Circuits II

8/3/2019 Circuits II

1/73

Circuit Analysis II

FE Examination

Preparation Courses

Engineering and Science ServicesThe American University in Cairo

Hanna Kirolous, 2009


2/73

Circuit Analysis II FE-Exam

Course Outline

Sinusoidal steady-state power calculations

Laplace transform Series and parallel resonance circuits

Frequency selective circuits

Two-port circuits


3/73


AC Power Calculations


4/73


Sinusoidal Steady State Power Calculation

)t(i)t()t(p

Instantaneous Power:

)tcos(I)t(i m )(t)t(i

PassiveCircuit

+

-

)2sin()2cos( tQtPPp

)tcos(V)t( m


5/73


Sinusoidal Steady State Power Calculation

)tsin(Q)tcos(PPp 22

)cos(IV)cos(IV

P rmsrmsmm

2Average Power (Watts)

)sin(IV)sin(IV

Q rmsrmsmm

2Reactive Power (VARs)


6/73


Average Power

)cos(IV

P mm 2

+ve -ve

The circuit in the boxabsorbs average power

from the terminals

The circuit in the boxdelivers average power

to the terminals

)(t

)(ti

Passive

Circuit


7/73


Reactive Power

)sin(IV

Q mm 2

+ve -ve

The circuit in the boxabsorbs reactive VARs

from the terminals

The circuit in the boxdelivers reactive VARs

to the terminals

)(t

)(ti

Passive

Circuit


8/73

00 Q )2cos(22

tIVIV

p mmmm

)(ti

)(tp

)(t


Resistive Circuits


9/73

090 P


Inductive Circuits

)2sin(2

)2sin( tIV

tQp mm

)(ti

)(tp

)(t


10/73

090 P


Capacitive Circuits

)tsin(IV

)tsin(Qp mm 22

2

)(ti)(tp

)(t


11/73

)cos()(cospf i

Leading

Current lags the voltageCurrent leads the voltage

Lagging

Capacitive Load Inductive Load

Used to characterize Loads


Power Factor


12/73

Quantity Units

Volt-amps (VA)

Watts (W)

Volt-ampsreactive (VAR)

QjPS

S

SP SQ


Complex Power


13/73

S

poweraverageP

powerreactiveQ

PowerApparent 22 QPS


Complex Power


14/73

)sin(IVj)cos(IVS imm

imm

22

ijm

j

m

eIeV

2

1

rmsrms IVS IVS

2

1


Complex Power & Phasors

)(jmm ieIV 2


15/73

rmsV

rmsI

Z+

-

rmsrms IZV )(

2

2

XjRI

ZI

IIZ

IVS

rms

rms

rmsrms

rmsrms

QjP

RIRIP mrms22

2

1

XIXIQ mrms22

2

1


Forms of Complex Power


16/73

rmsV

rmsI

Z+

-

rmsrms IZV

XjR

VZ

V

Z

V

I

IVS

rms

rms

rms

rms

rmsrms

2

2

QjP


Forms of Complex Power


17/73

THV

I

ZL

ZTH+-

THL ZZ

& THL RR THL XX

L

TH

R

VI

2

L

m

L

TH

LR

V

R

VRIP

84

22

2

max

Under maximum power condition:


Maximum Power Transfer


18/73


Laplace Transform


19/73

dte)t(f)s(F)t(fL ts 0

Time Domain Frequency Domain

DimensionlessQuantity

s has units ofsec-1 (frequency)

isUnilateral LaplaceTransform since

)(sF

0t

dte)t(f)s(F)t(fL ts


Laplace Transform


20/73

Type )(sF)(tf

step )(tus

1

impulse )(t 1

)(tutn

ramp

1

!ns

n

exponentialta

e

as

1

)(tut2

1

s

Integerexponent


Functional Laplace Transform


21/73

Type )(sF)(tf

sine tsin 22 s

cosine tcos

Dampedramp 2)(

1

as

teta

sin

22 s

s

taet

Dampedsine 22)(

as

Dampedcosine te

ta cos 22)(

as

as


22/73

(1) Multiplication bya constant

)s(F)t(fL

)s(FK)t(fKL


Operational Laplace Transform


23/73

(2) Addition & Subtraction

)s(F)t(fL 11

)s(F)s(F)s(F

)t(f)t(f)t(fL

321

321

)s(F)t(fL 22 )s(F)t(fL 33




24/73

(3) Differentiation & Integration

)(f)s(Fs)t(fL

0

)(f)(fs)s(Fs)t(fL 002

s

)s(Fdt)t(fL

t

0




25/73

(4) Translation in Time-Domain

)s(Fe)at(u)at(fLsa



(5) Translation in Frequency-Domain

)as(F)t(feL ta


26/73

(6) Time Scaling

)as

(Fa

)ta(fL1



(7) Derivative in s-domain

ds

)s(dF)t(ftL


27/73

Examples:atetL 2Find

3

2 2

stL)s(F

Let,

3

2 2

)as(etL at




28/73

tcostL Find

22 ss

tcosL)s(F

Let,

ds

)s(dF)t(ftL

222

22

)(

s

s222

22

)(

)2(

s

sss




29/73


S-Domain Analysis


30/73

A resistor in the s-Domain

Rtit )()(

R)t(iL)t(L

RsIsV )()(


Circuit Analysis in the s-domain


31/73

oILILsV s

IV

Ls

I o1

An Inductor in the s-Domain



Ci i A l i II FE E


32/73 oVCVCsI sV

ICsV

o1



A Capacitor in the s-Domain


33/73

)()()(

sXsYsH

The Transfer Function

depends on what is definedas X(s) and Y(s) X(s) Y(s)Circuit

The poles of H(s) must lie inthe left side of the s-plane

The zeros of H(s) may lie ineither the right or the leftside of the s-plane




34/73

)()()( sXsHsY

The completeresponse ofthe circuit

Poles of H(S) givethe transient partof the complete

response

Poles of X(s) givethe steady state

part of the

complete response




35/73

The Impulse Response

)t()t(x

)s(Y)s(H

)s(HL)t(y 1

1)s(X




36/73

The Exponential Response

)()( tuetxts js

)()( sHetyts

Where,




37/73

The Steady-State Frequency Response

)cos()( tAtx

jejHjH )()(

)cos()()( tjHAty



Circuit Analysis II FE Exam


38/73

Steps to Obtain Complete Response

Determine the Transfer Function

)()()( sXsHsY

)t(hL)s(H

Determine the Laplace Transform of the input )t(xL)s(X

Determine the Laplace Transform of the output

Breakup the output by partial fractions

Find the complete response )s(YL)t(y 1




39/73


Filters & ResonanceCircuits

l FE E


40/73

Circuits capable of stopping certain input signals basedon its frequency

Circuit)(sVi )(sVo

)(sH

)()()(

sVsVsH

i

o

We only change the frequency of the source withoutchanging its magnitude or phase

For pure sinusoidal signals the output depends on themagnitude and phase of the transfer function )( jH


Filters


41/73

Low-pass

Filters

High-pass Band-pass Band-reject


Filters

Frequency Response Plot

Magnitude Plot Phase Angle Plot


42/73

Low-pass FilterThe Series RL Circuit

sLRRsH )( LRs

LR

/

/

jLR

LRjH

/

/)(

22)/(

/)(

LR

LRjH

R

Lj

1tan)(


Filters


43/73

RL Low-pass Filter Cutoff frequency

22

)/(

/)(

LR

LRjH

22)/(

/

2

1)(

LR

LRjH

c

c

L

Rc

451tantan)( 11

R

Lj c

c

707.0

45


Filters


44/73

Low-pass Filter-The Series RC Circuit

sCR

sCsH /1

/1)( RCs

RC

/1

/1

jRC

RCjH

/1

/1)(

RCj 1tan)( 22)/1(

/1)(

RC

RCjH


Filters

l E E


45/73

RC Low-pass Filter - Cutoff frequency

22)/1(

/1)(

RC

RCjH

707.0

45

RCj 1tan)(

2

1

)/1(

/1)(

22

RC

RCjH

c

c

RCc

1

451tantan)( 11 RCjcc


Filters

Ci i A l i II FE E


46/73

High-pass FilterThe Series RL Circuit

LsRLssH )( LRs s /

jLR

jjH

/

)(

22)/(

)(LR

jH

R

Lj

1tan

2)(


Filters

Ci it A l i II FE E


47/73

707.0

45

RL High-pass Filter - Cutoff frequency

22)/(2

1)(

LRjH

c

cc

L

Rc

45tan2)(

1

R

L

j

c

c

22)/(

)(LR

jH

R

Lj

1tan

2)(


Filters

Ci it A l i II FE E


48/73

High-pass FilterThe Series RC Circuit

RC/ss

sC/RR)s(H

11

jRC

jjH

/1

)(

RCj

1tan2

)( 22)/1(

)(RC

jH


Filters

Ci it A l i II FE E


49/73

RC High-pass Filter - Cutoff frequency

2

1

)/1()(

22

RCjH

c

cc

RCc

1

45tan2

)( 1 RCjcc

707.0

45

RCj

1tan2

)( 22)/1(

)(RC

jH


Filters



50/73

Band-pass Filter - The Series RLC Circuit

)LC/(s)L/R(s

s)L/R()s(H

12

)/(])/1[(

)/()(

2

LRjLC

jLRjH


Filters



51/73

222)]/([])/1[(

)/()(LRLC

LRjH

21 )/1(

)/(tan90)(

LC

LRj


Filters



52/73

0)( oj1)( ojH

Center (Resonant) Frequency o

Def: The frequency for which the transferfunction is purely real

LCo1


Filters



53/73

Center & Cutoff Frequencies are Related

LCL

R

L

Rc

1

22

2

1

LCL

R

L

Rc

1

22

2

2

221 occ

21 cco


Filters



54/73

LCL

R

L

R

c

1

22

2

1

LCL

R

L

R

c

1

22

2

2

oQ

12 cc L

R

Bandwidth and Quality factor Q

2RC

LQ


Filters



55/73

Band-reject Filter - The Series RLC Circuit


Filters



56/73

211

ccoLC

oQ

12 cc L

R

CR

LQ

2

LCL

R

L

Rc

1

22

2

1

LCL

R

L

Rc

1

22

2

2

Band-reject Filter - Summary

Filters


57/73


Two-Port Circuits

C l FE E


58/73

CircuitInputport

1

2

Outputport

1i 2ia

b

c

d

All external connections has to be made at and dcba ,,

Conditions: 2i1i

No independent sources within the circuit.

Current input to a port is equal to current outputfrom the same port and11 ii 22 ii

No energy stored within the circuit


Two-Port Networks



59/73

2121111 IzIzV

2221212IzIzV

If the circuit details are known; the z-parameters can becalculated

If the circuit details are unknown; the z-parameters can be

measured

2

1

2221

1211

2

1

I

I

zz

zz

V

V

s-DomainCircuit

Inputport

1V

2V

Outputport

1I 2Ia

b

c

d


Z-Parameters



60/73

2221212 IzIzV Input impedance seen from port 1 whenport 2 is open

01

111

2

I

I

Vz

Transfer impedance when port 1 is open

02

112

1

I

I

Vz

Input impedance seen from port 2 whenport 1 is open

01

221

2

II

Vz Transfer impedance when port 2 is open

02

222

1

II

Vz

2121111 IzIzV

Z-Parameters Definitions



61/73

10

01

111

2II

Vz

02

112

1

II

Vz

01

221

2

I

I

Vz

02

222

1

II

Vz

102015

1

1

/V/V 5.7

15//)205( 375.9

3759

2520

2

2

./V

/V 5.7

1V 2

V

1I 2I5

20

15

Calculation of Z-Parameters



62/73

01

111

2

I

I

Vz

For a 2-port circuit operating in steady-state, with port 2 open, avoltage V is applied to port 1. The current into port 1 is

A and the voltage at port 2 is V.With port 1 open the same voltage is applied to port 2. The currentinto port 2 is measured to be A, while the voltage

at port 1 is measured to be

tcos150)45cos(25

t )15cos(50 t

)30cos(30t

)60cos(30

t

4564525

0150

01

221

2

I

I

Vz

6024525

1550

Measurement of Z-Parameters



63/73

02

112

1

I

I

Vz

9013030

6030

02

222

1

I

I

Vz

3053030

0150

Measurement of Z-Parameters

For a 2-port circuit operating in steady-state, with port 2 open, avoltage V is applied to port 1. The current into port 1 is

A and the voltage at port 2 is V.With port 1 open the same voltage is applied to port 2. The currentinto port 2 is measured to be A, while the voltage

at port 1 is measured to be

tcos150)45cos(25

t )15cos(50 t

)30cos(30t

)60cos(30

t



64/73

2121111 VyVyI

2221212 VyVyI

2

1

2221

1211

2

1

V

V

yy

yy

I

I

s-Domain

Circuit

Inputport

1V

2VOutput

port

1I 2Ia

b

c

d

y

Y-Parameters

Note: All parameters are admittances and have dimensions of S (-1)



65/73

2122111 IaVaV

2222211 IaVaI

2

2

2221

1211

1

1

I

V

aa

aa

I

V

s-Domain

Circuit

Inputport

1V

2VOutput

port

1I 2Ia

b

c

d

y

a-Parameters

Note: The parameters are inhomogeneous and havedifferent dimensions



66/73

Voltage transfer function seen from port 1when port 2 is open (dimensionless)

02

111

2

I

V

Va

-ve of the transfer impedance when port 2is short-circuited ()

02

112

2

V

I

Va

-ve of current transfer function seen fromport 1 when port 2 is short-circuited

(dimensionless)

02

121

2

IV

Ia

Transfer admittance when port 2 is open(S)

02

122

2

VI

Ia

2122111IaVaV

2222211IaVaI

a-Parameters Definitions



67/73

1121112 IbVbV

1221212 IbVbI

1

1

2221

1211

2

2

I

V

bb

bb

I

V

s-Domain

Circuit

Inputport

1V

2VOutput

port

1I 2Ia

b

c

d

y

b-Parameters

Note: The parameters are inhomogeneous and havedifferent dimensions



68/73

2121111 VhIhV

2221212 VhIhI

2

1

2221

1211

2

1

V

I

hh

hh

I

V

s-Domain

Circuit

Inputport

1V

2VOutput

port

1I 2Ia

b

c

d

h-Parameters



69/73

input impedance seen from port 1 when port2 is short-circuited

01

111

2

V

I

Vh

The transfer function seen from port 1when port 1 is open

02

112

1

I

V

Vh

Output admittance seen from port 2 when port1 is open

01

221

2

VI

I

hCurrent transfer function when port 2 isshort-circuited

02

222

1

IV

Ih

2121111 VhIhV 2221212 VhIhI

h-Parameters Definitions



70/73

2121111 IgVgI

2221212 IgVgV

2

1

2221

1211

2

1

IV

gggg

VI

g-Parameters Definitions

s-DomainCircuit

Inputport

1V

2V

Outputport

1I 2Ia

b

c

d



71/73

2112 zz

Condition:The interchange of an ideal voltage source at one port with an idealammeter at the other port produces the same ammeter reading

For all parameters:

2112 yy

1a

1b

2112 hh

2112

gg


Reciprocal Two-Port Circuits



72/73

Condition:NOTA reciprocal network in which the interchange of its ports do

disturb the values of the terminal voltages and currents

parameters:-For all

1h

2211 zz 2211 yy

2211 aa

2211 bb

1 g

Symmetric Two-Port Circuits



73/73

2

2

2221

1211

2221

1211

1

1

I

V

aa

aa

aa

aa

I

V

Cascaded Two-Port Circuits

2

2

2222122121221121

2212121121121111

1

1

)(

)(

I

V

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aaaaaaaa

I

V

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Circuits II