Frequency Response

Li ZhuSoutheast University

CH 16 FREQUENCY RESPONSE

The frequency response of a circuit is the variation in its behavior with change in signal frequency.

L j Lω=Z1

C jCω

= −Z

( )

( )

0 1 1 11 1 1 2

1 1

i

B

j CR j C j RC j fRC

j f f

ωωω ω π

= = = =+ + +

=+

VHV

( )2

11 ( / )B

H ff f

=+

12Bf RCπ

=

( ) 1tanB

fH ff

−∠ = −

16.1 Resonant Circuits

( )[ ] 0Im >ωjZ : inductive

( )[ ] 0Im <ωjZ : capacitive

( )[ ] 0Im =ωjZ : resistive

Resonance occurs in any circuit that has at least one inductor and one capacitor.

Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, thereby resulting in a purely resistive impedance.

1=ω

L=1 H

C=1 FResonance

16.1.1 Series resonance

1.Resonant frequency

( )[ ] 0Im =ωjZ

( )[ ] 0arg =ωjZ( )

CjLjRjZω

ωω 1−+=

01=−

CL

ωω rad/s 1

0 LC=ω

Hz 2

10 LC

fπ

=Resonant Frequency

2. Characteristics

Impedance

minimum:Z01

00 =−=

CLX

ωω

00

1LC

ωω

=def

Q =

Quality factor

0

0

1 1L LR CR R Cω

ω= =

R jX= +Z 22 XRZ +=

01 LC

ω =

LC

=

2. Characteristics

Current

s s

R= =

V VIZ I: maximum

Voltage s

sRR

= =V V

0L j Lω=V I

0C j Cω=

IV

0X L C= + =V V V ( )0C

s

VQ

Vω

=

, RV I,sV

LV

CV

R R=V I

0sj L

Rω=

VsjQ= V

0

1 sjC Rω

= −V

sjQ= − V

VR: maximum

+

-VX

2. Characteristics

Power and Energy

sin 0eff effQ V I ϕ= =2

effL

VQ Q

R=

2eff

C

VQ Q

R= −

pL<0

pc>0Reactive power

Quality factor

Power and Energy

( ) ( )tLitW LL2

21

= ( ) ( )212C CW t Cv t=

( ) ( ) ( )tWtWtW CLS +=

( ) 2 20

0

1 1 22 2R m mW I RT RI πω

ω= =

( )( )0

02ωω

πR

S

WW

Maximum energy stored in the circuit2Energy dissipated by the circuit in one period at resonance

Q π=

LV

CV

RV,V , I

( )0ωSW=

Q=R

L0ω=

2 2 21 12 2m mQ CV LI= =

3. Frequency response

The frequency response of a circuit is the variation in its behavior with change in signal frequency.

( )ωjZ

( ) 1j R j L jC

ω ωω

= + −Z

22 XR +=Z

( )R

X ωϕ 1tan −=

( )CL XXjR ++=

( )ωjXR +=

(Q ≥ 10)

3. Frequency response

( )ωI

( ) ( )mVI

Z jω

ω=

12 ωω −=B

( )22 1mV

R L Cω ω=

+ −

Half-power frequencies

The quality factor of a resonant circuit is the ratio of its resonant frequency to its bandwidth.

Selectivity

QLR 0ω==

( )( )22 1

mVIR L C

ωω ω

=+ −

22 2 2 0

0

mV

R Q R ωωω ω

=

+ −

22 0

0

1

1

mVR

Q ωωω ω

=

+ −

2

0

0

2

0

1

1

−+

=

ωω

ωωQ

I

ω

( )ωI 1. R

Q2. L , C

CLX

ωω 1

−=

−=

ωω

ω2

0L

QZ

Medium selectivity

Least selectivity

Greatest selectivity

1 LQR C

=rad/s 10 LC=ω

0

0

1LQR CRω

ω

= =

( )V ω

3. Frequency response

( ) ( )2 2

2 22 2

1 1 11L

LV QVV LI

R L QC

ωω ω ω

ωω η η

= = = + − + −

( ) ( )

( )2 22 2 22 1 1

C

I V QVVC QC R L

C

ωω

ω η ηω ωω

= = = + −+ −

0ωωη = Normalized frequency

0CdVdη

=

0LdVdη

=

max maxL CV V=

Q<0.707, no peak value

21 211 Q

−=η 1<2

1 >Q

( )max 1

211

4

C CQVV V

Q

η= =−

QV>

112

22

2

2 >−

=Q

Qη

( )max 2

211

4

L LQVV V QV

Q

η= = >−

21 >Q

VC/VVL/V

Note that at resonance:

The impedance is purely resistive, thus, Z=R. In other words, the LC series combination acts like a short circuit, and the entire voltage is across R.

The voltage V and the current I are in phase, so that the power factor is unity.

The magnitude of the input impedance Z(ω) is minimum.

The inductor voltage and capacitor voltage can be much more than the source voltage.

16.1.2 Parallel resonance

( ) 1 1j j C jR L

ω ωω

= + −Y

( )Im 0jω = Y

rad/s10 LC=ω ( )0

1YR

ω =

R R= =

VI I

jR C L= − I

( )0

RY ω

= =IV I

0L j Lω=

VI jQ= − I

0 0C j C j CR jR C L jQω ω= = = =I V I I I

0

1j RLω

= − I

×

×

( )2

2 0

0

1

1

V IR

Q

ωωω

ω ω

=

+ −

QRCB 0

121 ω

ωω ==−=

( )V I Z ω=

16.1.2 Parallel resonance

Frequency response:

(Q ≥ 10)

16.1.2 Parallel resonance

( )[ ] 0Im =ωjY

( )LjR

CjjYω

ωω+

+=1

( )002

02

0 =++

− CLR

Lω

ωω

LCR

LCCLCRL 2

2

2

0 11−=

−=ω C

LR <

( )( ) L

CRLR

RGjY eq =+

== 20

20 ωω ( ) ( )0

0

SS

I LV IY j CR

ωω

= =

( ) ( )2 0 2 0 0 2SLI V Y j I C

CRπω ω ω= = ∠

( ) ( ) ( )1 0 1 0 1 0 1SLI V Y j Y j I

CRω ω ω ϕ= = ∠

SIV

1I 2I1ϕ

+

-

V

16.2 FilterA filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.

Lowpass filter Highpass filter

Cutoff frequency

Bandpass filter Bandstop filter

16.2.1 Passive filters

1. Lowpass Filter

( )RCjCjR

Cj

i ωωωω

+=

+==

11

110

VV

H

( )2

11

1222=

+=

CRH

c

cω

ω

RCc1

=ω

A lowpass filter is designed to pass only frequencies from dc up to the cutoff frequency cω

2. Highpass Filter

( )RCj

RCjCjR

R

i ωω

ωω

+=

+==

110

VV

H

RCc1

=ω

A highpass filter is designed to pass all frequencies above its cutoff frequency cω

3. Bandpass Filter

A bandpass filter is designed to pass all frequencies within a band of frequencies, w1<w<w2 .

( ) ( )CLjRR

i ωωω

10

−+==

VV

H

LC1

0 =ω

4. Bandstop Filter

A bandstop filter is designed to stop or eliminate all frequencies within a band of frequencies, w1<w<w2 .

( ) ( )( )CLjR

CLj

i ωωωωω1

10

−+−

==VV

H

LC1

0 =ω

16.2.2 Active filters

1. First-order lowpass filterZf

( )i

f

i ZZ

−==VV

H 0ω

( )ffi

f

RCjRR

ωω

+−=

11H

ffc CR

1=ω

2. First-order highpass filter Zi

( )i

f

i ZZ

−==VV

H 0ω

( )ii

fi

ii

f

RCjRCj

CjRR

ωω

ωω

+−=

+−=

11H

iic CR

1=ω

3. Bandpass Filter

Low-pass filter (w2)

High-pass filter (w1<w2)

Invertervi vo

4. Bandreject (or notch) FilterLow-pass filter sets

w1

High-pass filter sets

w2>w1

Summing amplifier

vivo=v1+v2

v1

v2