1

Chapter 6Frequency Response & System Concepts

Jaesung Jang

AC steady state (frequency) response Phasor notation

Filter

2

Forced Response by Sinusoidal Excitation

• In a sinusoidally excited linear circuit, all branch voltages and currents are sinusoids at the same frequency as the excitation signal.

• The amplitudes of these voltages and currents are a scaled version of the excitation amplitude and the voltage and currents may be shifted in phase with respect to the excitation signal.

• Phasor notation to solve for frequency, amplitude, and phase with more ease.

( )

( ) ( )

( ) ( )( )

( ) ( )t

RC

Vt

RC

RCVtv

RC

V

RC

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tCtBtAtv

tVRCRC

v

dt

dv

RC

v

RC

v

dt

dv

tVtv

C

C

CCSCC

S

ωω

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22

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++

+=→

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=+→=+

→=

3

Complex Algebra

( )( )

e.convenienc almathematicjust istion simplifica This

formdomain - timeacosfor notation phasor Complex :

form (phasor)domain -frequency asincos

→+→∠=+=

θωθθθθ

tA

AjAAe j

• If either a sinusoidal voltage or current sourcesupplies a linear circuit consisting of resistance, inductances, and capacitances, all branch voltages and currents of those components will be sinusoids of the same frequency.

• We need a complex algebra to perform the addition, subtraction, division, and multiplication of voltages and currents in AC excited circuits.

• If an instantaneous voltage is described by a sinusoidal function of time such as

where V is a maximum value, and ω=2πf is an angular frequency of voltages.

( ) ( )( ) ( ) ( )( ) tjjtj eVeVetv

tVtvωθθω

θω ReRe

cos

==

+=+

( ) θθθ

θθθ

θ

∠=+=

+=

rjrre

jej

j

sincos

Identity sEuler' :sincos

Phasor concept has no real physical significance. It is just a convenient mathematical tool.

4

The combination of a real and an imaginary term is called a complex number.Complex numbers can be calculated as phasors.Graphically, the sum is the hypotenuse of the right triangle formed by the two phasors.

a

br

θ

z

Re

Im

( ) ( )( ) )coordinatepolar a(in sincos

)coordinatecartesian a(in Im Re

θθθθ ∠=+==+=+=

rjrrez

zjzjbazj

( ) ( )zrbzra

a

bbar

Imsin ,Recos

argument :tan ,magnitude : 122

====

=+= −

θθ

θ

Examples

2,1 -

2,1 -

0,1 - 1

πθπθ

θ

==>=−==>−=

==>=

rjz

rjz

rz

Complex Algebra (cont.)

( ) ( )

( ) ( ) zeronot is if ,,

of conjugate:Im Re2

ww

z

w

zwzzwwzwz

zzzzzjzjbaz

∗

∗∗∗∗∗∗∗∗

∗∗

=

=±=±

=→−=−=

5

2222111121 , jbaerzjbaerz jj +==+== θθ

Cartesian form is easier to use for addition and subtraction of complex numbers.

( ) ( ) ( )( )( ) ( ) ( )

( )( )

( )( )( )( )

( ) ( )( )2

222

21122121

2222

2211

22

11

2

1

21122121221121

2121221121

ba

babajbbaa

jbajba

jbajba

jba

jba

z

z

babajbbaajbajbazz

bbjaajbajbazz

+−++=

−+−+=

++=

++−=++=±+±=+±+=±

Polar form is easier to use for multiplication and division of complex numbers.

( ) ( )( ) ( )21

2

1

2

1

2

1

2

1

2121212121

2121

21

2

1

2121

21

θθ

θθ

θθθ

θ

θθθθ

θθ

−∠===

+∠===

±=±

−

+

r

re

r

r

er

er

z

z

rrerrererzz

ererzz

jj

j

jjj

jj

You have to be able to convert Cartesian form to polar form and vice versa.

Complex Algebra (cont.)

Multiplication: multiplication of both magnitudes & addition of both argumentsDivision: division of both magnitudes & subtraction of both arguments

6

• Here, phasor V is defined as . It has the essential information: amplitude and phase angle of an ac signal.

• The impedance Z is a measure of the opposition to the flow of AC current. That is, it is a complex resistance. The element equations that define the voltage-current characteristics of the passive elements are (the Ohm’s law of sinusoidal circuits)

where V, I, and Z are phasors of voltage, current, and impedance of an passive element, respectively.

• 1/ Z = Y is called admittance.• The passive elements has impedances to sinusoids• For resistance, • For inductance,

• For capacitance, `

ZI V =

( ) RRjZ R =°∠= 0ω

( ) reactance. inductive theis where9090 LXXLLjjZ LLL ωωωω =°∠=°∠==

( ) ( ) ( ) reactance. capacitive theis 1

where909011

CXX

CCjjZ CCC ωωωω =°−∠=°−∠==

θθ ∠== VVe jV

Impedance

The capacitors and inductors are frequency-dependent resistors.

7

Impedance (cont.)

• VR= ZR I = R I = The voltage across a resistance is in phase with the currentthrough it.

• VL= ZL I = jωL I = The voltage across an inductive reactance leads the current through it by 90o.

• VC= ZC I = (1/jωC) I = The voltage across a capacitive reactance lags the current through it by 90o.

8

• The total impedances are calculated just as resistances are. – In a series circuit, the total impedance is the sum of all the impedances.– In a parallel circuit, the reciprocal of the total impedance is the reciprocal sum of

individual impedances.

• Although the impedance of single circuit elements is either purely real (for resistors) or purely imaginary (for inductors and capacitors), the general definition of impedance should have both real and imaginary parts since the practical circuits consist of more or less complex combinations of different circuit elements.

++=

LCRT ZZZZ

1111

T

T

LCRT

Z

VI

ZZZZ

=

++=

Impedance (cont.)

9

I = 2 A

VT = 100R = 30 ΩΩΩΩ

XC = 40 ΩΩΩΩ

R

XC Z

ZT = ZR + ZC = R - j XC

( )

( ) ( )( ) ( ) ( )°−−∠=°−∠−∠==

−∠=°∠−∠==∠=

−∠=

−=∠

∠==

−∠=−=−=

−

−

9080902

6002

100

2

34

tan50

100

3

4tan504030

1

1

φθφθφθφθ

θ

φθφ

θ

CCC

RR

T

T

T

CT

XIZV

RIZV

V

Z

VI

jjXRZ

Note: The value of the total voltage (=100)is less than the sum of individual voltages (=140 = 60 + 80), which is possible because these voltage values have different phase angles.

Impedance (cont.)

10

( )

( ) ( )( ) ( ) ( )°+−∠=°∠−∠==

−∠=°∠−∠==∠=

−∠=

°≈

=∠

∠==

∠=+=+=

−

−

905.99900995.0

95.900995.0

100

0995.0

84100

1000tan1005

100

100

1000tan10051000100

1

1

φθφθφθφθ

θ

φθφ

θ

LLL

RR

T

T

T

LT

XIZV

RIZV

V

Z

VI

jjXRZ

( ) ( )222 1

1

1

1

RCCR

R

V

V

T

R

ωω+

=+

=

11

Summary• The resistive (due to DC) and reactive effects (due to AC) must be

combined by phasors (or complex numbers).

• For series or parallel circuits, all the impedances are combined to find ZT.

• The total impedances are calculated just as resistances are.

• For a “linear time-invariant circuit”, the Network theorems (mesh analysis, nodal analysis, Thevenin’s theorem etc), KVL, or KCL that were used in solving DC circuits can also be used for determining sinusoidal (AC) steady-state response using the phasor notations.

• For a linear time-varying circuits or nonlinear circuits, the steady-state response to a sinusoidal input is usually not sinusoidal.

– Circuit with R=R(t) or C=C(t)

• Time-invariant circuit: circuit elements are time-invariant (voltage or currents of circuit elements to inputs do not change with time) or independent source.

12

Nodal analysis with impedances

I2

Z1

Z2

Z3 IB

I4

I3IA

Z4

Z5

I1

I5

V1V2

V3

=

++−

++−

−−

+

=

++−+→−−+−=

=+

++−→=−−−−→−+=−

=−−

+→−+−=

B

A

BB

AA

I

I

V

V

V

ZZZZZ

ZZZZZ

ZZZZ

IVZZZ

VZ

VZZ

V

Z

VV

Z

VVI

VZ

VZZZ

VZZ

VV

Z

V

Z

VV

Z

VV

Z

V

Z

VV

IVZ

VZ

VZZZ

VV

Z

VVI

0

11111

11111

1111

11111

011111

0

1111

3

2

1

54141

44322

1221

3541

24

115

3

4

32

1

31

34

2432

124

32

3

2

2

21

4

32

3

2

2

21

31

22

1211

31

2

21

415

432

12

IIII

III

III

B

A

+=++=+=

KCL on the principle nodes

Inductor, capacitor, or resistor

13

Mesh analysis with impedances

Z1

Z2

Z3

VBVA

Mesh current IA Mesh current IB

B

( ) ( )( ) ( )

−=

+−−+

−=++−→−++==−+→−++−=

B

A

B

A

BBAABBB

ABABAAA

V

V

I

I

ZZZ

ZZZ

VZZIZIZIIZIV

VZIZZIZIIZIV

232

221

23223

22121

0

0

( )( ) 23

21

0

0

ZIIZIV

ZIIZIV

ABBB

BAAA

−++=−++−=

KVL on the two meshes

+_

+_

14

Frequency Response

( ) ( )( ) ( )

( ) ( )

( )

( )

( )( )

( ) ( ) ( ) ( )

( ) SSLS

L

SSSLS

SLT

LS

S

L

T

L

S

LT

LT

LL

SLS

L

S

LV

ZZZZZZZ

ZZ

ZZZ

Z

ZZZZZZZ

ZZZZj

ZZZZ

ZZZZ

jZ

ZZZ

Zj

ZZ

Zj

ZZZZZZZ

ZZ

j

jjH

V

VV

V

111

VV where

V

V

2121

2

21

2

2121

21

21

21

12

2121

2

++++=

++++++++

=+

+++

=

+

++

=+

=

++++==

ω

ωωω

ωωω

• Frequency response of a circuit provide a measure of how the circuit responds to sinusoidal inputs of arbitrary frequency.

• Given the input signal amplitude, phase, and frequency, knowledge of the frequency response of a circuit permits the computation of the output signal.

• To express the frequency response of a circuit in terms of variation in output voltage as a function of source voltage, we use

VL is a phase-shifted and amplitude-scaled version of VS

with the frequency unchanged.

15

Filters• Filtration system

– Filter: eliminate impurities from drinking water or outdoor air– Sunglass: filter out eye-damaging ultraviolet radiation– Electrical filter: attenuate (i.e. reduce in amplitude) or eliminate signals of unwanted

frequencies

• Filter types (in terms of amplitude response vs. frequency)– low-pass – pass low frequencies and attenuate high frequencies.– high-pass – pass high frequencies and attenuate low frequencies.– band-pass –pass only a specific band of frequencies from its input to its output.– band-stop –block or severely attenuate only a specific band of frequencies.– The filters can be thought of as a frequency-dependent voltage divider, since the

amount of output voltage is a function of frequency.

• Passive filters: filters made up of capacitor, inductor, and resistors, that is, passive components, which means they do not generate voltages or currents.

– Voltage or current source are not passive but active.

16

Low-Pass Filters

• low-pass filter – pass lower frequencies (-> pass band) and reduce the amplitude of higher frequencies.

• The filter passes the audio signal (lower frequency components, passband) but attenuates radio frequencies (higher frequency components, stopband).

• Cutoff frequency: the frequency at which the attenuation of a filter reduces the output amplitude to 70.7% of its value in the pass band.

( ) ( )( )

( ) ( )( )( )

( )( ) ( )

( )( ) ( ) ( )

RC

RCRCj

j

RCj

j

j

j

RCRC

RCjCjR

Cj

ZZ

Z

j

jjH

i

o

i

o

i

o

CR

C

i

oV

1filter a offrequency cutoff

arctanarctan0V

V of Phase

1

1

V

V

V

V of Magnitude

arctan1

01

1

1

1

1

V

V

0

2

2

==

−=−=

+==

∠+

∠→

+=

+=

+==

ω

ωωωω

ωωω

ωω

ωω

ωωω

ωωω

Voltage divider

17

Example of Low-Pass Filters

( )( ) ( )

( ) ( )

( ) ( ) ( )[ ]( )( )

( )( ) ( ) ( )

( )( ) ( )

( )( ) ( )

( )( )( )( ) ( )[ ] ( )[ ]

( ) ( ) ( )( )rad/s6001

1

1

1

2

1

1

1

1arctan1arctan0V

V of Phase

0V

V:

1

1

2

1

V

V:,

1

1

V

V:0

1

1

V

V

V

V of Magnitude

1arctan1

01

1

1

1

1

11

1

V

V

1

1|| and

0222

20

2

22

22

1

=

′+

′=→

′+=

′+′+

′+′−=′+′−=

→∞→

′+==

′+==

′+′+==

′+′∠′+′+

∠→

+=

′+′+=

′+′+

=++′

=

++′

+=′+′

′=

+==′+=′

LLL

LLS

L

S

L

LS

L

LS

L

LS

L

S

L

LL

LLL

L

LL

L

L

L

L

L

S

L

L

LLS

R

R

CRRRCRRR

RRCRRRCRj

j

j

j

RRj

j

RRj

j

CRRRj

j

j

j

RRCRCRRR

RCjCRjRRCRRjRR

R

RCRjR

R

CRj

RR

CRj

R

ZR

Z

j

j

CRj

R

CjRZRRR

ωω

ωωωωωωω

ωωωω

ωωω

ωωω

ωω

ωω

ωωω

ωω

ωωω

ωω

The magnitude of frequency response of this filter is not equal to 1 at the passband.

18

High-Pass Filters• high-pass filter – pass higher frequencies (-> pass band) and reduce the amplitude of lower frequencies (-> stop band).

( ) ( )( )

( )( ) ( )

( )( )

( )( ) ( )

( )( ) ( )

( ) RCRC

RCRC

j

j

RC

RC

j

j

j

j

RCRC

RC

RCj

RCj

CjR

R

ZZ

Z

j

jjH

i

o

i

o

i

o

CR

R

i

oV

1

2

1

1

arctan2

V

V of Phase

1

V

V

V

V of Magnitude

arctan1

2

11V

V

02

2

2

=→=+

−=

+==

∠+

∠→

+=

+=

+==

ωω

ωωπωω

ω

ωωω

ωω

ωω

πωω

ωωω

ωω

19

Band-Pass Filters, & Resonance

( ) ( )( ) ( )

( )( )

( )( )

( )( ) ( ) ( )

( ) ( )( )( )

( ) ( ) arctanarctan2or

1

arctan2V

V of Phase

11or

1

V

V

V

V of Magnitude

constant :A

filter, theof passband thedetermine that sfrequencie two:,

111

11V

V

21

2

22

21

222

21

212

2

ωωωωπω

ωπωω

ωωωω

ωωω

ωωω

ωω

ωωωωωω

ωωω

ωωω

ωωωω

ωω

−−

−−=

++

+−==

++→

++−=

++=

++=

++==

LC

RC

j

j

A

RCLC

RC

j

j

j

j

jj

jA

CRjLC

CRj

CRjLCj

CRj

LjCjR

R

ZZZ

Z

j

jjH

i

o

i

o

i

o

CLR

R

i

oV

• band-pass filter – pass frequencies within a certain frequency range (-> pass band) and reduce the amplitude of the other frequencies (-> stop band).

20

Band-Pass Filters, & Resonance (cont.)

( ) ( )( ) ( )

( )( ) ( )

( )( ) ( )

1/2. offactor aby decreased hasfilter theofoutput at thepower the

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12

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1

11

1

12

2

1V

V

22

2

QB

Q

RCQ

LCf

LC

Qjj

Qj

jj

j

CRjLCj

CRj

j

jjH

n

n

nn

nn

n

nn

n

i

oV

ωζ

ωζ

πω

ωωωωωω

ωζωωωωζω

ωωω

ωωω

≈=

→

==

=→=

++=

++=

++==

The response of second-order filters can be explained by rewriting the frequency response function in the following forms:

21

Tuning• Band-pass filter: Tuning circuits (-> choose desired frequency you want to hear) employed in conventional AM radio by using a variable capacitor.

•As illustrated in the figure below, the variable capacitance C can be adjusted to tune the series LC circuit to resonance at any one of five different frequencies. -> The desired frequency component can be collected by changing the capacitance.

Five AC inputs

( )

( )( ) kHz5001042410239

1

2

1

1

2

1

10424

126

12

=××

==

×=

−−

−

πLCπf

FC

n

Amplitude frequency response function

Frequencies with 500kHz are visible and the others are eliminated or attenuated at the output.

22

Bode Plots

• Bode plots: Frequency response plots of linear systems displayed in the form of logarithmic plots.

• The amplitude ratio is expressed in units of decibel (dB), wherei

o

i

o

A

A

A

A10

dB

log20=

( )( ) ( ) ( ) ( )

( )

0

0

02

02

1

filter a offrequency power,-halfor cutoff,

arctan1

1

arctan1

01

1

1

V

V

ωτω

ωωωωωωωω

ω

====

−∠+

=∠+

∠→+

=

RC

RC

RCRCRCjj

j

i

o

( )( )

( )( ) dB 3

2

1log20

V

V,

dB 01log20V

V,0

10

dB

0

10dB

−===

===

ωωωω

ωωω

j

j

j

j

i

o

i

o

3-dB frequency