RC Charging Circuit

The Time Constant

All Electrical or Electronic circuits or systems suffer from some form of "time-delay" between its input and output, when a signal or voltage, either continuous, (DC) or alternating (AC) is firstly applied to it. This delay is generally known as the time delay or Time Constant of the circuit and it is the time response of the circuit when a step voltage or signal is applied. The resultant time constant of any circuit or system will mainly depend upon the reactive components either capacitive or inductive connected to it and is a measurement of time with units of, Tau - τ

When an increasing DC voltage is applied to a Capacitor the capacitor draws a charging current and "charges up", and when the voltage is reduced, the capacitor discharges. Because capacitors store electrical energy they act like small batteries and are able to store or release the energy as required. This charging (storage) and discharging (release) of a capacitors energy is never instant but takes a certain amount of time to occur with the time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value being known as its Time Constant (τ).

If a resistor is connected in series with the capacitor forming a RC charging circuit, the capacitor will then charge up gradually through the resistor until the voltage across the capacitor reaches that of the supply voltage. The time required for this to occur is equivalent to about 5 time constants or 5T. This time constant T, is measured by τ = R x C, in seconds, where R is the value of the resistor in ohms and C is the value of the capacitor in Farads. This then forms the basis of an RC charging circuit where 5T can also be thought of as "5 x RC".

RC Charging Circuit

The figure below shows a Capacitor, (C) in series with a Resistor, (R) forming a RC Charging Circuit connected across a DC battery supply (Vs) via a mechanical switch. When the switch is closed, the capacitor will gradually charge 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 also shown below.

RC Charging Circuit

RC Charging Curves

Let us assume that the Capacitor, C is fully "discharged" and the switch is open. 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, (Vc = 0) the capacitor appears to be a short circuit and the maximum current flows through the circuit restricted by resistor R. This current is called the Charging Current and is found by using the formula: i = Vs/R.

The capacitor now starts to charge up with the actual time taken for the charge on the capacitor to reach 63% of its maximum possible voltage, in our curve 0.63Vs is known as the Time Constant, (T) of the circuit and is given the abbreviation of 1T.

So we can say that the time required for a capacitor to charge up to one time constant is given as:

Where, R is in Ω's and C in Farads.

The value of the voltage across the capacitor, (Vc) at any instant in time during the charging period is given as:

• Where:
• Vc is the Voltage across the Capacitor
• V is the Supply Voltage
• t  is the elapsed time since the application of the supply voltage
• RC is the Time Constant of the RC Charging Circuit

After a period equivalent to 4 time constants, (4T) the capacitor in this RC charging circuit is virtually fully charged and the voltage across the capacitor is now approx 99% of its maximum value, 0.99Vs. The time period taken for the capacitor to reach this 4T point is known as the Transient Period. After a time of 5T the capacitor is now fully charged and the voltage across the capacitor, (VC) is equal to the supply voltage, (Vs). As the capacitor is fully charged no more current flows in the circuit. The time period after this 5T point is known as the Steady State Period.

As the voltage across the capacitor Vc changes with time, and is a different value at each time constant up to 5T, we can calculate this value of capacitor voltage, Vc at any given point, for example.

Example No1.

Calculate the time constant of the following circuit.
	The time constant τ is found using the formula T = R x C in seconds. Therefore the time constant τ is: T = R x C = 47k x 1000uF = 47 Secs
a) What value will be the voltage across the capacitor at 0.7 time constants?
At 0.7 time constants (0.7T) Vc = 0.5Vs. Therefore, Vc = 0.5 x 5V = 2.5V
b) What value will be the voltage across the capacitor at 1 time constant?
At 1 time constant (1T) Vc = 0.63Vs. Therefore, Vc = 0.63 x 5V = 3.15V
c) How long will it take to "fully charge" the capacitor?
The capacitor will be fully charged at 5 time constants.         1 time constant (1T) = 47 seconds, (from above). Therefore, 5T = 5 x 47 = 235 secs
d) The voltage across the Capacitor after 100 seconds?
The voltage formula is given as Vc = V(1 - e-t/RC)         which equals: Vc = 5(1-e-100/47)    RC = 47 seconds from above, Therefore, Vc = 4.4 volts