14.4 THE BODE PLOT FOR RESONANT FUNCTIONS * In this section, we will extend the Bode method for plotting approximate fre- quency responses (Section 13.4.3) to resonant system functions. Recall that a Bode plot is an approximate sketch of the frequency response, which can be drawn by intuition without the use of a computer. Section 13.4.3 discussed a simple and intuitive method for sketching the Bode plots for general circuits. The method is based on the intuition that a general system function can be written in the form shown in Equation 13.94, which contains four types of terms: 1. A constant term, 2. s terms, 3. real terms of the form (s + a), and 2 + α + ω2 4. quadratic terms of the form (s 2 s o) with complex roots. The Bode method proceeded by drawing the individual magnitude and angle curves for each of the four types of terms in the numerator and denominator of Equation 13.94. The magnitude curves are drawn on log-log scales and the phase curves on log-linear scales. Observing that log-magnitudes and phases add (Equations 13.95 and 13.96), the method concluded by constructing the overall magnitude and phase plots by simply adding together the individual curves. Section 13.4.3 discussed how the real terms (types 1, 2, and 3) could be plotted. This section discusses how we can plot type 4 terms, namely, quadratic terms with complex roots. Once we know how to sketch the plots for each of the four types of terms, we can then sketch any general system function by superposition (see Section 13.4.3). The Bode plot for a quadratic term with complex roots is easily drawn from the insight gained in Section 14.2. There we showed that the low- and high- frequency magnitude and phase asymptotes of second-order system functions yielded insight into the general form of response. It turns out that for a quadratic 2 + α + ω2 term of the form (s 2 s o), the low- and high-frequency asymptotes can be combined to yield a good approximation of the actual curve. Accordingly, the following is a procedure for sketching the form of the 2 + α + ω2 frequency response for a quadratic term of the form s 2 s o, which has complex roots:
Magnitude Plot 1. Sketch the low-frequency asymptote. For our quadratic term, the low-frequency asymptote is given by the horizontal line: | ω |≈ω2 H( j ) o.

808a 1010 180 170 |H| 160 150 140 9

10

FIGURE 14.28 Sketching the frequency response of the 2. Sketch the high-frequency asymptote. The high-frequency asymptote resonant function s2 + 2αs + ωo2. is given by: |H( jω)|≈ω2. This asymptote appears as a line of slope 2 in log-log scales. Figure 14.28a shows these two asymptotes in dashed lines, assuming

4 ωo = 10

α = 500. For comparison, the actual magnitude is also shown as a solid curve. The two straight line asymptotes intersecting at ωo are a good approximation of the magnitude curve. It is also clear from Figure 14.28a that our approximation and the actual curve differ in the vicinity of ωo, and amount by which they differ relates to the peakiness of the curve, which in turn relates to the 4 value of Q. For ωo = 10 and α = 500, ω Q = o = 10. 2α Figure 14.29 plots the frequency response for several values of Q (keeping ωo constant). It is easy to see that the difference between the

808b 180 170 160 10 10 150

|H| 140

actual magnitude (solid curves) and the approximate value from the FIGURE 14.29 Frequency s2 + αs + ω2 Bode splot (dashed curve) at ω becomes substantial for large values response of 2 o for o Q of Q. The exact difference is computed in Equation 14.74. different values of .

Phase Plot 1. Sketch the low-frequency asymptote. The low-frequency asymptote is given by: ◦ ∠H( jω) ≈= 0 . 2. Sketch the high-frequency asymptote. The high-frequency asymptote is given by: ◦ ∠H( jω) ≈= 180 . ◦ 3. Mark ∠H( jωo) = 90 , the angle of the system function at the frequency ωo. 4. Draw a smooth line starting with the low-frequency asymptote, ◦ passing through 90 at ωo, and finishing off at the high-frequency asymptote. Figure 14.28b shows these two asymptotes in dashed lines. For comparison, the actual phase curve is also shown as a solid line.

808c example 14.6 bode plot example Let us sketch the frequency response of the admittance of the second-order circuit in Figure 14.25 using the Bode method. From Equation 14.66, the desired system function is

2 R 1 I s + s + H(s) = z = L LC . V s + R z C LC

For

L = 1mH

C = 10 µF R = 1

we get 4 ωo = 10 rad/s, and Q = 10. Since Q > 0.5, the roots of the characteristic equation are complex and the circuit is resonant. Substituting the numerical quantities into our system function, we get

s2 + 1000s + 108 H(s) = . 105(s + 103)

The system function has three terms: a constant term, a term of the form (s + a), and 2 + α + ω2 a quadratic term of the form (s 2 s o). The Bode construction of the magni- tude curve of the frequency response for the preceding transfer function is shown in Figure 14.30. The corresponding phase construction is shown in Figure 14.31. For ref- erence, the actual frequency response generated using a computer is shown using solid curves.

808d | 12 5 10 12 1012 12 | 10 3 10 10 |10 10 10 10 108 8 108 8 106 6 |s + 10 106 6 104 4 104 4 102 2 102 2 100 0 100 0 10-2 -2 10-2 -2 102 103 104 105 106 102 103 104 105 106 ω ω

| 12 1 8 10 12 10 1

1010 10 |H| 108 8 100 0 106 6 4 + 1000s 10 10 4 2 10-1 -1 |s 102 2 100 0 10-2 -2 10-2 -2 102 103 104 105 106 102 103 104 105 106 ω ω

FIGURE 14.30 Construction of the magnitude curve of the Bode plot. The composite magnitude curve for the transfer function is obtained by subtracting the sum of the magnitude curves of 105 and (s + 103) from the magnitude curve of (s2 + 1000s + 108).

180 180 5 150 3 150

<10 120 120 90 90 60

180 180 8 150 150 120

-90 -90 | 101 102 103 104 105 106 101 102 103 104 105 106 ω ω

FIGURE 14.31 Construction of the phase curve of the Bode plot. 808e