RC Charging Circuit

Yaser Rahmati | یاسر رحمتی

Capacitor Charging

The figure below shows a capacitor, ( $C$ ) in series with a resistor, ( $R$ ) forming an RC Charging Circuit connected across a DC battery supply ( $V_{s}$ ) via a mechanical switch.

RC Charging Circuit

At time zero, when the switch is first closed, the capacitor gradually charges up through the resistor until the voltage across it reaches the supply voltage of the battery. The manner in which the capacitor charges up is shown below.

Charging the capacitor

Let us assume above, that the capacitor, $C$ is fully “discharged” and the switch (S) is fully open. These are the initial conditions of the circuit, then $t=0$, $i=0$ and $q=0$. When the switch is closed the time begins at $t=0$ and current begins to flow into the capacitor via the resistor.

Since the initial voltage across the capacitor is zero, ( $V_{C}=0$ ) at $t=0$ the capacitor appears to be a short circuit to the external circuit and the maximum current flows through the circuit restricted only by the resistor $R$. Then by using Kirchhoff’s voltage law (KVL), the voltage drops around the circuit are given as:

$$V_{s}-Ri(t)-V_{c}=0$$

So, we have:

$$V_{s}-R\frac{dQ}{dt}-\frac{Q}{C}=0$$

RC Time Constant, Tau

This RC time constant only specifies a rate of charge where $R$ is in $\Omega$ and $C$ in Farads.

$$\tau = R\times C$$

Since voltage V is related to charge on a capacitor given by the equation, $V_{C}=Q/C$ , the voltage across the capacitor ( $V_{C}$ ) at any instant in time during the charging period is given as:

$$V_{c}=V_{s}(1-e^{-\frac{t}{RC}})$$

RC Charging Table

Time Constant	RC Value	Percentage of Maximum Voltage	Percentage of Maximum Current
0.5 time constant	0.5T = 0.5RC	39.3%	60.7%
0.7 time constant	0.7T = 0.7RC	50.3%	49.7%
1.0 time constants	1T = 1RC	63.2%	36.8%
2.0 time constants	2T = 1RC	86.5%	13.5%
3.0 time constants	3T = 1RC	95.0%	5.0%
4.0 time constant	4T = 1RC	98.2%	1.8%
5.0 time constants	5T = 1RC	99.3%	0.7%

Example (Transient Analysis)

1. Circuit Diagram

Circuit Diagram

1. Voltage of Capacitor

Voltage of Capacitor

1. Current

Current

Python

import sympy as sp

# Define the symbols
t = sp.symbols('t')

# Define the time constant tau = R * C
C = 100 * 10**(-6)
R = 2000
tau = R * C

# Define the expression for the voltage across the capacitor
V = 5
Vc = V * (1 - sp.exp(-t / (tau)))

# Print the voltage across the capacitor
sp.plot(Vc , (t, 0, 5*tau))

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