Massachusetts Institute of Technology Department of Electrical Engineering and omputer Science 6.002 ircuits & Electronics Spring 2006 Problem Set #9 Issued 4/12/06 Due 4/19/06 Exercise 9.1: Determine i(t) andv(t) fort 0 in the network shown below. Note that the inductor and capacitor both have nonzero states at t = 0, and that both sources are impulse sources. Hint: use superposition to establish initial conditions at t =0. Qδ(t) v(t) L i(t) Λδ(t) v(0) = V i(0) = I Exercise 9.2: Shown below is a series resistor-inductor-capacitor network. Also shown below is a graph of the resistor voltage v(t) as a function of time; the voltage is the homogeneous response of the network to some set of initial conditions. From the voltage graph, determine the inductance of the inductor and the capacitance of the capacitor given that the resistance of the resistor is 10 Ω. Note: the next to last page of this problem set contains a larger graph of the common voltage. It can be turned in with your problem set solutions. 1 Homogeneous Response: Resistor Voltage 0.8 0.6 L Resistor Voltage [V] 0.4 0.2 0 0.2 R 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [ms]
Problem 9.1: The network shown below is driven in steady state by the sinusoidal input voltage v I (t) =V I cos(ωt). The output of the network is the voltage v O (t), which takes the form v O (t) =V O cos(ωt φ). Find V O and φ as functions of ω as follows. R 1 v I (t) R 2 v O (t) (A) Using the Taylor Series expansions for e x, cos(x) and sin(x), show that e jx = cos(x)j sin(x). Following this, recognize that cos(x) =R { e jx}. (B) Show that A Bj = A 2 B 2 e j arctan(b/a). Thus, the magnitude and phase of A Bj are A 2 B 2 and arctan(b/a), respectively. () Find a differential equation that can be solved for v O (t) givenv I (t). (D) Following Part A, let v I (t) =V I e jωt. Also, let v O (t) = ˆV O e jωt where ˆV O is a complex function of the circuit parameters, ω and V I. With these substitutions, use the differential equation to find ˆV O. (E) Following Parts A and B, first express v O from Part (D) in the form v O (t) = ˆV O e j(ωt ˆVO ), and determine ˆV O and ˆVO as functions of the circuit parameters, ω and V I. Then, find V O and φ for the original cosine input, again both as functions of the circuit parameters, ω and V I. (F) Sketch and clearly label V O /V I and φ as functions of ω. Identify the low-frequency and highfrequency asymptotes on the sketch.
Problem 9.2: This problem concerns the sinusoidal-steady-state behavior of the networks shown below, both of which have two ports. Port #1 R 1 R 2 L Port #2 Port #1 L R 1 R 2 Port #2 (A) Determine the impedance of each network as viewed into Port #1 under the assumption that Port #2 is open. (B) Assume that Port #1 of each network is driven in sinusoidal steady state by the voltage V 1 cos(ωt), and that Port #2 is open. Determine the current into the positive terminal of each network at Port #1. Express the current in the form I 1 cos(ωt φ 1 ) where I 1 is an amplitude and φ 1 is a phase angle. () Assume that Port #1 of each network is again driven in sinusoidal steady state by the voltage V 1 cos(ωt), and that Port #2 is again open. Determine the voltage that appears at Port #2. Express the voltage in the form V 2 cos(ωt φ 2 ) where V 2 is an amplitude and φ 2 is a phase angle. Note that the results of this problem are useful when completing the pre-lab exercises to Lab #3.
Problem 9.3: When the two-port network shown below is driven with the input voltage v IN (t) =V IN cos(ωt), its output voltage takes the form v OUT (t) =V IN A(ω) cos(ωt φ(ω)). The normalized output voltage magnitude A(ω) is also shown below for a specific combination of R 1, R 2, R 3 and. Note: the last page of this problem set contains a larger graph of the normalized output voltage magnitude. It can be turned in with your problem set solutions. (A) Assuming that =1µF, and that the smallest resistor has a resistance of 1 kω, determine R 1, R 2 and R 3. Hint: consider first the low-frequency and high-frequency behavior of the network. (B) There is a second but different network topology that also contains three resistors and one capacitor, and that also provides the same input-output voltage relation as does the network shown below. Determine the second network topology. You need not determine component values for that network. 10 0 Normalized Output Voltage Magnitude A(ω) R 1 R 2 v IN R 3 v OUT A(ω) [1] 10 1 10 2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Frequency ω [rad/s] Problem 9.4: The input-output voltage relation of the network shown in Problem 9.3 can also be produced by a network containing three resistors and one inductor. As in Problem 9.3, there are actually two such network topologies. Determine both topologies; you need not determine resistor and inductor values to match the graphical relation given in Problem 9.3.
1 Homogeneous Response: Resistor Voltage 0.8 0.6 0.4 Resistor Voltage [V] 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [ms]
10 0 Normalized Output Voltage Magnitude A(ω) A(ω) [1] 10 1 10 2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Frequency ω [rad/s]