University of Southern C alifornia School Of Engineering Department Of Electrical Engineering

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1 University of Southern alifornia School Of Engineering Department Of Electrical Engineering EE 348: Homework Assignment #02 Spring, 2001 (Due 02/01/2001) homa Problem #05: The amplifier module in Fig. (P5) has a finite voltage gain, A, a finite input resistance, R in, and a non-zero output resistance, R out. A voltage source whose Thévenin resistance is R s excites the amplifier, and the load is the indicated capacitance,. R in R out R s V i A R 2 Fig. (P5) (a). Derive a generalized expression for the low frequency voltage gain, A v (0) A vo = /. In the process of arriving at this relationship, you may wish to introduce the algebraic substitution, Rout R2 A Rin K, Rout R2 1 R1 which physically represents the ratio of the voltage developed across resistance to the voltage, V i. (b). Derive a generalized expression for the low frequency voltage transfer function, V i /. (c). Using the expression deduced in Part (b), can you deduce a generalized relationship for the input resistance, say R i, seen by the signal source? (d). Derive an expression for the Thévenin resistance, say R thc, seen by the capacitor,. Assume here that the amplifier input resistance, R in, is sufficiently large to enable its approximation by an infinitely large resistance. (e). Give an expression, in terms of R thc and, for the 3-dB bandwidth, B, of the entire circuit.

2 (f). If the amplifier module is an idealized operational amplifier, R in and A are both infinitely large, and R out is zero. To what do the results of Parts (a), (b), (c), (d), and (e) reduce under this idealized operating constraint? What can specifically be asserted with respect to the input voltage, V i, and the current supplied by the signal source? (g). an you make use of the observations in Part (f) to find the voltage gain, A vo, and the input resistance, R in, virtually by inspection? What happens to the bandwidth, B? an you use this bandwidth observation to state a general strategy for designing wide bandwidth circuits that must drive capacitive loads? (h). Take R s = 300 Ω, R in = 4 MegΩ, R out = 100 Ω, = 1 KΩ, R 2 = 9 KΩ, = 20 pf, and A = 100. Numerically calculate the results of Parts (a) through (e). Verify the accuracy of these calculations by simulating the circuit on SPIE. Submit SPIE results in the form of a gain magnitude frequency response and a frequency response of the input impedance seen by the signal source. Problem #06: The amplifier module in Fig. (P6) has a finite voltage gain, A, an infinitely large input resistance, R in, and zero output resistance, R out. A voltage source whose Thévenin resistance is R s excites the amplifier, and the load is the indicated resistance, R l. f R f R s V i R in A R out Fig. (P6) (a). Derive a generalized expression for the low frequency voltage gain, A v (0) A vo = /. (b). Derive a generalized expression for the low frequency voltage transfer function, V i /. (c). Using the expression deduced in Part (b), deduce a generalized relationship for the input resistance, R i, seen by the signal source. (d). Derive an expression for the Thévenin resistance, say R thc, seen by the capacitor, f. (e). Give an expression, in terms of R thc and, for the 3-dB bandwidth, B, of the entire circuit. (f). If the amplifier module is an idealized operational amplifier, to what do the results of Parts (a), (b), (c), (d), and (e) reduce? What can be asserted specifically with respect to the input voltage, V i, and the current supplied by the signal source? Homework #02 15 Spring Semester, 2001

3 (g). an you make use of the observations in Part (f) to find the voltage gain, A vo, and the input resistance, R in, virtually by inspection? What happens to the bandwidth, B? an you use this bandwidth observation to state a general strategy for designing wide bandwidth circuits having the architecture in the subject schematic diagram? (h). Take R s = 300 Ω, R in = 4 MegΩ, R out = 10 Ω, R l = 1 KΩ, R f = 12 KΩ, f = 10 pf, and A = 98 db. Numerically calculate the results of Parts (a) through (e). Verify the accuracy of these calculations (in light of the idealized input and output resistance approximations) by simulating the circuit on SPIE. Submit SPIE results in the form of a gain magnitude frequency response and a frequency response of the input impedance seen by the signal source. Problem #07: The circuit in Fig. (P7) is an example of a lowpass filter realized with an operational amplifier (op-amp) and several passive elements. R R Op-Amp R 2 Fig. (P7) (a). Assuming that the op-amp is ideal in the senses of infinitely large gain, infinitely large input resistance, and zero output resistance, show that the voltage transfer function, A v (s) = (s)/v i (s), is of the form, A(0) v A(s) v = ; 2 s s 1 Qω o ω o specifically, derive expressions for the zero frequency gain, A v (0), the circuit quality factor, Q, and the natural, or undamped resonant, frequency,? o. (b). Must a limit be imposed on the resistor ratio, R 2 /? What is the basis of this limiting value? What is the maximum resistance ratio value and what is he corresponding maximum possible low frequency gain? (c). With R = = 500 Ω, design the circuit for a 10 MHz bandwidth and a maximally flat frequency response. Homework #02 16 Spring Semester, 2001

4 (d). Verify the accuracy of your design by simulating the circuit on SPIE. For this simulation, consider the op-amp to have a gain of 98 db, an input resistance of 10 MegΩ, and an output resistance of 10 Ω. Submit your simulation results as a frequency response that clearly shows the value of the 3-dB bandwidth. Problem #08: Bandpass filters, such as the unit drawn in Fig. (P8), are used to provide amplification in a narrow range, or band, of frequencies, while providing attenuation outside of this band. In other words, the bandpass filter provides frequency selective amplification. As such, these filters are useful in radio receiver and other communication system applications. 1 R 2 2 R 3 Op-Amp Fig. (P8) (a). Assuming an ideal operational amplifier, show that the transfer function of the filter in Fig. (P8) is expressible as, V(j o ω) A(j v ωo ) A(j v ω ) = = ; V(j s ω ) ω ω o 1 jq ω o ω specifically, give expressions for the center frequency,? o, the center frequency gain, Av(j? o ), and the circuit quality factor, Q. (b). Using EXEL or suitable other software, plot the normalized gain magnitude, A v (j?)/a v (j? o ) -versus- the normalized signal frequency,?/? o, for Q = 1, 3, 5, and 10. If you were to use this filter in a radio receiver application, would you want large or small Q? Explain your answer. (c). Study the plots deduced in Part (b). Note in particular that a gain that is 3-dB below the center frequency gain is achieved at two signal frequencies. One such frequency, say? low, is smaller than the center frequency,? o, while the other frequency, say? high, is larger than? o. Derive expressions for? low and? high. (d). The 3-dB bandwidth of a bandpass filter is defined as the difference between the aforementioned frequencies,? low and? high. Show that B? high? low =? o /Q. Homework #02 17 Spring Semester, 2001

5 University of Southern alifornia School Of Engineering Department Of Electrical Engineering EE 348: Homework Assignment #02 Spring, 2001 (SOLUTIONS: Due 02/01/2001) homa Problem #05: Homework #02 18 Spring Semester, 2001