6.002 Circuits and Electronics Final Exam Practice Set 1

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE Circuits and Electronics Set 1 Problem 1 Figure 1 shows a simplified small-signal model of a certain type of transistor amplifier driven from a non-ideal source. source e 2 i C amplifier e 1 i D R A R C + V A ki C R D v D - Figure 1 (1A) Find the node voltages e1 and e2 in terms of VA, RA, RC, RD, and k. (Assume id = 0.)

2 e1 = e2 = Page 2 of 17

3 (1B) Find a Thévenin equivalent model of the amplifier system viewed from the vd, id port. (That is, find the Thévenin voltage and resistance VTH, RTH that characterize the system.) VTH = RTH = Page 3 of 17

4 Problem 2 Figure 2 shows a relay driver circuit. The relay is modeled as the series connection of a resistor RR = 100 Ω and an inductor LR = 200 μh. To protect the switch S driving the relay, a resistor RF = 100 Ω is placed across the relay as shown. R F R R Relay model 5V L R + i sw v sw - S Figure 2 After being open for a long time, the switch S is closed at t = 0, then opened at t = 10 μs. On the axes provided, plot the switch voltage vsw and the switch current isw from t = -5 μs to 15 μs. Clearly label and specify (numerically) all steady-state levels and time constants. v sw t i sw -5 s 5 s 10 s 15 s t Page 4 of 17

5 Problem 3 In the circuit of Fig. 3, a resistor RS is used to sense the current ix(t) by converting it to a voltage measured at vo. To attenuate high-frequency components in the sensed signal a low-pass filter comprising resistor RF and capacitor CF is inserted between RS and vo. + R F v o C F R S i x - Figure 3 Page 5 of 17

6 (3A) Assume that ix(t) = Ixe st and vo(t) = Voe st Vo. Determine the ratio H ( s) as a I x function of s. In other words, find the input-to-output transfer function of the system H(s). H(s) = Page 6 of 17

7 (3B) On the axes below, plot the magnitude and phase of H(jω) vs. ω. Use a logarithmic scale for magnitude and frequency and a linear scale for phase. Clearly label and specify all important breakpoints and asymptotes, and label and dimension all axes. Page 7 of 17

8 (3C) Consider the circuit with parameters RS = 0.1 Ω, RF = 10 Ω, and CF = 33 nf. If the system is driven with the steady-state input (in Amperes) ix(t) = 5 + 1sin(2π 10 6 t) find the steady-state output vo(t). vo(t) = Page 8 of 17

9 Problem 4 (15 Points) Figure 4 shows a parallel RLC circuit driven by a current source. The circuit parameters are L = 1 μh, C = 1 μf, R = 1 Ω, I = 1 A. The switch is closed for t < 0, and it opens permanently at t = 0. (You may assume that the inductor current is zero at t = 0.) Find v(t) for t > 0. + I v S C L R - Figure 4 Page 9 of 17

10 v(t) = Page 10 of 17

11 Problem 5 (25 Points) (5A) Figure 5 shows a widely-used amplifier configuration incorporating an operational amplifier. Find an expression for the output voltage vo in terms of the input voltages va and vb, assuming that the op amp is ideal. v B v A R 1 R 1 R R 2 v o Figure 5 vo(t) = Page 11 of 17

12 (5B) Figure 6 shows an operational amplifier circuit incorporating a diode. For the operating range of interest we will model the i-v characteristic of the diode as: vd VT I s e 1 i d. Find the relationship between the output voltage and the input voltage for this circuit, assuming that the op amp is ideal, the diode acts as modeled above, and the input voltage is greater than zero. i d v I R - + v d - + v O Figure 6 vo(t) = Page 12 of 17

13 (5C) Assume the input voltage vi in Fig. 6 is composed of a large positive dc term VI plus a small ac term vi. Find the small-signal gain of the circuit as a function of VI, R, IS, and VT. (That is, vi = VI + vi and vo = VO + vo. Find the gain from the small variations vi in vi to the resulting small variations vo in vo.) vo/vi = Page 13 of 17

14 Problem 6 Assume that the network in Figure 7 is operating in the sinusoidal steady state. Determine the response vout (t) to the input voltage vin (t) = VSI cos(ωs t). Note that vout (t) will take the form vout (t) = VSO (ωs) cos(ωs t + φ(ωs) ). Figure 7 vout (t) = Page 14 of 17

15 Problem 7 This problem concerns the MOSFET inverter circuit of Figure 8. When operating in the saturation region (vgs > VT and vds vgs VT) the MOSFET has the characteristic: i DS K ( v 2 GS V The circuit parameters are Vs = 8 V, RD = 1 kω, K = 1 ma/v 2, VT = 1 V. T 2 ) Figure 8 (5A) Find the dc bias input voltage VI that yields a dc operating point output voltage VOUT = 4 V. Page 15 of 17

16 VI = Page 16 of 17

17 (5B) A small-signal model for the MOSFET in the amplifier circuit is shown in Fig. 9 below. Determine the value of the input bias voltage VI such that gm = 2 ma / V. Figure 9 VI = Page 17 of 17