Summary Week 8

1. Connection with Transfer Function:
Given:  a transfer function G(s),
Amplitude Ratio,  AR  = B/A = |G(iw)| , magnitude of G(iw)  and
Phase Shiftf(w) = arg( G(iw) ) , argument of G(iw)
2. Frequency Response Plots of Elementary Transfer Functions
a) Gain

 Transfer Function : K LM vs w : horizontal line at 20 log |K| dB Phase vs w : if K>0,   horizontal line at 0o  if K<0,   horizontal line at -180o

b) First Order Lag

 Transfer Function : 1/(ts+1) LM vs w : -low frequency approx a horizontal line at 0 dB  -high frequency approximated by a line sloping at         -20dB/ decade  -at w=1/t, LM = -3dB Phase vs w : -low frequency approx a horizontal line at 0o  -high frequency approx a horizontal line at -90o  -at w=1/t, phase = -45o with a slope= -66o/decade

 Transfer Function : ts+1 LM vs w : -low frequency approx a horizontal line at 0 dB  -high frequency approximated by a line sloping at         +20dB/ decade  -at w=1/t, LM = +3dB Phase vs w : -low frequency approx a horizontal line at 0o  -high frequency approx a horizontal line at +90o  -at w=1/t, phase = +45o with a slope= +66o/decade

d) Second Order Underdamped Lag (z<1)

 Transfer Function : 1/ ( t2s2 + 2zts +1) LM vs w : -low frequency approx a horizontal line at 0 dB  -high frequency approximated by a line sloping at         -40dB/ decade  -at w=(1/t)(1-2z2)1/2 LM attains maximum         where the peak increases as z goes towards zero Phase vs w : -low frequency approx a horizontal line at 0o  -high frequency approx a horizontal line at -180o  -at w=1/t, phase = -90o with a slope= (-132/z)o/decade

e) Delay

 Transfer Function : exp(-td s) LM vs w : horizontal line at 0 dB Phase vs w : starts at 0o and drops exponentially.     ( the larger td is, the earlier the drop occurs )

f) Integrator

 Transfer Function : 1/(ts) LM vs w : -one line sloping at -20 dB/decade  -0 dB at w=1/t Phase vs w : horizontal line at -90o

g) Differentiator

 Transfer Function : t s LM vs w : -one line sloping at +20 dB/decade  -0 dB at w=1/t Phase vs w : horizontal line at +90o
3. Transfer Functions in Series.

Let G(s) = G1(s) G2(s)

then

|G(iw)| = |G1(iw)| |G2(iw)|
or
LM(G) = LM(G1) + LM(G2)
and
arg( G(iw) ) = arg( G1(iw) ) + arg( G2(iw) )
or
f( G ) = f( G1 ) + f( G2 )