CHAP. 8]

8.11

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

175

HIGHER-ORDER ACTIVE CIRCUITS

Application of circuit laws to circuits which contain op amps and several storage elements produces, in general, several ﬁrst-order diﬀerential equations which may be solved simultaneously or be reduced to a higher-order input-output equation. A convenient tool for developing the equations is the complex frequency s (and generalized impedance in the s-domain) as used throughout Sections 8.5 to 8.10. Again, we assume ideal op amps (see Section 7.16). The method is illustrated in the following examples. EXAMPLE 8.13 Find HðsÞ ¼ V2 =V1 in the circuit of Fig. 8-41 and show that the circuit becomes a noninverting integrator if and only if R1 C1 ¼ R2 C2 . Apply voltage division, in the phasor domain, to the input and feedback paths to ﬁnd the voltages at the terminals of the op amp. At terminal A: At terminal B: But VA ¼ VB .

1 V 1 þ R1 C1 s 1 R2 C2 s V VB ¼ 1 þ R2 C2 s 2

VA ¼

Therefore, V2 1 þ R2 C2 s ¼ V1 ð1 þ R1 C1 sÞR2 C2 s

Only if R1 C1 ¼ R2 C2 ¼ RC do we get an integrator with a gain of 1=RC ð V2 1 1 t ; v2 ¼ ¼ v dt RC 1 1 V1 RCs EXAMPLE 8.14 The circuit of Fig. 8-42 is called an equal-component Sallen-Key circuit. Find HðsÞ ¼ V2 =V1 and convert it to a diﬀerential equation. Write KCL at nodes A and B. VA  V1 VA  VB þ þ ðVA  V2 ÞCs ¼ 0 R R VB  VA þ VB Cs ¼ 0 At node B: R Let 1 þ R2 =R1 ¼ k, then V2 ¼ kVB . Eliminating VA and VB between the above equations we get At node A:

V2 k ¼ V1 R2 C 2 s2 þ ð3  kÞRCs þ 1 R2 C 2

d 2 v2 dv þ ð3  kÞRC 2 þ v2 ¼ kv1 dt dt2

EXAMPLE 8.15 In the circuit of Fig. 8-42 assume R ¼ 2 k, C ¼ 10 nF, and R2 ¼ R1 . By substituting the element values in HðsÞ found in Example 8.14 we obtain

Find v2 if v1 ¼ uðtÞ.

V2 2 ¼ V1 4  1010 s2 þ 2  105 s þ 1 d 2 v2 dv þ 5  104 2 þ 25  108 v2 ¼ 5  109 v1 dt dt2 The response of the preceding equation for t > 0 to v1 ¼ uðtÞ is v2 ¼ 2 þ et ð2 cos !t  2:31 sin !tÞ ¼ 2 þ 3:055et cos ð!t þ 130:98Þ where  ¼ 25 000 and ! ¼ 21 651 rad/s. EXAMPLE 8.16 Find conditions in the circuit of Fig. 8-42 for sustained oscillations in v2 ðtÞ (with zero input) and ﬁnd the frequency of oscillations. In Example 8.14 we obtained V2 k ¼ V1 R2 C 2 s2 þ ð3  kÞRCs þ 1

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