Page 283 - Dynamics and Control of Nuclear Reactors

APPENDIX E Frequency response analysis of linear systems 285

Note that the magnitude is constant at low frequencies and it decreases at a rate of
one decade per decade of frequency at high frequencies. The frequency, at which
the low frequency and high frequency asymptotes intersect, is called the break fre-
quency, ω b , of the frequency response magnitude plot. The magnitude at ω¼1rad/s
1
is equal to p ¼ 0:707. This frequency is called the half-power frequency since
ffiffi
2
2
1
ð
j GjωÞj ¼ . The magnitude of 0.707 is often referred to as the RMS (root-mean-
2
squared) value.
The units for frequency in the above developments are rad/s, but it is also
common practice to use frequency in cycles/s or Hertz (Hz). Frequency ω (rad/s)
is converted to frequency f (Hz or cycles/s) using the relationship.
ω ¼ 2π f (E.21)
There is a direct relationship between system poles and zeroes and the asymptotic
magnitude and phase for systems described by ordinary differential equations.
For example, consider the contribution of a term, 1/(s+a) in a transfer function. Sub-
stitute s¼jω to obtain 1/(jω+a). The term goes to a constant value, 1/a, at low fre-
2
quencies and to 1/jω or equivalently, jω/(ω ), at high frequencies. The term,
2
jω/(ω ), is a negative imaginary number for all values of ω, corresponding to a
phase shift of 90 degrees. Other poles and zeroes would likewise have asymptotic
contributions to the total system frequency response. Table E.1 shows asymptotic
magnitudes and phases for various terms in system transfer functions. These systems,
characterized by the asymptotic relations as shown in Table E.1 [2], are called
minimum phase systems. All systems described by ordinary differential equations
are minimum phase systems except for those containing pure time delays. Some
systems require partial differential equations for their description and they are not
minimum phase systems.

Remark
1. In the literature on control systems analysis, it is common practice to define the magnitude in
terms of the deciBel (dB) given by
2
ð
j GjωÞj in deciBelsÞ ¼ 10log 10 Gjωj (E.22)
ð
ð

dBmagnitude ¼ 20log 10 Gjωð Þj (E.23)
j
One deciBel or 1dB¼one-tenth of a Bel. The unit Bel was named in honor of Graham Bell
and is used to express a power level with respect to a standard power level. Thus the magnitude
2
in Bels is log 10 jG(jωj .
2. If the transfer function has the general form G(s)¼1/(s+a), the parameter a is the break fre-
quency of the magnitude plot. The inverse of this parameter is called the time constant of
the first order system. That is, time constant, τ¼1/a sec.