**What is an RC Circuit?**

- An RC circuit contains both resistance and capacitance. The capacitor, as described in Capacitance, is an electrical component that stores electric charge, storing energy in an electric field.
- A dc (direct current) voltage source, a resistor R, a capacitor C, and a two-position switch are shown in the diagram below. Depending on the position of the switch, the circuit allows the capacitor to be charged or discharged. The capacitor charges when the switch is moved to position (A), resulting in the circuit shown in the Figure below. The capacitor discharges through the resistor when the switch is moved to position B.

- The capacitor stores energy and the resistor in the circuit regulates the rate of charging and discharging.
- The charging and discharging of the capacitor is a slow process that takes time. When the resistor and capacitor are connected in series, the capacitor gradually charges through the resistor until the voltage across the resistor equals the supply voltage.

## RC Circuit with Time Delay or Time Constant

- Before we proceed to the RC charging circuit and capacitor charging equation, we should first define the term Time Constant. This time delay or time constant can be found in any electrical or electronic circuit.
- When an electrical circuit is powered by a voltage or signal in direct current (DC) or alternating current (AC), there will be some “time delay” between the input and output terminals (AC).
- Moving on, this Time Constant represents the first-order time response of the signal or voltage-supplied circuit. This time constant value is affected by the reactive components in the circuit, such as the capacitor and inductor.
- If we try to solve an equation for capacitor charging, we will come across the time constant quite frequently.
- The Time Constant unit is Tau, with the symbol –
- Let’s start with a circuit that has an “empty” capacitor. This is referred to as a “discharged” capacitor. The current begins to flow after we apply a DC voltage to the circuit. The capacitor draws this current, which we refer to as “charging current.”
- As long as the DC voltage source is applied, the capacitor begins to “charge up.” As soon as the voltage is reduced, the capacitor begins “discharging” in the opposite direction of the voltage source.
- The stored electrical charge between the conductor plates can be calculated using the following equation:

Q = C.V - This charging and discharging of a capacitor takes time. For calculating the required time, we use the term “Time Constant.”
- This will also serve as the capacitor charging equation.
- In summary, the Time Constant is the time required to charge a capacitor through a resistor from zero to around 63.2% of the applied DC voltage source. The Time Constant is also used to calculate the time required to discharge the capacitor using the same resistor, which is approximately 36.8% of the initial charge voltage.
- The RC circuit is made up of a series connection of the previously mentioned resistor, capacitor, and voltage source. In an ideal world, the capacitor will gradually charge up its charge voltage until it matches the voltage source.
- The time it takes for the capacitor to fully charge is also known as the transient response time.

- The value can be calculated by taking the product of resistance and capacitance. Hence,Where:Time constant expressed in seconds (s)R denotes resistance in ohms (ohm)C denotes capacitance in Farads (F)

## Capacitor Charging RC Circuit

- To understand how the capacitor charges, we can apply Kirchhoff’s loop rule. This gives rise to the equation ϵ−VR−VC=0. As the capacitor charges, this equation can be used to model the charge as a function of time. Because capacitor capacitance is defined as C=q/V, the voltage across the capacitor is VC=qC. The potential drop across the resistor is defined by Ohm’s law as VR=IR, and the current is defined as I=dq/dt.

- This differential equation can be integrated to obtain an equation for the capacitor charge as a function of time.

- Let,

- then,

- The result is,

- Simplifying yields the following equation for the charging capacitor charge as a function of time:

- Figure below depicts a graph of the charge on the capacitor versus time. First, as the time approaches infinity, the exponential approaches zero, so the charge approaches the maximum charge Q=C and has coulomb units. The units of RC are seconds or time units. This is referred to as the time constant:

- At time t=τ=RC, the charge equal to 1−e−1=1−0.368=0.632 of the maximum charge Q=Cϵ. Notice that the time rate change of the charge is the slope at a point of the charge versus the time plot. The slope of the graph is large at time t−0.0s and approaches zero as time increases.
- The current through the resistor decreases as the charge on the capacitor increases, as shown in the Figure below. Taking the time derivative of the charge yields the current through the resistor.

- The current through the resistor is I0=R at time t=0.0s. The current approaches zero as time approaches infinity. The current through the resistor at time t= is I(t=)=I0e1=0.368I0.

## Example of RC Charging Circuit

- Determine the RC time constant for the following circuit.

- T = R x C in seconds is the formula for calculating the time constant.
- As a result, the time constant is given as T = R x C = 47k x 1000uF = 47 Secs.

**a) What is the voltage across the capacitor plates at precisely 0.7-time constants?**

- Vc = 0.5Vs at 0.7 time constants (0.7T). As a result, Vc = 0.5 x 5V = 2.5V.

**b) What is the voltage across the capacitor at one time constant?**

- Vc = 0.63Vs at 1 time constant (1T). As a result, Vc = 0.63 x 5V = 3.15V.

**c) How long will it take for the capacitor to “fully charge” from the supply?**

- We discovered that after 5-time constants, the capacitor will be fully charged (5T).
- One time constant (1T) equals 47 seconds (from above). As a result, 5T = 5 x 47 = 235 secs.

**d) After 100 seconds, what is the voltage across the capacitor?**

Because the voltage formula is Vc = V(1 – e(-t/RC)), this becomes Vc = 5(1 – e(-100/47)).

Where V equals 5 volts, t equals 100 seconds, and RC equals 47 seconds from above.

Vc = 5(1 – e(-100/47)) = 5(1 – e-2.1277) = 5(1 – 0.1191) = 4.4 volts as a result.

- We’ve seen that the charge on a capacitor can be calculated using the formula Q = CV, where C is the fixed capacitance value and V is the applied voltage. We’ve also learned that when a voltage is applied to the capacitor’s plates for the first time, it charges up at a rate determined by its RC time constant and is considered fully charged after five time constants, or 5T.

## Applications of RC Charging Circuit

- Low pass and high-pass filters can be implemented using RC circuits in general.
- The LM555 timer employs an RC circuit.
- They are commonly found in traffic lights.
- In pacemakers, RC circuits are used.
- It is suitable for use in audio equipment.
- Signal filtering can be accomplished with RC circuits.
- It can function as both an integrator and a differentiator.