Question

What happens in an RC circuit when the switch is suddenly closed?

Answer 1

When a switch in an RC circuit is suddenly closed, the circuit will undergo a transient response. The behavior of the circuit during this transient period depends on the values of the resistor (R) and capacitor (C) and the initial conditions.

An RC circuit consists of a resistor and a capacitor connected in series or parallel. When the switch is closed, the capacitor begins to charge up through the resistor. Let's look at what happens in each case:

Charging in Series RC Circuit:
In a series RC circuit, the capacitor charges up to the applied voltage (V) gradually. Initially, when the switch is closed, the capacitor behaves like a short circuit, and the current flows through the resistor and the capacitor. The charging current starts high and gradually decreases as the capacitor voltage approaches V.

The charging process is governed by the following equation:


(

)
=

×
(
1
−

−

/
(

×

)
)
V(t)=V×(1−e
−t/(R×C)
)

Where:


(

)
V(t) is the voltage across the capacitor at time

t.

V is the applied voltage.

R is the resistance in ohms.

C is the capacitance in farads.

e is the base of the natural logarithm.
Charging in Parallel RC Circuit:
In a parallel RC circuit, the capacitor charges up similarly to the series case, but the voltage across the capacitor increases instantly. When the switch is closed, the capacitor starts to charge up rapidly through the resistor, reaching its final voltage almost immediately.

The voltage across the capacitor during charging is given by:


(

)
=

×
(
1
−

−

/
(

×

)
)
V(t)=V×(1−e
−t/(R×C)
)

The time it takes for the capacitor to charge to approximately 63.2% of the applied voltage (V) is called the time constant (τ) of the circuit and is given by τ = R × C.

It's essential to note that during the transient period, the current and voltage change continuously, eventually reaching a steady-state condition where the capacitor is fully charged (in the case of a charging circuit) and the current becomes negligible (in the case of a discharging circuit).

If the circuit was previously in a steady state with the switch open for a long time, the initial voltage across the capacitor would be 0V (discharged), and it would start charging or discharging from this initial condition based on the direction of the current when the switch is closed.