FREQUENCY RESPONSES

CHE302 LECTURE IX

FREQUENCY RESPONSES

Professor Dae Ryook Yang

Fall 2001

Dept. of Chemical and Biological Engineering

Korea University

DEFINITION OF FREQUENCY RESPONSE

• For linear system

– “The ultimate output response of a process for a sinusoidal input at a frequency will show amplitude change and phase shift at the same frequency depending on the process

characteristics.”

– Amplitude ratio (AR): attenuation of amplitude, – Phase angle ( ): phase shift compared to input – These two quantities are the function of frequency.

Process

Input Output

sin

A ωt Aˆsin(ω φt + )

After all transient effects are decayed out.

t φ

( ) u t

( ) y t

-A Aˆ A

Aˆ

−

φ

ˆ/

A A

BENEFITS OF FREQUENCY RESPONSE

• Frequency responses are the informative representations of dynamic systems

– Audio Speaker

– Equalizer

– Structure

Elec. Signal Sound wave

Raw signal

Processed signal

Expensive speaker

Cheap speaker

Old Tacoma bridge New Tacoma bridge

Old Tacoma bridge

New Tacoma bridge

Adjustable for each frequency

band

– Low-pass filter

– High-pass filter

– In signal processing field, transfer functions are called “filters”.

• Any linear dynamical system is completely defined by its frequency response.

– The AR and phase angle define the system completely.

– Bode diagram

• AR in log-log plot

• Phase angle in log-linear plot

– Via efficient numerical technique (fast Fourier transform, FFT), the output can be calculated for any type of input.

• Frequency response representation of a system dynamics is very convenient for designing a

feedback controller and analyzing a closed-loop system.

– Bode stability

– Gain margin (GM) and phase margin (PM)

• Critical frequency

– As controller gain changes, the amplitude ratio (AR) and the phase angle (PA) change.

– The frequency where the PA reaches –180° is called critical frequency ( ).

– The component of output at the critical frequency will have the exactly same phase as the signal goes through the loop due to comparator (-180 °) and phase shift of the process (-180 °).

– For the open-loop gain at the critical frequency,

• No change in magnitude

• Continuous cycling

– For

• Getting bigger in magnitude

• Unstable

– For

• Getting smaller in magnitude

• Stable

ωc

( ) 1

OL c

K ω =

R ⁺ G_c… G_P

-

Y E

Sign change

Sign change

( ) 1

OL c

K ω >

( ) 1

OL c

K ω <

• Example

– If a feed is pumped by a peristaltic pump to a CSTR, will the fluctuation of the feed flow appear in the output?

– V=50cm³, q=90cm³/min (so is the average of q_i)

• Process time constant=0.555min.

– The rpm of the peristaltic pump is 60rpm.

• Input frequency=180rad/min (3blades)

– The AR=0.01

t q_i

V c_Ai, q_i

Peristaltic pump c_A, q

If the magnitude of fluctuation of q_i is 5% of nominal flow rate, the fluctuation in the output concentration will be about 0.05% which is almost unnoticeable.

( constant)

A

i Ai A

V dc q c qc q

dt = − ≈

/ ( )

( ) ( / ) 1

Ai Ai

A i

C C q

C s

q s =Vs q = V q s

+ +

(ωτ = 100)

OBTAINING FREQUENCY RESPONSE

• From the transfer function, replace s with

– For a pole, , the response mode is .

– If the modes are not unstable ( ) and enough time elapses, the survived modes becomes . (ultimate response)

• The frequency response, is complex as a function of frequency.

jω

( )

( )

G s 

→ G j ω

Transfer function Frequency response

( ) G jω

( ) Re[ ( )] Im[ ( )]

G jω = G jω + j G jω

2 2

( ) Re[ ( )] Im[ ( )]

AR = G jω = G jω + G jω

( )

( ) tan 1 Im[ ( )]/ Re[ ( )]

G j G j G j

φ = R ω = ⁻ ω ω

s = +α jω e^(α+jω)t α ≤ 0

ej t^ω

Re Im

φ

( ) G jω

AR

ω↑

Nyquist diagram

Bode plot

• Getting ultimate response

– For a sinusoidal forcing function – Assume G(s) has stable poles b_i.

– Without calculating transient response, the frequency response can be obtained directly from .

– Unstable transfer function does not have a frequency response because a sinusoidal input produces an unstable output

response.

2 2

( ) ( ) A Y s G s

s ω

= ω

+

1

2 2 2 2

1

( ) ( ) ⁿ

n

A Cs D

Y s G s

s s b s b s

α

ω α ω

ω ω

= = + + + +

+ + L + +

( ) ( ) D C

G j A Cj D G j j R jI

A A

ω ω = ω + ω ⇒ ω = + = +

Decayed out at large t

, ( cos_ul sin ) ˆsin( ) C = IA D = RA ⇒ y = A I ωt + R ωt = A ω φt +

2 2 1

ˆ/ = ( ) and tan ( / ) ( )

AR A A R I G jω φ ⁻ I R G jω

∴ = = + = = R

( ) G jω

• First-order process

• Second-order process

( ) ( 1) G s K

τ s

= +

2 2

( ) (1 )

(1 ) (1 )

K K

G j j

ω j ωτ

ωτ ω τ

= = −

+ +

2 2

( ) 1

N 1

AR G jω

= = ω τ

+

( ) tan (1 ) φ = RG jω = − ⁻ ωτ

( ) 2 2

( 2 1)

G s K

s s

τ ζτ

= + +

2 2

( )

(1 ) 2

G j K ω j

τ ω ζτω

= − +

2 2 2 2

( )

(1 ) ( 2 )

A R G jω K

ω τ ζωτ

= =

− +

1 1

2 2

Im( ( ) ) 2

( ) tan tan

Re( ( )) 1

G j G j

G j

ω ζωτ

φ ω

ω ω τ

− −

= = = −

R −

ζ >1

Corner freq.

• Process Zero (lead)

• Unstable pole

( ) _a 1 G s =τ s +

( ) 1 _a

G jω = + jωτ

2 2

( ) 1

N a

AR = G jω = +ω τ

( ) tan (1 _a) φ =RG jω = ⁻ ωτ

( ) 1

( 1)

G s = τs

− +

2 2

1 1

( ) (1 )

1 1

G j j

ω j τω

τω τ ω

= = +

− +

2 2

( ) 1

1 A R G jω

= = ω τ

+

1 I m ( ( )) 1

( ) tan tan

Re( ( ) ) G j G j

G j

φ ω ω ωτ

ω

− −

= R = =

• Integrating process

• Differentiator

• Pure delay process

( ) 1

G s = As ^{G j( ) 1 1 j}

jA A

ω = ω = − ω ( ) 1

ARN G j ω A

= = ω

1 1

( ) tan ( )

0 2

G j π

φ ω

ω

= = − − = −

R ⋅

( ) ^s G s =e^−θ

( ) ^j cos sin

G jω = e^{− θ ω} = θω − j θω

( ) 1

AR = G jω =

( ) tan 1 tan

φ = RG jω = − ⁻ θω = −θω ( )

G s = As ^{G j(} ω⁾= ^jAω ( )

ARN = G jω = Aω

1 1

( ) tan ( )

0 2

G j π

φ ω

ω

= = − =

R ⋅

SKETCHING BODE PLOT

• Bode diagram

– AR vs. frequency in log-log plot – PA vs. frequency in semi-log plot – Useful for

• Analysis of the response characteristics

• Stability of the closed-loop system only for open-loop stable

systems with phase angle curves exhibit a single critical frequency.

1 2 3

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

a b c

G s G s G s G s = G s G s G s L

L 1 2 3

( ) ( ) ( )

( )

( ) ( ) ( )

a b c

G j G j G j G j G j G j G j

ω ω ω

ω = ω ω ω L

L

1 2 3

( ) ( ) ( )

( )

( ) ( ) ( )

a b c

G j G j G j G j G j G j G j

ω ω ω

ω = ω ω ω L

L

1 2 3

( ) ( ) ( ) ( )

( ) ( ) ( )

a b c

G j G j G j G j

G j G j G j

ω ω ω ω

ω ω ω

= + + +

− − − −

R R R R L

R R R L

• Amplitude Ratio on log-log plot

– Start from steady-state gain at =0. If G_OL includes either integrator or differentiator it starts at or 0.

– Each first-order lag (lead) adds to the slope –1 (+1) starting at the corner frequency.

– Each integrator (differentiator) adds to the slope –1 (+1) starting at zero frequency.

– A delays does not contribute to the AR plot.

• Phase angle on semi-log plot

– Start from 0° or -180° at =0 depending on the sign of steady- state gain.

– Each first-order lag (lead) adds 0° to phase angle at =0, adds -90° (+90°) to phase angle at = , and adds -45° (+45°) to

phase angle at corner frequency.

– Each integrator (differentiator) adds -90° (+90°) to the phase angle for all frequency.

– A delay adds – to phase angle depending on the frequency.

∞

∞

ω

ω

ω

θω

ω

Examples

1. 2.

(10 1)(5 1)( 1) G s K

s s s

= + + +

5(0.5 1) 0.5

( ) (20 1)(4 1) s e s

G s s s

+ −

= + +

G₁

G₂ G₃

G₄ G₅

G₁ G₂ G₃

3. PI: 5. PID:

4. PD:

( ) _C 1 1

I

G s K

τ s

 

=  + 

 

( ) _C 1 1 _D

I

G s K s

s τ τ

 

=  + + 

 

Slope=-1 Slope=+1

1/ at 0

Notch I D

ω = τ τ φ = °

1/ at 45

b I

ω = τ φ = − °

Slope=-1

( )

( ) _C 1 _D G s = K +τ s

Slope=+1

1/ at 45

b D

ω = τ φ = − °

NYQUIST DIAGRAM

• Alternative representation of frequency response

• Polar plot of ( is implicit)

– Compact (one plot)

– Wider applicability of stability analysis than Bode plot

– High frequency characteristics will be shrunk near the origin.

• Inverse Nyquist diagram: polar plot of

– Combination of different transfer function components is not easy as with Nyquist diagram as with Bode plot.

Re Im

φ

( ) G jω

AR

ω↑

Nyquist diagram

( ) Re[ ( )] Im[ ( )]

G jω = G jω + j G jω ( )

G jω ω

1/G j( ω)