The electrical engineering handbook CH011

The electrical engineering handbook

Neudorfer, P “Frequency Response”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

11 Frequency Response

11.1 Introduction 11.2 Linear Frequency Response Plotting 11.3 Bode Diagrams

11.4 A Comparison of Methods

11.1 Introduction

systems to be “the frequency-dependent relation in both gain and phase difference between steady-state sinu-soidal inputs and the resultant steady-state sinusinu-soidal outputs” [IEEE, 1988] In certain specialized applications,

back to the fundamental definition The frequency response characteristics of a system can be found directly

H(jw) is a complex quantity Its magnitude, *H(jw)*, and its argument or phase angle, argH(jw), relate, respectively, the amplitudes and phase angles of sinusoidal steady-state input and output signals Using the terminology of Fig 11.1, if the input and output signals are

x ( t ) = X cos ( w t + Qx)

y ( t ) = Y cos ( w t + Qy)

Y = * H ( j w ) * X

Qy = arg H ( j w ) + Qx

H s K N s

D s

( )

=

-1

H j K N j

( )

w

Paul Neudorfer

Seattle University

The phrase frequency response characteristics usually

implies a complete description of a system’s sinusoidal

steady-state behavior as a function of frequency Because

frequency response characteristics cannot be graphically

dis-played as a single curve plotted with respect to frequency

sep-arately plotted as functions of frequency Often, only the

magnitude curve is presented as a concise way of

character-izing the system’s behavior, but this must be viewed as an incomplete description The most common form

magnitude versus phase for the system function Frequency again is a parameter of the resultant curve, sometimes shown and sometimes not

Frequency response techniques are used in many areas of engineering They are most obviously applicable

to such topics as communications and filters, where the frequency response behaviors of systems are central to

an understanding of their operations It is, however, in the area of control systems where frequency response techniques are most fully developed as analytical and design tools The Nichols chart, for instance, is used exclusively in the analysis and design of feedback control systems

The remaining sections of this chapter describe several frequency response plotting methods Applications

11.2 Linear Frequency Response Plotting

example, consider the transfer function

phase found

FIGURE 11.1 A single-input/single-output lin-ear system.

H s

( )

,

=

160,000

2

220 160 000

H j

H j

H j

arg ( ) tan

,

w

w

w

=

=

=

-160 000

220 160 000

160 000

160 000 220

220

160 000

2

2 2 2

1

2

* *

11.3 Bode Diagrams

A Bode diagram consists of plots of the gain and phase of a transfer function, each with respect to logarithmically

definition

This definition relates to transfer functions which are ratios of voltages and/or currents The decibel gain

information was popularized by H.W Bode in the 1930s There are two main advantages of the Bode approach The first is that, with it, the gain and phase curves can be easily and accurately drawn Second, once drawn, features of the curves can be identified both qualitatively and quantitatively with relative ease, even when those features occur over a wide dynamic range Digital computers have rendered the first advantage obsolete Ease

of interpretation, however, remains a powerful advantage, and the Bode diagram is today the most common method chosen for the display of frequency response data

A Bode diagram is drawn by applying a set of simple rules or procedures to a transfer function The rules relate directly to the set of poles and zeros and/or time constants of the function Before constructing a Bode

gain of one For instance:

Figure 11.2 Linear frequency response curves of H(jw).

* * * * H dB dB= H = 20 log ( )10 H j w

H s K s

s s

s

s s z

p

z p

z p

z p

( )

/ ( / ) ( )

+

+ +

w w

w w

w w

t t

1 1

1 1

In the last form of the expression, tz =1/wz and tp =1/wp tp is a time constant of the system and s = –wp is the

corresponding natural frequency Because it is understood that Bode diagrams are limited to sinusoidal

of the four terms K¢, stz + 1, 1/s, and 1/(stp + 1) As described in the following paragraph, the frequency response

effects of these individual terms are easily drawn To obtain the overall frequency response curves for the transfer

function, the curves for the individual terms are added together

and denominator polynomials of the transfer function The factorization results in four standard forms These

are (1) a constant K; (2) a simple s term corresponding to either a zero (if in the numerator) or a pole (if in

the denominator) at the origin; (3) a term such as (st + 1) corresponding to a real valued (nonzero) pole or

pair of complex conjugate poles or zeros The Bode magnitude and phase curves for these possibilities are

displayed in Figs 11.3–11.5 Note that both decibel magnitude and phase are plotted semilogarithmically The

equal distance The magnitude axis is given in decibels Customarily, this axis is marked in 20-dB increments

Positive decibel magnitudes correspond to amplifications between input and output that are greater than one

(output amplitude larger than input) Negative decibel gains correspond to attenuation between input and

output

Figure 11.3 shows three separate magnitude functions Curve 1 is trivial; the Bode magnitude of a constant

K is simply the decibel-scaled constant 20 log10 K, shown for an arbitrary value of K = 5 (20 log10 5 = 13.98).

Phase is not shown However, a constant of K > 0 has a phase contribution of 0° for all frequencies For K <

0, the contribution would be ±180° (Recall that –cos q = cos (q ± 180°) Curve 2 shows the magnitude frequency

response curve for a pole at the origin (1/s) It is a straight line with a slope of –20 dB/decade The line passes

through 0 dB at w = 0 rad/s The phase contribution of a simple pole at the origin is a constant –90°, independent

of frequency The effect of a zero at the origin (s) is shown in Curve 3 It is again a straight line that passes

through 0 dB at w = 0 rad/s; however, the slope is +20 dB/decade The phase contribution of a simple zero at

s = 0 is +90°, independent of frequency

Figure 11.3 Bode magnitude functions for (1) K = 5, (2) 1/s, and (3) s.

of the form 1/(s/w p + 1) Exact plots of the magnitude and phase curves are shown as dashed lines Straight

line approximations to these curves are shown as solid lines Note that the straight line approximations are so

good that they obscure the exact curves at most frequencies For this reason, some of the curves in this and later figures have been displaced slightly to enhance clarity The greatest error between the exact and approximate magnitude curves is ±3 dB The approximation for phase is always within 7° of the exact curve and usually much closer The approximations for magnitude consist of two straight lines The points of intersection between

of Bode gain curves always correspond to locations of poles or zeros in the transfer function

In Bode analysis complex conjugate poles or zeros are always treated as pairs in the corresponding quadratic

(Greek letter zeta) is within the range 0 < z < 1 Quadratic terms cannot always be adequately represented by straight line approximations This is especially true for lightly damped systems (small z) The traditional approach was to draw a preliminary representation of the contribution This consists of a straight line of 0 dB

depending on whether the plot refers to a pair of poles or a pair of zeros Then, referring to a family of curves

contribution of the quadratic term was similarly constructed Note that Fig 11.5 presents frequency response contributions for a quadratic pair of poles For zeros in the corresponding locations, both the magnitude and phase curves would be negated Digital computer applications programs render this procedure unnecessary for purposes of constructing frequency response curves Knowledge of the technique is still valuable, however, in the qualitative and quantitative interpretation of frequency response curves Localized peaking in the gain curve

value of z) is a direct indication of the degree of resonance

Bode diagrams are easily constructed because, with the exception of lightly damped quadratic terms, each contribution can be reasonably approximated with straight lines Also, the overall frequency response curve is found by adding the individual contributions Two examples follow

Example 1

In Fig 11.6, the individual contributions of the four factored terms of A(s) are shown as long dashed lines.

The straight line approximations for gain and phase are shown with solid lines The exact curves are presented with short dashed lines

Example 2

Note that the damping factor for the quadratic term in the denominator is z = 0.35 If drawing the response curves by hand, the resonance peak near the breakpoint at w = 100 would be estimated from Fig 11.5

Figure 11.7 shows the exact gain and phase frequency response curves for G(s).

Figure 11.5 Bode diagram of 1/[(s/wn) 2 + (2z/wn )s + 1]

10

0

-10

-20

-30

-40 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 3 4 5 6 7 8 9 10

z = 0.05

0.10 0.15 0.20 0.25

0.3 0.4 0.5 0.6 0.8 1.0

u = w/wn = Frequency Ratio

w/wn = Frequency Ratio

(a)

(b)

-20

0

-40

-60

-80

-100

-120

-140

-160

-180

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 3 4 5 6 7 8 9 10

z = 0.05

0.10 0.15 0.20 0.25 0.3

0.4 0.5 0.6 0.8 1.0

s

s

4

4

1

1100 10

10

=

s

,

( / ) ( / ) ( )( / )

+

1000 500

70 10 000

50 500 1

100 2 0 35 100 1

11.4 A Comparison of Methods

This chapter concludes with the frequency response of a simple system function plotted in three different ways

Example 3

Figure 11.8 shows the direct, linear frequency response curves for T(s) Corresponding Bode and Nyquist

Figure 11.7 Bode diagram of G(s).

T s

( )

=

10

100 200 300

7

Figure 11.8 Linear frequency response plot of T(s).

Figure 11.9 Bode diagram of T(s).

Figure 11.10 Nyquist plot of T(s).

Decade: Synonymous with power of ten In context, a tenfold change in frequency.

ratio of two powers is 10 log10(P1/P2)

steady-state sinusoidal inputs and the resultant steady-steady-state sinusoidal outputs

transfer function referred to as ordinates of logarithmic loop gain and abscissae of loop phase angle

and the imaginary part of the transfer function on the ordinate

the excitation frequency is near a natural frequency of the system

Related Topics

2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals • 100.3 Frequency Response Methods: Bode Diagram Approach

References

R.C Dorf, Modern Control Systems, 4th ed., Reading, Mass.: Addison-Wesley, 1986.

IEEE Standard Dictionary of Electrical and Electronics Terms, 4th ed., The Institute of Electrical and Electronics

Engineers, 1988

D.E Johnson, J.R Johnson, and J.L Hilburn, Electric Circuit Analysis, 2nd ed., Englewood Cliffs, N.J.:

Prentice-Hall, 1992

B.C Kuo, Automatic Control Systems, 4th ed., Englewood Cliffs, N.J.: Prentice-Hall, 1982.

K Ogata, System Dynamics, Englewood Cliffs, N.J.: Prentice-Hall, 1992.

W.D Stanley, Network Analysis with Applications, Reston, Va.: Reston, 1985.

M.E Van Valkenburg, Network Analysis, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1974.

Further Information

Good coverage of frequency response techniques can be found in many undergraduate-level electrical engi-neering textbooks Refer especially to classical automatic controls or circuit analysis books Useful information can also be found in books on active network design

Examples of the application of frequency response methods abound in journal articles ranging over such diverse topics as controls, acoustics, electronics, and communications