# Time Constant

A unit-step function u(t) can be considered a causal constant source (e.g. a battery in a circuit if the units of u(t) is volts).

(a) From basic principles consider the response of an RC circuit to u(t) - that is, a battery connected in series with the resistor and the capacitor. Plot what would be the voltage across the capacitor for t > 0 (assume the capacitor has no initial voltage at t=0).

(b) What would be the voltage across the capacitor in the steady state? Explain.

(c) Suppose that the capacitor is disconnected from the circuit at some time t0 >> 0. Ideally, what would be the voltage across the capacitor from then on?

(d) If you disconnect the capacitor, again at t0 >> 0, but somehow it is left connected to the resistor, so they are in parallel, what would happen to the voltage across the capacitor? Plot approximately the voltage across the capacitor for all times and explain the reason for your plot.

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#### Solution Summary

This posting contains the solution to the given problems.

DC Circuit Properties

See attachment for better symbol representation and diagrams.

1. The diagram of FIGURE 1 represents a meter shunted by a parallel resistance RS.

(a) Determine the required value of the shunt resistance if the maximum value of the current I is 200 A. The meter can read a maximum of 1 mA and has a

resistance of 0.1 Ω.

Meter

IM R I

IS RS

Shunt

FIG. 1

(b) If the shunt is made of copper and has a cross-sectional area of

25 cm2 calculate its required length.

(For copper take ρ as 1.7 × 10-8 Ωm.)

2. The circuit of FIGURE 2 shows a 10 kΩ potentiometer with a 5 k Ω load. Determine the position of the slider on the

'pot' when the voltage across points 'XX'

is 3 V.

9 V 10 kΩ

5 kΩ

FIG. 2

3. (a) Calculate the value of the current through the 12 V battery shown in

FIGURE 3.

(b) Calculate the power dissipated in R1, R2 and R.

I1 I2

13 V

R = 3 Ω

14 V

R1 = 1 Ω

12 V

R2 = 2 Ω

FIG. 3

4. If the magnetic flux linking all the turns of a 50 turn coil changes from

10 mWb to 20 mWb and induces an e.m.f. of 62.5 V in the coil, calculate the time over which the flux changes.

5. A 500 mm conductor inside an electric motor has a force of 1.5 newtons exerted on it. If it is at right angles to a magnetic field of flux density

0.6 T, calculate the current flowing in the conductor.

6. FIGURE 4 show the construction of a multi-plate variable capacitor having 4 pairs of plates. The plates, when closed, are separated in air by

0.01 mm and a capacitance range of 10 to 400 pF is required. (a) Estimate the required radius, R, of each plate.

(b) The capacitor is set to the maximum 400 pF and is charged to 10 V

through a 50 kΩ resistor. Determine:

(i) the initial value of current flowing. (ii) the time constant for the circuit.

(c) Having fully charged, the capacitor is then discharged through the

50 kΩ resistor. Determine:

(i) the current flowing when the capacitor has been discharging for

5 µs

(ii) the voltage drop across the resistor when the capacitor has been discharging for 10 µs.

Rotating

Moveable plates mounted on a spindle

Spindle

Fixed

FIG. 4

7. An inductor of negligible resistance and an inductance of 0.2 H is connected in series with a 330 Ω resistor to a 12V d.c. supply. Determine:

(a) the time constant of the circuit

(b) the voltage drop across the inductor after two time constants

(c) the voltage drop across the resistor after three time constants

(d) the resistance of a 0.2 H coil used to replace the inductor if the circuit's time constant falls to 0.55ms.

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