Refer to Fig. 16.9, where the op amp is ideal and the excitation nu_SνS is a halfwave rectified 800kHz sinusoid having peak amplitude x0=200mV. (a) Obtain an expression for the Fourier coefficients for the response νL. (b) Calculate the peak amplitudes of the first four non-zero terms in the

Question

Refer to Fig. 16.9, where the op amp is ideal and the excitation [latex] u_S [/latex] is a halfwave rectified [latex] 800 \mathrm{kHz}[/latex] sinusoid having peak amplitude [latex] x_0=200 \mathrm{mV} [/latex]. (a) Obtain an expression for the Fourier coefficients for the response [latex] u_L [/latex]. (b) Calculate the peak amplitudes of the first four non-zero terms in the amplitude-phase series for the response.

Accepted Answer

(a) From Table 16.2, the Fourier coefficients for the excitation are given by

[latex] \begin{aligned} X_k &=\frac{x_0}{4}\left\{\mathrm{sa}\left[\frac{(k+1) \pi}{2} ight]+\operatorname{sa}\left[\frac{(k-1) \pi}{2} ight] ight\}, \\ k &=0, \pm 1, \pm 2, \cdots (16.42) \end{aligned} [/latex]

Using frequency-domain (phasor) methods, we find that the voltage transfer function for the circuit is

[latex] H_ u(j 2 \pi f)=\frac{ ilde{V}_L}{ ilde{V}_S}=\frac{R_2 / R_1}{1+j 2 \pi f R_2 C}. [/latex] (16.43)

The Fourier coefficients for the response are given by

[latex] \begin{aligned} Y_k &=H_ u\left(2 \pi k f_0 ight) X_k, \\ k &=0, \pm 1, \pm 2, \cdots . (16.44) \end{aligned} [/latex]

Table 16.5 gives the peak amplitudes and relative phases of the first four zero terms, where

Table 16.5 Amplitude and phase of the first five terms of the Fourier series for the response of the circuit considered in Example 16.11. The graph does not change perceptibly if the number of terms is doubled
k	0	1	2	3	4
[latex]A_k(\mathrm{~V}) [/latex]	3.183	3.896	1.12	0.000	0.126
[latex] heta_k [/latex]	0.000	-0.677	-1.015	0.000	1.872

[latex] \begin{aligned} &A_0=\left|Y_0 ight|, \quad heta_0=∡Y_0=0, \\ &A_k=2\left|Y_k ight|, \quad heta_k=∡ Y_k ; \quad k=0,1,2, \cdots . ] \end{aligned} [/latex]

Figure 16.10 shows a graph of one period of the excitation and of the approximate response, given by

[latex]\hat{y}(t)=\sum_{k=-4}^4 Y_k \exp \left(2 j k \pi f_0 t ight) [/latex]

Question 16.11: Refer to Fig. 16.9, where the op amp is ideal and the excita...

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