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Name: 1 1. See the 2nd order linear circuit shown in Figure 1 which consists of one current source (Iin), three resistors (R1 to R3), one capacitor (C), and one inductor (L). Figure 1 – 2nd order Linear Circuit a) Find the 2nd order differential equation which defines the inductor current iL(t) in terms of the input current Iin and circuit elements R1, R2, R3, C, and L. (10 pts) b) Solve the differential equation you derived in part (a) and find iL(t) assuming that R1=R2=R3=1Ω, C=1F, L=1H, and Iin(t)=sin(ωt)u(t). The initial conditions for the inductor current are given as ?? ? = 0 = 0 ? and ??? ?? ? = 0 = 0 ?/?. (10 pts) c) Now consider the steady state condition for the circuit shown in Figure 1. Work in the phasor domain and find the steady state inductor current, iL(t) using the parameter values given in part (b). (10 pts) d) In frequency (phasor) domain, the transfer function for the circuit shown in Figure 1 is given as ? ? = ?? ??? , i.e. the inductor current divided by the input current. Answer the following questions using the parameter values given in part (b): Calculate the transfer function. (3 pts) What kind of filter is this circuit? (3 pts) What are the critical frequencies of the circuit? (cutoff and/or center frequencies)? (5 pts) Plot the Bode diagram for the magnitude of the transfer function. (7 pts) Plot the Bode diagram for the phase of the transfer function. (7 pts) (Note: Indicate all critical frequencies, slopes, magnitude and phase values, units etc. in your Bode plots and label the axes properly) e) For this part, consider the circuit in Laplace Domain. A convenient way would be computing the Laplace transform of the differential equation you found in part (a). Find the Laplace transform of the inductor current, IL(s), using the parameter values and initial conditions given in part (b). (10 pts) Calculate the inverse Laplace transform of IL(s) and find iL(t). Compare your results with what you found in part (b). Do they match? (10 pts) iL(t) Name: 2 2. See the Bode Diagram in Figure 2 for the gain (magnitude of the transfer function) of a band- pass filter. Figure 1 – Gain Bode Plot for the Band-pass Filter a) Design a circuit to implement this filter. Clearly indicate the circuit parameter values and discuss the logic behind your design. (20 pts) b) Plot the Bode diagram for the phase of the transfer function for your filter. (5 pts) (Note: Indicate all critical frequencies, slopes, magnitude and phase values, units etc. in your Bode plots and label the axes properly) Name: 1 1. See the 2nd order linear circuit shown in Figure 1 which consists of one voltage source (Vin), two resistors (R1 and R2), one capacitor (C), and one inductor (L). Figure 1 – 2nd order Linear Circuit (a) Find the 2nd order differential equation which defines the output voltage Vout(t) in terms of the input voltage Vin(t) and circuit elements R1, R2, C, and L. (10 pts) iL(t) Name: 2 (b) Solve the differential equation you derived in part (a) and find Vout(t) assuming that R1=2Ω, R2=3Ω, C=1F, L=1H, and Vin(t)=cos(ωt)u(t). The initial conditions for the inductor current are given as ????(?? = 0) = 0 ?? and ?????? ???? (?? = 0) = 0 ??/??. (10 pts) Name: 3 Name: 4 (c) Now consider the steady state condition for the circuit shown in Figure 1. Work in the phasor domain and find the steady state output voltage, Vout(t) using the parameter values given in part (b). (10 pts) 5 (d) In frequency (phasor) domain, the transfer function for the circuit shown in Figure 1 is given as ??(??) = ???????? ?????? , i.e. the output voltage divided by the input voltage. Answer the following questions using the parameter values given in part (b): (d-1) Calculate the transfer function. (3 pts) (d-2) What kind of filter is this circuit (highpass, lowpass, bandpass, bandstop etc.)? (3 pts) (d-3) What are the critical frequencies? (cutoff and/or center frequencies)? (5 pts) (d-4) Plot the Bode diagram for the magnitude of the transfer function. (7 pts) (d-5) Plot the Bode diagram for the phase of the transfer function. (7 pts) (Note: Indicate all critical frequencies, slopes, magnitude and phase values, units etc. in your Bode plots and label the axes properly) Name: 6 Name: 7 (e) For this part, consider the circuit in Laplace Domain. A convenient way would be computing the Laplace transform of the differential equation you found in part (a). (e-1) Find the Laplace transform of the output voltage, Vout(s), using the parameter values and initial conditions given in part (b). (10 pts) (e-2) Calculate the inverse Laplace transform of Vout(s) and find Vout(t). Compare your results with what you found in part (b). Do they match? (10 pts) Name: 8 Name: 9 2. See the Bode Diagram in Figure 2 for the gain (magnitude of the transfer function) of a band- pass filter. Figure 2 – Gain Bode Plot for the Band-pass Filter a) Design a circuit to implement this filter. Clearly indicate the circuit parameter values and discuss the logic behind your design. (20 pts) b) Plot the Bode diagram for the phase of the transfer function for your filter. (5 pts) (Note: Indicate all critical frequencies, slopes, magnitude and phase values, units etc. in your Bode plots and label the axes properly) Name: 10