Even though we have focused on frequencyindependent feedback, we can easily generalize to frequency-dependent cases because stability is determined by T( jf ) 5 a( jf )( jf ) regardless of whether...

Even though we have focused on frequencyindependent feedback, we can easily generalize to frequency-dependent cases because stability is determined by T( jf ) 5 a( jf )( jf ) regardless of whether frequency dependence is due to a( jf ), or ( jf ), or both. A classic example is the op amp differentiator of Fig. P7.54. Suppressing Vi , we fi nd the feedback factor as v( jf ) 5 VnyVo 5 1y(1 1 jfyf0), f0 51y(2RC). Ideally, a differentiator gives Aideal( jf ) 5VoyVi52jfyf0, but because of T( jf ) Þ`, the actual gain A( jf ) will depart from the ideal. (a) Assuming a single-pole op amp with av(jf) 5 av0y(1 1 jfyfb), sketch and label the linearized Bode plots of uav( jf )u and u1yv( jf )u if av0 5 105 V/V, fb 5 10 Hz, R 5 10 kV, and C 5 15.9 nF. Hence, use visual inspection for an initial estimation of the frequency fx at which the two curves intersect. (b) Obtain an expression for T(jf) 5 av(jf)v(jf), use trial and error for a more accurate calculation of fx, x, and m, and verify that the circuit is on the verge of oscillation. (c) Calculate D(jfx) and A(jfx), and compare with Aideal(jfx). What are the magnitude and phase errors at this frequency?