When a discharged capacitor is suddenly connected across a DC supply, such as E_{s} in figure 1 (a), a current immediately begins to flow. At time t_{1} (Figure 1 (b)), the moment the circuit is closed, the capacitor acts like a short circuit. The in-rush current i_{C} is at its maximum value and is limited only by the series resistor R. however, as soon as the capacitor begins o charge, electrons from the supply being to build upon the lower plate of the capacitor. Simultaneously, electrons from the upper plate are attracted to the positive terminal of the supply. A difference of potential now begins to appear across the capacitor. The polarity, being the same as that of the source, opposes E_{s}.

**Exponential Curve**

A curve that varies by the square or some other power of a factor instead of linearity.

The net voltage available to charge the capacitor is the difference between the capacitor voltage v_{C} and E_{s}. (Note: lowercase letters are used to designate voltage and current values that are changing). As V_{C}, capacitor’s emf increases, this net voltage diminishes. Consequently, the rate of charge slows down. This cumulative effect continues until V_{C} is approximately equal to E_{s}, at which time the charging current i_{C} is reduced to nearly zero. See figure 1 (b). Theoretically, the capacitor never reaches a full charge, but most capacitors can be considered fully charged in several seconds or less. Curves i_{C} and V_{C} in figure 1 (b) are examples of exponential curves. An exponential curve has the property of dropping or rising very quickly toward a limiting value. The closer it goes to the limit, the more gradual its approach becomes.

Fig.1: Effect in an RC Circuit: (a) RC Circuit (b) Current and Voltage Waveforms for Closed Switch

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**Effect of R and C on Charging Time**

Inasmuch as the charging current must flow through the series resistor, this current inevitably has an effect on the rate of charge.

**Changing Resistance**

If the resistance is increased (C unchanged), the IR drop produced by the charging current in greater, and the net voltage charging the capacitor is reduced. Therefore, the capacitor takes a longer period of time to reach full charge when the series resistance in increased. Of course, the reverse is true if R is made smaller.

**Changing Capacitance**

Consider what happens when the series resistance remains constant but the capacitance increases. The initial in-rush of current is the same as illustrated in figure 1, assuming the same supply voltage. Since the capacitor’s charge capacity Q is equal to CV, it is apparent that more time will be required to charge that capacitor because C is larger. The emf across the capacitor builds up more slowly, causing the rate of charge to be slowed down. Hence, the charging time of a capacitor is directly proportional to its capacitance.

**Time constant $\tau =RC$**

Whenever a voltage or current constantly changes value, it exhibits transient effects. The voltage across the resistance and capacitance in an RC circuit have these characteristics. They are of a transient nature until reaching steady-state values.

**Time constant τ**

In a capacitor, the time required for a voltage to reach 63.2 % of the steady-state or full charge value. In an inductor, the time required for a current to reach 63.2 % of full or steady-state value.

When analyzing the amount of time it takes an RC circuit to reach a steady state condition, we must deal with a term referred to as circuit’s time constant. Expressed mathematically, the time constant τ is as follows:

$\tau =RC$

The time constant τ (Greek lowercase letter tau) is expressed in seconds when R is in ohms and C is in farads. That τ is expressed in seconds can be derived as follows:

\[\tau =R*C=ohms*farads=\frac{coulombs}{volts}*\frac{volts}{amperes}\]

\[\text{ }\tau \text{ =}\frac{\text{coulombs}}{\text{amperes}}\text{=}\frac{\text{coulombs}}{\text{coulombs/seconds}}\text{=seconds}\]

Now, the circuit’s time constant τ represents the time required for the voltage across the capacitor to reach 63.2 % of the steady-state or full-charge value. It takes four more time constants for V_{C} to reach a charge value negligibly different from its full-charge values, demonstrated by the graph in figure 2.

Fig.2: Curve of Time Constant in RC Circuit

**Example**

Calculate the time constant of a series RC circuit when R=200 kΩ and C=3 μF.

**Solution**

$\tau =RC=(200*{{10}^{3}})*(3*{{10}^{-6}})=0.6s$

**Example**

What value of resistance must be connected in series with a 20μF capacitor to provide a τ of 0.1 s?

**Solution**

$R=\frac{\tau }{C}=\frac{0.1}{20*{{10}^{-6}}}=5*{{10}^{3}}\Omega $

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