RC Circuit

Timers are a vital instrument in almost all the things we do on a daily basis. Whether in the kitchen to keep track of how long the turkey has been in the oven, or in a physics lab when conducting experiments to measure the speed of sound. Electrical devices that allow us to track how much time has elapsed over a period are made up of minuscule circuits containing an array of resistors and capacitors, also referred to as Resistor-Capacitor (RC) Circuits.

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Team RC Circuit Teachers

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    Resistor-Capacitor (RC) Circuits Traffic light StudySmarterFig. 1 - The electrical system behind a traffic light is maintained by an RC circuit.

    An especially important role these circuits are involved in is the maintenance and timing of traffic lights. These devices, which indicate to drivers when it is safe for them to continue, are vital to road safety worldwide. By varying the number of resistors and the charge on the capacitor in RC circuits, engineers can use these to change the time allocated to a red light or a green light. To learn more about RC circuits and how they work, keep reading!

    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 \(R\).

    Resistor-Capacitor (RC) Circuits RC circuit setup StudySmarterFig. 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.

    The time constant of an RC circuit is the time it takes for the capacitor to increase or decrease its voltage to a fraction of \(\frac{1}{e}\approx0.632\) (where \(e\) is the famous Euler's number) of its initial or final value, depending on whether the system is charging or discharging.

    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.

    A charging RC circuit is when the capacitor and resistor are connected to a power source, allowing for opposite charges to build up on the plates of the capacitor.

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

    A discharging RC circuit is a charged capacitor connected to a resistor, with the capacitors electrical energy flowing through the circuit as current.

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

    Resistor-Capacitor (RC) Circuits Discharging voltage graph StudySmarter

    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.

    A steady state is defined as a state in which a system, on average, has no significant fluctuations.

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

    Resistor-Capacitor (RC) Circuits Voltage charging graph StudySmarter

    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.

    A low-pass filter is a circuit that allows the lower frequencies in a signal to go through while filtering out and removing the higher frequencies in the signal.

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

    The impedance of an electrical component measures how much the component opposes an alternating current.

    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 \(R = \frac{V}{I}\).

    Resistor-Capacitor (RC) Circuits Low-pass filter StudySmarterFig. 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.

    The cut-off frequency is the point at which a signal with frequencies lower than this value is removed from the 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.

    Consider an RC circuit comprising of a resistor with resistance \(R = 4.5 \, \Omega\) and a capacitor of capacitance \(C = 7.2 \, \mathrm{\mu F}\). The initial voltage in the circuit is \(V_0 = 5.0 \, \mathrm{V}\).

    1. What is the voltage at time \(t = 1.4 \times 10^{-6}\, \mathrm{s}\)?
    2. What is the time constant of this RC circuit?
    3. What would the cut-off frequency be if this RC circuit was used as a low-pass filter?

    1. In order to find the voltage at a specific time, we can use our equation derived from the first-order differential equation. Substituting our values results in

    \[ \begin{align} V(t) &= V_0 e^{\frac{-t}{CR}} \\ V(t = 1.4 \times 10^{-6}\, \mathrm{s} ) &= 5.0 \, \mathrm{V} \times e^{- \frac{1.4 \times 10^{-6}\, \mathrm{s}}{7.2 \times 10^{-6} \, \mathrm{F} \times 4.5 \, \Omega }} \\ V(t = 1.4 \times 10^{-6} \, \mathrm{s} ) &= 4.8 \, \mathrm{V} \end{align} .\]

    2. To find our time constant, we can use our derived expression for the time constant as

    \[ \begin{align} \tau &= CR \\ \tau &= 7.2 \times 10^{-6} \, \mathrm{F} \times 4.5 \, \Omega \\ \tau &= 3.2 \times 10^{-5} \, \mathrm{s} . \end{align} \]

    3. Finally, using our expression defined in the section before, we can find the cut-off frequency as

    \[ \begin{align} f_{\text{C}} &= \frac{1}{2\pi RC} \\ f_{\text{C}} &= \frac{1}{2 \pi \times 4.5 \, \Omega \times 7.2 \times 10^{-6} \, \mathrm{F} } \\ f_{\text{C}} &= 4900 \, \mathrm{Hz} . \end{align} \]

    RC Circuit - Key takeaways

    • An RC circuit consists of a resistor connected to a capacitor.
    • The voltage of an RC circuit can be derived from a first-order differential equation, and is given by \(V(t) = V_0 e^{\frac{-t}{CR}}\).
    • An RC circuit can be in a charging state when connected to a power source, allowing for the capacitor to build up electrical energy.
    • When disconnected from the power source, the RC circuit is in a discharging state as the electrical energy stored in the capacitor discharges as a current.
    • The RC circuit eventually reaches a steady-state and has a time constant.
    • An RC circuit can act as a low-pass filter, filtering out higher frequencies in signals.

    References

    1. Fig. 1 - Traffic light, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Traffic_Light,_Vienna.jpg) Licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/)
    2. Fig. 2 - RC circuit, StudySmarter Originals.
    3. Fig. 3 - Discharging voltage graph, StudySmarter Originals.
    4. Fig. 4 - Charging voltage graph, StudySmarter Originals.
    5. Fig. 5 - Low-pass filter, StudySmarter Originals.
    Frequently Asked Questions about RC Circuit

    What is an RC circuit?

    A circuit with a resistor and capacitor.

    What are RC circuits used for?

    Low-pass filters.

    What are examples of RC circuits?

    An example of an RC circuit is the timers for traffic lights.

    How to find resistance in an RC circuit?

    Finding the total resistance of all the resistors in the circuit.

    What is the formula and applications of RC circuit?

    The formula of an RC circuit is given by solving a first-order differential equation.

    How to calculate current in an RC circuit?

    The current in an RC circuit can be found by calculating the total charge, then relating that to current.

    How to calculate impedance in an RC circuit?

    The impedance of an RC circuit is given by the impedance of the individual electrical components.

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