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

c) First Order Lead
 

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 )