Analog VLSI and Neural Systems

We have done integration and differentiation with simple, single–timeconstant circuits that had τs + 1 in the denominator of their transfer functions. These systems gave an exponentially damped response to step or impulse inputs. In Chapter 8, we showed how a second-order system can give rise to a sinusoidal response. In this chapter, we will discuss a simple circuit that can generate a sinusoidal response. We call this circuit the second-order section; we can use it to generate any response that can be represented by two poles in the complex plane, where the two poles have both real and imaginary parts. With this circuit, we can adjust the positions of the complex-conjugate poles anywhere in the plane. The second-order circuit is shown in Figure 11.1; it contains two cascaded follower–integrator circuits and an extra amplifier. The capacitance C is the same for both stages (C1 = C2 = C), and the transconductance of the two feed-forward amplifiers, A1 and A2, are the same: G1 = G2 = G (approximately—if G is defined as the average of G1 and G2, small differences will have no first-order effect on the parameters of the response). We obtain an oscillatory response by adding the feedback amplifier A3. This amplifier has transconductance G3, and its output current is proportional to the difference between V2 and V3, but the sign of the feedback is positive; for small signals, I3 is equal to G3(V2 − V3). If we reduce the feedback to zero by shutting off the bias current in A3, each follower–integrator circuit will have the transfer function given