Lecture 10 Sinusoidal steady-state and frequency response

S. Boyd

EE102

Lecture 10 Sinusoidal steady-state and frequency response

? sinusoidal steady-state ? frequency response ? Bode plots

10?1

Response to sinusoidal input

convolution system with impulse response h, transfer function H

PSfrag replacements

u

y

H

sinusoidal input u(t) = cos(t) = ejt + e-jt /2

t

output is y(t) = h( ) cos((t - )) d

0

let's write this as

y(t) = h( ) cos((t - )) d - h( ) cos((t - )) d

0

t

? first term is called sinusoidal steady-state response

? second term decays with t if system is stable; if it decays it is called the transient

Sinusoidal steady-state and frequency response

10?2

if system is stable, sinusoidal steady-state response can be expressed as

ysss(t) =

h( ) cos((t - )) d

0

= (1/2)

h( ) ej(t-) + e-j(t-) d

0

= (1/2)ejt

h( )e-j d + (1/2)e-jt

h( )ej d

0

0

= (1/2)ejtH(j) + (1/2)e-jtH(-j)

= ( H(j)) cos(t) - ( H(j)) sin(t)

= a cos(t + )

where a = |H(j)|, = H(j)

Sinusoidal steady-state and frequency response

10?3

conclusion

if the convolution system is stable, the response to a sinusoidal input is asymptotically sinusoidal, with the same frequency as the input, and with magnitude & phase determined by H(j)

? |H(j)| gives amplification factor, i.e., RMS(yss)/RMS(u) ? H(j) gives phase shift between u and yss

special case: u(t) = 1 (i.e., = 0); output converges to H(0) (DC gain)

frequency response

transfer function evaluated at s = j, i.e.,

H(j) =

h(t)e-jtdt

0

is called frequency response of the system

since H(-j) = H(j), we usually only consider 0

Sinusoidal steady-state and frequency response

10?4

Example

? transfer function H(s) = 1/(s + 1)

? input u(t) = cos t

? SSS output has magnitude |H(j)| = 1/ 2, phase

H(j) = -45

u(t) (dashed) & y(t) (solid)

1

0.5

0

-0.5

PSfrag replacements

-1 0

5

10

15

20

t

Sinusoidal steady-state and frequency response

10?5

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