Page 122 - Modern Control of DC-Based Power Systems

86 Modern Control of DC-Based Power Systems

(Continued)
Factor Impact on Magnitude Plot Impact on the Phase Plot
1
First order factors in For ω . : Straight line For 1 , ω # 10 : gradual
T 10T T
the denominator with slope of 220 dB/ decrease from 0 to

21
dec 290
ð 11jωTÞ
For ω . 10 :0
T
Quadratic factors in the For ω{ω n : 0 dB For ω{ω n :0
numerator
2 Near ω n : Resonant peak, Near ω n : gradual increase
½112ξ jω 1 jω 11
ω n ω n Š whose magnitude is from 0 to 1180 ;the

determined by the value increased profile heavily
of ξ dependent on the value
of ξ
For ωcω n : 140 dB/dec For ωcω n : 1180
Quadratic factors in the For ω{ω n : 0 dB For ω{ω n :0
denominator
2 Near ω n : A resonant peak, Near ω n : gradual decrease
½112ξ jω 1 jω 21
ω n ω n Š whose magnitude is from 0 to 2180 ;the

determined by the value decreased profile heavily
of ξ dependent on the value
of ξ
For ωcω n : 240 dB/dec For ωcω n : 2180
For systems with nonminimum-phase transfer functions, i.e., transfer
functions with neither zeros nor poles in the right-half s-plane, the concepts
of phase margin and gain margin can be used to determine the stability.
The phase margin can be defined as follows:

PM 5 180 1[ (3.3)

where [ is the phase angle of the open-loop transfer function at the frequency
in which the magnitude of the open-loop transfer function is equal to zero.
Gain margin can be also defined as the inverse of the gain of the
open-loop transfer function at ω c , which is the frequency at which the
phase angle of the open-loop transfer function is equal to 2180 degrees:
1
(3.4)
GM 5

ð
Gjω c ÞHjω c Þ
ð
In decibels, GM can be expressed as:
1
(3.5)

GM dB 5 20log 52 20log Gjω c ÞHjω c Þ
ð
ð

ð
ð
Gjω c ÞHjω c Þ