

1R

( )Y jω

2π

2π

−

0 0ω

0ω

ω

( )Y jω

ω = −∞

0

0ω ω=

( )G jω

( )B jω

( )Y jω

Magnitude function←

Phase function←

1R

( )Y jω

ω = ∞

Resonance

( 2) 1 , ( 2) 1Y j Z j= =

CLRRi Li Ci

1Ω 14 H 1 F

v+

−

sitωπ

2π 3

2π 2π

cosSi tω=

1

0

1−

CYLYRYRI LI CI

V+

−01 iI e=

( 2) ( 2) ( 2) 1V j Z j I j= =

( 2) ( 2) ( 2) 1R RI j Y j V j= =

( 2) ( 2) ( 2) 2 90L LI j Y j V j= = ∠ −

( 2) ( 2) ( 2) 2 90C CI j Y j V j= = ∠

2π0

π

32π

2π tω

( ) ,Ci t A

⇓2ω =

01 2LC

ω = =

CI

LI

RI

2

2−

10

2C L RI I I= =

no-gain propertydoes not hold forRLC circuits.

Resonance

1( 1) 10 71.6 , ( 1) 71.610

Y j Z j= ∠ − = ∠

CLRRi Li Ci

1Ω 14 H 1 F

v+

−

sitωπ

2π 3

2π 2π

cosSi tω=

1

0

1−

CYLYRYRI LI CI

V+

−01 iI e=

1( 1) ( 1) ( 1) 71.610

V j Z j I j= = ∠

1( 1) ( 1) ( 1) 71.610R RI j Y j V j= = ∠

4( 1) ( 1) ( 1) 18.410L LI j Y j V j= = ∠ −

1( 1) ( 1) ( 1) 161.610C CI j Y j V j= = ∠

2π

0π

32π

2πtω

( ) ,Ci t A

⇓1ω =

01 2LC

ω = =

CI

LI

RI

10

1 0I = ∠71.618.4−

161.6

Resonance

3( 2) 4 10 , ( 2) 250Y j Z j−= × = Ω

CLRRi Li Ci

250 Ω 14 H 1 F

v+

−

sitωπ

2π 3

2π 2π

cosSi tω=

1

0

1−

CYLYRYRI LI CI

V+

−01 iI e=

( 2) ( 2) ( 2) 250V j Z j I j V= =

( 2) ( 2) ( 2) 1R RI j Y j V j A= =

( 2) ( 2) ( 2) 500 90L LI j Y j V j A= = ∠ −

( 2) ( 2) ( 2) 500 90C CI j Y j V j A= = ∠

2π0

π

32π

2π tω

( ) ,Ci t A

⇓2ω =

01 2LC

ω = =

CI

LI

RI

500

500−

10

500−

500

Instantaneous, Average, Complex Power

+−+

−v

( )( ) cosv t V t Vω= +

Instantaneous Power

N

i

( )( ) cosi t I t Iω= +

( ) ( ) ( ) cos( ) cos( )p t v t i t V I t V t Iω ω= = + +

1 1cos( ) cos(2 )2 2

V I V I V I t V Iω= − + + +

constant Sinusoid of twice the frequency

Average Power

0

1 1( ) cos( )2

T

avP p t dt V I V IT

= −∫

t

( )v t

2T πω

=

t

( )i t

( )p t

avP

0

Power being returned by N

V Z I=j V j Z j IV e Z e I e= i

( )j Z IZ I e +=

V Z I Z V I∴ = + ⇒ = −

INDUCTOR1. 0 in CAPACITORavP =

Average Power 1 cos2avP V I Z=

Remarks:

pa2. ssIf N ive contains only elements, then0 cos 0avP Z≥ ⇒ ≥

90 90 for passive NZ∴ − ≤ ≤

I

IZ

V Z I=

0

Complex Po2

er 1w P V I

( )1 12 2

j V j I j V IP V e I e V I e− −= =i

Active power

Z

1 1cos sin2 2

P V I Z j V I Z = +

ReavP P ImQ PReactive power

1Re Average (Active) power2 in Watts1Im Reactive power2

avP V I

Q V I

= = ∴ = =

Effective (RMS) Value

21 ( )T

RMS oX x t dt

T ∫

Definition : Given any periodic waveform x(t) of period T,

the RMS (root-mean-square) or effective value of x(t)

is defined as

R( )i t

Interpretation

RRMSI

Let WR = average power dissipated in Resistor

( )( ) 2 2

0 0

1 1( ) ( ) ( )T T

R RMSW Ri t i t dt R i t dt R IT T

= = = ∫ ∫

The same average power is dissipated if the resistor is driven

by a dc current source of value IRMS ; hence IRMS is called

the effective value of i(t).

∴

2RMSI

Sinusoidal waveforms: ( ) cos( )x t X t Xω= +

2RMS

XX =

1. Since most instruments measure RMS values;

hence our 60Hz-sinusoidal voltages are rated in

RMS values.

2.

: Line voltage =110 magnitude=Exam 2(11 )le .p 0V V⇒

cos cos2 2av RMS RMS

V IP Z V I Z= =i

Note:

Significance of Complex Power

Most electrical machines are designed to withstand a

maximum voltage magnitude |V| and a maximum current

magnitude |I|. Hence, electrical machines are rated in

maximum in KVA, and not in maximum

average power dissipation Pav.

12

P V I=

12

Power factor cosav V I ZPPF

V I V I=

cosPF Z∴ =

1 for Resistors0 for INDUCTORS and CAPACITORSPF =

Note:

1. For power generation companies, it is important to keep the

PF of the load (customers) be as close to unity as possible.

2. The Watthour meter measures Pav, not |P|.

Example: If a factory dissipates 10KW of power with 50 %

PF, then the power company must generate 20 KVA of

power.

A penalty is usually levied for low PF customers.∴