RC Circuit

RC Circuits Definition

First, let's define an RC circuit and how it works. Referring to the figure below, we have a simple circuit connecting a charged capacitor with capacitance \(C\), to a resistor with resistance \(R\).

Fig. 2 - The setup of an RC circuit.

When a charged capacitor is connected to the resistor, it discharges its stored electrical energy through the electric current in the circuit. However, the value of the current is determined by the resistance of the resistor; adding more resistors in series and parallel will alter the resultant current and voltage in the circuit. Over a period of time, the electrical energy stored in the capacitor will be depleted, resulting in zero current. Applications of multiple resistor RC circuits include low-pass filters, which will be discussed in later sections of this article.

RC Circuits Equations

Now that we understand the setup of an RC circuit, let's look into how we can describe what's going on mathematically. Firstly, let's think about the current flowing across both components. According to Kirchoff's current law, the current flowing across the capacitor must be equal and opposite to that of the current flowing across the resistor. We can write this as

\[ I_{\text{C}} + I_{\text{R}} = 0 ,\]

where \(I_{\text{C}}\) is the current running across the capacitor and \(I_{\text{R}}\) is the current running across the resistor, both of which are measured in amperes \(\mathrm{A}\). Now we can substitute in our equations relating current to the voltage across the component, which for a capacitor is

\[ I_{\text{C}} = C \frac{\mathrm{d} V}{\mathrm{d} t} ,\]

where \(C\) is the capacitance measured in farads \(\mathrm{F}\), \(V\) is the voltage measured in volts \(\mathrm{V}\), and \(t\) is the time measured in seconds \(\mathrm{s}\). For a resistor, this is given by Ohm's law

\[ I_{\text{R}} = \frac{V}{R} ,\]

where \(R\) is the resistance measured in ohms \(\Omega\).

Now we can substitute these expressions into our current law equation to give

\[ C \frac{\mathrm{d} V}{\mathrm{d} t} + \frac{V}{R} = 0,\]

which is a first-order differential equation.

Solving this will give us the expression of the voltage in the circuit with respect to time. Firstly, we rearrange the equation slightly as

\[ \frac{\mathrm{d}V}{\mathrm{d}t} = - \frac{1}{CR} V ,\]

where we have separated the differentials from the constants and variables on either side. You may solve this differential equation with whatever method you feel comfortable with. In this article, we will use the separation of variables method, resulting in

\[ \begin{align} \int^{V}_{V_0} \frac{1}{V'} \mathrm{d} V' &= \int^{t}_0 - \frac{1}{CR} \mathrm{d} t \\ \ln(V) - \ln(V_0) &= \frac{-t}{CR} \\ \ln(\frac{V}{V_0}) &= \frac{-t}{CR} \\ \frac{V}{V_0} &= e^{\frac{-t}{CR}} \\ V(t) &= V_0 e^{\frac{-t}{CR}} . \end{align} \]

Thus resulting in the equation of the voltage within an RC circuit.

Time Constant in an RC Circuit

An important characteristic of an RC circuit is its time constant.

Another way to think about it is the time taken for the system's initial value to reach \(\frac{V_0}{e} \). Thus, we can use our derived equation for voltage and substitute that expression to determine our time constant for an RC circuit. This is given by

\[ \begin{align} \frac{V_0}{e} &= V_0 e^{\frac{-\tau}{CR}} \\ \frac{\bcancel{V_0}}{e} &= \bcancel{V_0} e^{\frac{-\tau}{CR}} \\ e^{-1} &= e^{\frac{-\tau}{CR}} \\ -1 &= \frac{-\tau}{CR} \\ \tau &= CR , \end{align} \]

where we have denoted the time constant as \(\tau\) measured in units of seconds \(\mathrm{s}\). As a result, we can see that the characteristic time constant of a circuit is dependent on the total resistance and total capacitance within the circuit.

Discharging RC Circuit

An RC circuit can be in two orientations: charging and discharging.

On the other hand, we also have a discharging capacitor.

We have already seen how to derive the voltage within a discharging RC circuit, thus we can represent it graphically in the figure below.

Fig. 3 - The voltage across the capacitor in a discharging RC circuit.

Thus we can see the exponential decay of the amount of voltage present in the circuit. After a period of time, as the capacitor runs out of electrical energy, the voltage in the circuit will reach a steady state and approach zero.

On the other hand, a charging RC circuit will follow an inverse pattern, resulting in the graph below.

Fig. 4 - The voltage across the capacitor of a charging RC circuit.

Here we can see that a charging RC circuit obeys an inverse pattern, increasing in voltage exponentially over time and plateauing after a certain period. The maximum voltage reached by the capacitor is determined by the total resistance and capacitance within the circuit. This graph can be represented mathematically as

\[ V(t) = V_0 \left( 1 - e^{\frac{-t}{CR}} \right) .\]

Cutoff Frequency of an RC Circuit

An application of RC circuits is a low-pass filter.

In order to fully understand how a low-pass filter works, we must understand impedance and how it is similar to resistance.

This indeed sounds similar to the definition of resistance, and impedance even has the same units of resistance, the ohm \(\Omega\). Impedance is a more generalized notion of resistance, whilst resistance is specifically defined as \(R = \frac{V}{I}\).

Fig. 5 - The set-up of a low-pass filter using an RC circuit.

In the figure above, we have the layout of a low-pass filter that uses an RC circuit. The \(V_{\text{in}}\) represents the input signal into the circuit. It then passes through the resistor and then into a capacitor and load that are placed in parallel with respect to one another. The load in this case represents the output signal \(V_{\text{out}}\).

If we had an input signal with a high frequency, there would be a higher impedance across the resistor than the capacitor, resulting in a higher potential difference across the resistor than the capacitor. Since the capacitor is in parallel with the load, that results in a low potential difference across the load as well, reducing the amount of output signal from the circuit.

On the other hand, if our signal was made up of lower frequencies, there would be a higher impedance across the capacitor than the resistor. Thus we would have a higher potential difference between the capacitor and the load, resulting in a larger output signal.

We can also define the cut-off frequency mathematically as

\[ f_{\text{C}} = \frac{1}{2\pi RC} ,\]

where \(f_{\text{C}}\) is the cut-off frequency measured in hertz \(\mathrm{Hz}\), \(R\) is the resistance of the resistor measured in ohms \(\Omega\), and \(C\) is the capacitance of the capacitor measured in farads \(\mathrm{F}\).

RC Circuits Examples

Finally, let's consider an example question concerning an RC circuit.