RC circuits are a ubiquitous part of many electronic devices, from simple filters to complex devices: use our RC circuit calculator to find the best parameters to build the suitable circuit for your needs. Use our calculator for low-pass filters and high-pass filters as well; the math is safe for both!

In this short article, we will explain to you:

  • What is an RC circuit, how does it operate, and which are the two types of RC circuits;
  • How to calculate the RC circuit time constant;
  • What is the cut-off frequency of an RC circuit and its formula: calculate high-pass filter and low-pass filter characteristic values.

What is an RC circuit? Why do we use RC circuits?

RC circuits are simple electronic circuits made of the two most basic passive components:

  • A resistor; and
  • A capacitor.

RC circuits have a comprehensive set of applications in circuitry: their simple construction and characteristic make them a fundamental part of timing and filtering elements in more complex circuits.

You can see resistors as voltage-to-current transducers. In an RC circuit, the resistor defines the resistance on one side of the capacitor, surpassing the capacitor's wires resistance by an amount large enough to allow it to be fixed and known. The resistance "generates" the current feeding the capacitor (the charging current) and defines both the charging and the discharging processes.

How does an RC circuit operate?

An RC circuit operates in different ways according to the position of the resistor and the capacitor. However, the general mechanism is the same.

Let's see the two types of RC circuits first.

If resistor and capacitor are in series, we feed the circuit with a voltage source. In this configuration, we sample the voltage across one of the two components accordingly to their relative position.

  • If the resistance is placed between the capacitor and the power source positive pole, we sample the signal across the capacitor.
  • If the capacitor is placed before the resistance, we sample across the resistance.

Due to the high resistance of the capacitor to low frequencies, the different placement of the two components allows for two vastly different behaviors of the circuit.

If we sample across the capacitor, we allow the higher frequencies to pass quickly from source to ground, leaving on the sides of the capacitor the low frequencies. We built a low-pass filter.

If the capacitor is placed before the sampling on the resistance, we oppose a great resistance to the passage of the lower frequencies, hence letting through only the higher ones. On the resistance, we will sample only the latter: this is the simplest possible passive high-pass filter.

If resistor and capacitance are in parallel to each other, we can't feed the circuit with a voltage anymore since the drops would be the same across both components. It is convenient to feed the circuit with a current and measure the variation in the related signal.

Parallel RC circuits are vastly less common than their series counterparts but still operate using the same physics.

In the following sections, we will learn how to calculate the time constant of an RC circuit (a formula valid for all the previously seen configurations) and then calculate the high-pass filter/low-pass filter cut-off frequency.

How to calculate the time constant of an RC circuit

The time constant of an RC circuit defines the time required by the capacitor to charge. In particular, since charging a capacitor is an exponential process that slows with time, the time constant corresponds to the time required by the charge to reach a value of approximately 1e163.21-e^{-1}\approx 63.2% the value of the applied voltage.

We calculate the RC circuit time constant with an extremely simple formula that relates the values of resistance and capacitance in our circuit:

τ=RC\tau = R\cdot C

The measurement units of the involved quantities are the SI ones:

  • τ\tau is measured in seconds;
  • RR, the resistance, is measured in ohm, Ω\text{Ω}; and
  • CC, the capacitance, is measured in farad, F\text{F}

Why do we calculate the RC circuit time constant? Its value defines the characteristic operating times of the circuit. If you turn on the current, the circuit will reach its steady state after 5τ5\tau, even in front of an instantaneous rise in voltage. This smoothing effect is also visible when the power source is turned off. Rather than vanishing immediately, an RC circuit turns off gradually, following an exponential decline in the output voltage across its terminals.

If fed with a square wave, an RC circuit returns a smoothed albeit non-constat output. If fed with a signal, the smoothing results in the reduction of noise.

Calculate the high-pass filter/low-pass filter cut-off frequency

From calculating the time constant of an RC circuit to finding the cutoff frequency, the step is short. The formula to calculate the low-pass filter frequency is the same used for the high-pass filter circuit: two sides of the same coi- of the same spectrum.

f=12πRCf = \frac{1}{2\pi\cdot R\cdot C}

The measurement unit of the cut-off frequency is, of course, the Hertz. As you can see, only the values of capacitance and resistance matter in calculating the value of ff. This means there is no way to identify the disposition of the two components: the cut-off is one for both the high-pass filter and the low-pass filter. If we use a low-pass filter, we will save the frequencies below the cut-off. If we want to let through only the higher frequencies, we switch the position of the capacitor and resistance.

Davide Borchia
Charging time
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