- Capacitor Transient Response Definition: The transient response of a capacitor is the period during which it charges or discharges, changing its voltage and current over time.
- Charging Behavior: When a voltage is applied, the capacitor charges, with the current starting high and decreasing to zero as the voltage across it increases.
- Time Constant ( 𝜏 ): The time constant in an RC circuit, given by 𝜏 = 𝑅 𝐶 τ=RC, is the time it takes for the voltage to reach 63.2% of its final value.
- Discharging Behavior: When disconnected from the power source and short-circuited, a capacitor discharges, with the voltage and current decreasing exponentially to zero.
- Kirchhoff’s Laws in Capacitor Circuits: Kirchhoff’s Voltage Law helps determine the relationship between voltage and current in a capacitor during its transient response.
When a voltage is suddenly applied through a resistor to an initially uncharged capacitor, charge begins to accumulate on its plates. The capacitor voltage cannot change instantaneously in the ideal model. It rises toward the supply voltage while the charging electric current falls toward zero. The changing voltage and current form the circuit’s transient response.
For an initially uncharged series RC circuit, the current starts at its maximum value V/R. It then decays exponentially as the capacitor voltage rises.
The circuit below shows the resistor R and capacitor C connected to a DC source.
Close switch S at t = 0 and denote the instantaneous current by i(t).
Denote the instantaneous voltage across the capacitor by Vc(t).
Applying Kirchhoff’s Voltage Law around the loop gives:
If q is the charge transferred by time t, the current is its rate of change:
Therefore:
Substitute this expression for i(t) into equation (i):
Integrating both sides with respect to time gives:
The integration constant K follows from the initial condition.
Set t = 0 at the switching instant. Substitution into the equation gives:
Because the capacitor is initially uncharged, its voltage is zero at t = 0.
Therefore:
At t = RC, the equation becomes:
The product of resistance and capacitance is the RC circuit’s time constant, τ. It has units of seconds. After one time constant, the capacitor voltage has completed 63.2% of its change from the initial value toward the final value. For an initially uncharged capacitor connected to V, this is 0.632V.
At t = 0, the ideal initially uncharged capacitor has zero voltage. Equation (ii) gives the same result.
The initial current is V/R; denote it by I0.
At any later time, the current is:
At t = RC, the circuit current is:
After one time constant, the current through the capacitor is about 36.8% of its initial magnitude.
The Greek letter τ (tau) normally denotes the time constant:
Transient Response While a Capacitor Discharges
Suppose the capacitor has an initial voltage equal to the source voltage. Disconnect the voltage source and complete a discharge path through resistance R rather than through the battery. Charge then moves between the plates through the resistor, and the capacitor voltage approaches zero exponentially. The following circuit shows the transient behavior of capacitor voltage and current during discharge.
Using Kirchhoff’s loop rule together with the conservation principles discussed under Kirchhoff Current Law gives:
Integrating both sides gives:
The initial capacitor voltage determines the constant K. At the instant the discharge path is completed:
Substituting t = τ = RC into equation (iii) gives:
The current magnitude at the same instant is:
After one time constant, the capacitor voltage ϑc and the magnitude of current i are about 36.8% of their initial values. The current direction is opposite to the charging-current reference direction.
