2 Problem Set for Dr. Todd Huffman Michaelmas Term I. Introduction to Simple ircuits of esistors 1. For the following circuit calculate the currents through and voltage drops across all resistors. The resistances are: 1 = 1kΩ, 2 = 3kΩ, 3 = 3kΩ and 4 = 6kΩ. 2. The structure of a cube is soldered by using resistors for its edges. What is the resistance between two most distant corners (diametrically opposite)? 3. Harder (for the enthusiast): In problem 2 above, what is the resistance between two adjacent corners and between two corners on the same face but not adjacent? 4. For the voltage divider, where 1 = 1kΩ, 2 = 4kΩ and V0 = 5V find the voltage drop V2 across 2. When a load resistor is fitted in parallel with 2, what minimum value must have in order not to change V2 by more than 5%? 1
5. onsider the following circuit: I 1 I 2 V 1 + Ω Ω I 3 Ω V 2 + Given V1 =5V and V2 =10V, find I1, I2 and I3. Hint: you may wish to tackle the problem using mesh currents. 6. onsider the following circuit with resistors 1 = 2 = 1kΩ: 10V + 1 2 A 2mA Find the voltage between A and and the currents I1 and I2 through the resistors 1 and 2. onsidering Thevenin s theorem, what are Veq and eq in the equivalent circuit? What is the equivalent circuit according to Norton s theorem? Find Ieq and eq in this case. 7. Find the currents through resistors 1 to 4. Give magnitudes and directions. The values are: V0 = 2V, I1 = 1mA, I2 = 4mA, 1 = 1kΩ, 2 = 1.5kΩ, 3 = 0.5kΩ and 4 = 2kΩ. I1 V0 + 4 1 2 3 I2 2
II. esponse of inear ircuits to Transients. 8. The capacitor is initially uncharged. At time t=0 the switch is moved from position A to position. Derive an expression for the current flowing through at time t. y performing an integration over time, derive an expression for the total energy dissipated in the resistor. What is the final energy stored in the capacitor? Hence show that the total energy supplied by the battery is V 2. v + A 10 kω 9. Initially the switch in the circuit below is open and the capacitor is uncharged. At time t=0 the switch is closed. Show that the voltage 6V V (t) across the capacitor as a 20μF function of time goes as: t 4.125 1 exp volts 0.1375 (where t is in seconds) When the steady state has been reached, what is i) the power dissipated in each resistor and ii) the energy stored in the capacitor? 22 kω 10. At time t=0 the switch is moved from position A to position. Derive an expression for the current flowing through at time t. v + A 11. At times t < 0 the switch is open and the capacitor is charged with 1V across its terminals. At t =0 the switch is closed. Show that the subsequent time response of the circuit is oscillatory and damped, and sketch (quantitatively) the response of the circuit for t > 0, given =150Ω, =10 mη, and =10 nf.
P P 12. In the circuit shown, at time t=0 the switch is closed. y solving the appropriate differential equation, show that the voltage V out is oscillatory and damped with an exponential decay-time constant given by t0 =2. If the component values are =100 kω, =0.01 F and =4 mh, show that the resonant frequency, 0, is equal to 1.6 10 5 rad/s. 13. The switch in the figure has been closed for a long time, ie. a steady state has been reached, it is then suddenly opened. Demonstrate that VA = V = 0V before the switch is opened, VA = 10V, V = 1010V immediately after the switch is opened, then that V falls exponentially to 10V with a time constant of 10 5 s After many time constants have elapsed, the switch is again closed. Show that the voltage across the 100Ω resistor rises exponentially with time with a time constant of 1 ms. 4
III. omplex Impedances and esponse of inear ircuits to A 14. MS-values of voltages: a) What is the MS-voltage of a constant voltage V(t) = V? b) What is the MS-voltage of a square wave between 0 and V volts? c) What is the MS-voltage of a square wave between V/2 and +V/2 volts? Why is the answer to b) and c) different? t d) alculate the MS-value of a saw-tooth voltage V t V of period 2T. 0 T 15. a) d) 100 kω 1kΩ 1µF 10µH b) e) 0.1mH 10mH 100pF 20pF c) 5kΩ 100µH f) 470nF 20µH μf For each of the above networks: 47Ω i) alculate the (complex) impedances; ii) A voltage of the form V=V 0 sin (ωt) is applied across the network. Evaluate the phase shift between the voltage and the current flowing through the network. State whether the current leads or lags; iii) alculate the peak voltage drop across the resistors in circuits a) and d), the capacitors in circuits b) and e), and the inductors in circuits c) and f). Take the driving voltage to be of the form V=V0 sin(ωt) with V0 =10V and frequency (ω/2π) =10 khz. 5
16. The two circuits below are driven by sinusoidal input voltages V1 =V0 sin t V1(t) V2(t) V1(t) v2(t) a) Draw the phasor diagrams of all voltages. b) With the help of the phasor diagrams determine the ratio of the amplitudes of voltages V2 and V1 as well as the phases of the output voltages V 2 for the two circuits. c) Sketch the ratios of amplitudes and the phases as a function of normalised frequency x =, with = 1/ and = / respectively for the two circuits. 17. An A current I(t) = I 0 sin (ωt) is flowing through the circuit below made from a series combination of capacitance, inductance and resistance. The numerical values are: = 10nF, = 0.2mH, = 100Ω, I0 = 0.1A and = 10 6 s -1. I(t) V(t) V (t) V (t) V (t) a) Find the amplitudes and phase angles of the voltages across the capacitor, inductor and resistor V (t), V (t) and V (t) and of the total voltage V(t). b) Find the total power dissipated by the circuit. c) Find expressions for the energy contents W (t) and W (t) of the inductor and capacitor. d) What are the maximum values for these energies? e) Sketch W (t) and W (t). f) For which value of (provided all other values remain constant) would the sum of W (t) and W (t) be constant? 6
18. At what frequency does the network below have its minimum impedance? If the driving voltage is of the form V=V0 sin (ωt) with V0 =10V, show that the voltage across the capacitor at the frequency of minimum impedance is 3.16 V. 100 1 H 1000 pf 5 1 19. Show that there are two frequencies, 1 and 2, for 2 2 which the impedance between points A and in the network below is zero. 2 A 2 20. In the following bridge circuit 2, 3, 4 and 4 are fixed and Z1 is variable. Vi(t) Z1 Vo(t) 3 2 4 4 a) Find the complex value of Z1 for which the bridge is balanced. b) When Z1 is a series combination of 1 and 1 what values must they have to balance the bridge? c) Given that Z1 is the parallel combination of a resistor 1 and a capacitor 1, for which values of 1 and 1 is the bridge balanced? 7
21. A voltage VA =V0 cos (ωt) is applied between points A and in the circuit below. and have the values (ω 3) -1 and ( 3)/ω respectively. A X Z Y i) Show that the total impedance between A and is 3; ii) Verify that voltages of equal amplitude are developed between the points X-A, the points X-Y and the points X-Z; iii) Show that the phases of these three voltages relative to V A are 0, + 3 and - 3. iv) Fix the values of and such that the total impedance between A and is 3 at 10 khz. Plot the magnitude and phase of the voltage across A if a current source of value I( ) = cos( t) were connected across the points A. 22. In the circuit below, show that the amplitude of the voltage V XY between points X and Y is independent of, and show that if =1/ω, the phase of V XY with respect to the applied voltage V is π V=V 0 sin(ωt) 0 0 X Y 8
IV. Operational Amplifier ircuits 23. For circuit (a) (c) below, calculate V OUT as a function V IN, 1 and 2 assuming the ideal op-amp model. When 2 = 4 1, give the value of V IN which would produce an output voltage of V OUT =2V 24. For the circuit below, calculate V OUT as a function of time for an input voltage of 0 t 0 V IN 1V t 0 When the constant reference voltage, the resistance, and capacitance have the following values: V 1 =10V, =100kΩ, =10μF, sketch the value of V OUT against time. What happens as t? 9
25. For the circuit below, calculate the ratio V V OUT angular frequency ω. At what frequency does IN as a function of, and the V V of an active filter such as this, over a simple filter? OUT IN 1 2? What are the benefits 26. Show that the circuit below functions as a DA (digital to analogue converter) by calculating the output voltage V OUT as a function of V 0 and the switch settings S 0, S 1, S 2, and S 3 (which can have a value of 0 or 1). When V 0 =10V, what is the range of V OUT? Design a simple circuit to rescale the output to the range 0 to +5V 10