Transient Response | Additional Documents & Tests for IIT JAM PDF Download

Consider a circuit consisting of a capacitor and a resistor that are connected as shown in Figure A. What will be the voltage at point b if the voltage at a is increased suddenly from V_a = 0 to V_a = +50 volts? Closing the switch produces such a voltage because it connects the positive terminal of a 50-volt battery to point a while the negative terminal is at ground (point c). Figure B graphs this voltage V_a as a function of the time. 

Figure A: An RC circuit. This type of electric circuit consists of both a resistor and a capacitor connected as shown (see text).

Figure B: Voltage as a function of time (see text).

Initially, the capacitor has no charge and does not affect the flow of charge. The initial current is obtained from Ohm’s law, V = iR, where V = V_a − V_b, V_a is 50 volts and V_b is zero. Using 2,000 ohms for the value of the resistance in Figure A, there is an initial current of 25 milliamperes in the circuit. This current begins to charge the capacitor, so that a positive charge accumulates on the plate of the capacitor connected to point b and a negative charge accumulates on the other plate. As a result, the potential at point b increases from zero to a positive value. As more charge accumulates on the capacitor, this positive potential continues to increase. As it does so, the value of the potential across the resistor is reduced; consequently, the current decreases with time, approaching the value of zero as the capacitor potential reaches 50 volts. The behaviour of the potential at b in Figure B is described by the equation V_b = V_a(1 − e^−t/RC) in volts. For R = 2,000Ω and capacitance C = 2.5 microfarads, V_b = 50(1 − e^−t/0.005) in volts. The potential V_b at b in Figure B increases from zero when the capacitor is uncharged and reaches the ultimate value of V_a when equilibrium is reached.
How would the potential at point b vary if the potential at point a, instead of being maintained at +50 volts, were to remain at +50 volts for only a short time, say, one millisecond, and then return to zero? The superposition principle (see above) is used to solve the problem. The voltage at a starts at zero, goes to +50 volts at t = 0, then returns to zero at t = +0.001 second. This voltage can be viewed as the sum of two voltages, V_1a + V_2a, where V_1a becomes +50 volts at t = 0 and remains there indefinitely, and V_2a becomes −50 volts at t = 0.001 second and remains there indefinitely. This superposition is shown graphically on the left side of Figure C. Since the solutions for V_1b and V_2b corresponding to V_1a and V_2a are known from the previous example, their sum V_b is the answer to the problem. The individual solutions and their sum are given graphically on the right side of Figure C. 

Figure C : Application of the superposition principle to a problem concerned with voltages as a function of time (see text).

The voltage at b reaches a maximum of only 9 volts. The superposition illustrated in Figure C also shows that the shorter the duration of the positive “pulse” at a, the smaller is the value of the voltage generated at b. Increasing the size of the capacitor also decreases the maximum voltage at b. This decrease in the potential of a transient explains the “guardian role” that capacitors play in protecting delicate and complex electronic circuits from damage by large transient voltages. These transients, which generally occur at high frequency, produce effects similar to those produced by pulses of short duration. They can damage equipment when they induce circuit components to break down electrically. Transient voltages are often introduced into electronic circuits through power supplies. A concise way to describe the role of the capacitor in the above example is to say that its impedance to an electric signal decreases with increasing frequency. In the example, much of the signal is shunted to ground instead of appearing at point b.