2010-11-15
Problems in circuit theory
DC circuits: 1. Calculate U and I in the figure on the right if a R = 1000 Ω, b R = 15 Ω, c R = 0.1 Ω. 2. Taking a more general case, we can call the 10 Ω resistor in the figure above Rin. How should we choose R, in terms of U and Rin, if we want to dissipate the largest possible power in R ? 3. Calculate U, Ix and Iy in the figure on the right if a R = 1000 Ω, b R = 15 Ω, c R = 0.1 Ω.
4. What are the currents through the two resistors in the figure on the right?
5. What is the voltage at point A (relative to ground)?
6. What is the voltage at point A (relative to ground)?
7. In problem 4, replace the series combination of a 1 V voltage source and its closest 10 Ω resistor by a current source in parallel with a resistor in such a way that you do not change any currents, voltages, or other properties in the rest of the circuit!
8. What are the voltages UA and UB at points A and B below, if R = 10 Ω?
9. Suppose the voltage and current generators in the circuit above produce some kind of signals. Suppose further that the resistor R represents a device which will receive these signals. We want to know the output signal and the properties of the circuit under various conditions, and we thus want to have a simpler model. Assume that points A and B are the two output terminals, and that the ground connection is moved to point B! a. Replace the circuit by an equivalent Thevenin circuit. b. Replace the circuit by an equivalent Norton circuit. c. How should R be chosen if we want to dissipate the largest possible power in it? 10.
a. Replace the circuit below with an equivalent Thevenin two-pole circuit! b. Replace the circuit below with an equivalent Norton two-pole circuit! E=5V I =2A R1 = 40 Ω R2 = 60 Ω R3 = 6 Ω R4 = 20 Ω
11.
a. Replace the circuit below with an equivalent Thevenin two-pole circuit. b. Replace the circuit below with an equivalent Norton two-pole circuit. c. If A and B are shorted, what current would run through the points? E1 = 12 V E2 = 10 V E3 = 6 V R1 = 80 Ω R2 = 40 Ω R3 = 80 Ω
12.
In the circuit below, what is the potential difference between a. points A and B, b. points A and C ?
U1 = 2 V U2 = 21 V R1 = 100 Ω R2 = 100 Ω R3 = 10 Ω R4 = 10 Ω R5 = 100 Ω
13.
a. Replace the circuit below with an equivalent Thevenin two-pole circuit. b. Replace the circuit below with an equivalent Norton two-pole circuit. I = 1A E1 = 6 V E2 = 12 V R1 = 8 Ω R2 = 6 Ω R3 = 12 Ω R4 = 8 Ω R5 = 8 Ω
14. The circuit below illustrates a very common industrial installation where the signal is transformed into a current and sent by the current source (left) through a double wire (10 Ω resistances) to a measurement instrument (represented by the resistance right). An interference voltage, capacitively or inductively coupled to the signal wires, is also shown. It can be represented by a voltage source in series with the signal. a. What is the output voltage UAB between points A and B due to the current source alone? b. What is the output voltage UAB due to the interfering voltage source?
I = 10 mA U = 10 V
R1 = 100 Ω R2 = 1 M Ω
R3 = R4 = 10 Ω
AC Circuits: 15.
Calculate
j!
16.
If Z = (1 + 2j)/(1 - j), what are |Z| and arg (Z) ?
17. a. Calculate the ratio G = Uout/Uin as a function of frequency! b. Exchange R and C and repeat the calculation.
18.
Calculate the ratio G = Uout/Uin as a function of frequency for the circuit below!
19. a. Calculate the total impedance Z for the LC-R series resonance circuit shown on the right. b. Find the resonance frequency ωo, defined as the frequency at which Z is real ! c. What are the voltages over L, R, and C, respectively, at ωo? d. For coils, and for all circuit containing coils, the quality factor Q is defined as Q = ωοL/R. Use the results of c) to find the magnitude of the voltage over L at ωo in terms of Q and Uin! e. The quality factor also defines how "sharp" the resonance is. Assuming that Q >> 1, show that |Z| increases by a factor of 2 when ω increases from ωo to (ωo+∆ω), where ∆ω = ωo/2Q ! 20. a. Calculate the total impedance Z as a function of frequency for the parallel resonance circuit shown on the right. b. What is the resonance frequency ωo? Show that in the limit of very large values of the factor Q defined above, the expression for ωo is identical to that for the series circuit!
21. The figure below shows a Maxwell bridge, which is still often used for measuring inductances. The unknown coil, with inductance Lx and resistance Rx, is connected as shown; arrows denote variable resistances. a. Show that the balance condition gives Lx = R1 R4 C3 ! b. What is the corresponding relation giving Rx ?
22. The figure below shows a Schering bridge, used for measuring capacitances. The unknown capacitance Cx, with resistance rx, is connected as shown; arrows denote variable impedances. What are the balance conditions for Cx and rx?
Operational amplifier circuits:
23. Calculate the gain G = Uout/Uin for the circuit on the right.
24. Calculate the gain G = Uout/Uin as a function of frequency for the circuit on the right. (This can be considered as a very simple active low-pass filter.)
25. Calculate the gain G for the circuit on the right (the "voltage follower").
26. Calculate the gain G for the circuit on the right.
27. The diagram below shows a differential amplifier, with two inputs (A and B) and one output. Assume that there are two input voltages, UA and UB. What is the output voltage, expressed as a function of these input voltages, a. if all resistors are equal, i.e. R1 = R2 = R3 = R4 ? b. if R1 = R2 = 10 kΩ and R3 = R4 = 30 kΩ ?
28. The circuit above gives an output voltage (relative to ground) which is proportional to the difference between the two input voltages. The circuit below is in many ways similar, but gives an output current I to ground. Show that the output current I is proportional to the difference between the input voltages and that it does not depend on the “load resistance” RL attached! (This circuit is one possible way to realize the type of current source shown in example 13.)
Answers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
a. 4.95 V, 4.95 mA b. 3 V, 0.2 A c. 50 mV, 0.495 A 10 Ω a. 4.95 V, 5 mA, 495 mA b. 3 V, 0.2 A, 0.3 A c. 50 mV, 495 mA, 5 mA 100 mA, 200 mA 0V 0.67 V 0.1 A, 10 Ω UA = 8.75 V, UB = 7.5 V a. 3.33 V, 16.67 Ω b. 200 mA, 16.67 Ω c. 16.67 Ω a. 20.4 V, 12 Ω b. 1.7 A, 12 Ω a. 2 V, 20 Ω b. 100 mA, 20 Ω c. 100 mA a. 1 V b. 10 V a. - 2 V, 16 Ω b. -0.125 A, 16 Ω a. 1 V b. 1 mV +(1 + j)/21/2 1.581, 108.4o a. G = jωRC/(1+jωRC) b. G = 1/(1+jωRC) G = R2/(R1+R2+jωL) a. Z = R + j(ωL - (1/ωC)) b. ωo = (LC)-1/2 c. (jωL/R)Uin, Uin, - jUin/(ωRC) d. |UL| = QUin a. Z = (R + jωL)/[(1 - ω2LC) + jωRC] b. ωo = [(L - R2C)/L2C]1/2 b. Rx = (R1 R4)/R3 Cx = (R3 C4)/ R1, rx =(C3 R1)/ C4 -5 -5/(1 + 5⋅10-4jω) 1 10 a. Uout = UB-UA b. Uout = 3(UB-UA) I = (U2-U1)/R