What Is the Impedance of an RLC Circuit?

January 28, 2021

Key Takeaways

• RLC circuits are fundamental as basic circuits found in many systems.

• RLC circuits are also used as tools to understand and explain complex electrical behavior.

• Some RLC circuits require a circuit simulator to understand electrical behavior as these circuits can be rather complex. Here’s how to determine the impedance of an RLC circuit.

RLC circuits are basic building blocks of more complex analog systems and provide many useful features. Passive amplification, filtering, impedance matching, and more can be accomplished with RLC circuits. Also, RLC circuits are used as fundamental models for more complex portions of electrical systems, such as the PDN in a PCB. In all of these areas, a designer needs to know the impedance of an RLC circuit as they create their design.

How to Determine the Impedance of an RLC Circuit

Series and parallel arrangements of RLC components are the easiest to address as the common formulas for equivalent resistance can be used with the impedances of RLC elements. Only 3 simple mathematical tools are needed to simulate RLC circuits:

• Kirchoff’s current law

• Kirchoff’s voltage law

• Ohm’s law

More complex RLC circuits may not have the same form of impedance equation as the series and parallel circuits. This is because the circuit may not reduce to a simple equation using the series and parallel rules, but Kirchhoff’s Laws and Ohm’s Law can still be used to determine the power dissipated throughout the circuit.

Let’s look at common series and parallel circuits first, as these are prevalent in many systems.

Series RLC Circuit

In the series RLC circuit shown below, the impedance can be easily derived using Kirchoff’s voltage law. According to Kirchoff’s Current Law, the current is the same in each element in the series RLC circuit. Using Ohm’s Law, we can write out the characteristic differential equation for this circuit and solve it in the frequency domain. The formula for the impedance of this circuit is shown below. Note that this is the same impedance you would find if you used the series rule for calculating the equivalent impedance. Series RLC circuit impedance.

This circuit is a damped oscillator, where damping is provided by the series resistor. When the circuit is underdamped, there is a resonant frequency, which occurs when the impedance is minimized. In this circuit (or any other frequency-dependent circuit), the resonant frequency is determined by calculating the critical points for the impedance function and solving for frequency. In this case, the impedance is minimized at the resonant frequency for a series RLC circuit.

Parallel RLC Circuit

The circuit diagram below shows a parallel RLC circuit. The impedance in this case is easily defined by calculating the total current flowing into the circuit using Kirchoff’s current law. The impedance of each element and an equivalent impedance for the total circuit can be defined using Ohm’s law. With some algebra, you’ll arrive at the formula below for the total impedance of the circuit. Note that this is the same impedance you would find if you used the parallel rule for calculating the equivalent impedance. Parallel RLC circuit impedance.

This formula is more complex than the formula for a series circuit, and there is also a resonant frequency in this circuit. For a given set of R, L, and C values, the parallel and series RLC circuits will have the same resonant frequency. However, the impedance in a parallel RLC circuit is maximized at resonance, whereas it is minimized in the series RLC circuit at resonance. In this way, the two types of RLC circuits provide two different types of filtering behavior: bandpass and bandstop.

Bandpass vs. Bandstop Filtration

The equation below is the value of the resonant frequency in a series or parallel RLC circuit. What is interesting is that, although the two types of circuits are laid out differently, they have the same resonant frequency. This is because resonance occurs when power supplied by the discharging capacitor is balanced by the power generated by the inductor. This leaves the resistor as the only element left to provide net power dissipation in each circuit. Series and parallel RLC resonant frequency.

The table below shows how resonance in each type of circuit is related to the filtering behavior provided by the circuit. From this table and from substituting in the value of the resonant frequency, we can see that the impedance of both circuits is equal to R at resonance.

 Parallel Series Filtration type Bandstop Bandpass Total impedance at resonance Maximized, equal to R Minimized, equal to R Impedance of LC portion Infinity Zero

As a final note, it helps to see physically how each type of circuit provides filtration. At resonance in the series circuit, the L and C elements have equal and opposite reactance, so their total impedance is zero and they provide no reactive power. In the parallel circuit, the net current flowing into and out of these two elements at resonance is zero, so the only low impedance path back to the ground is through the resistor.

More Complex RLC Circuits

Complex circuits involving RLC elements may not have such simple impedance characteristics. They could be composed of the following elements:

• Nonlinear components, including diodes and transistors

• Mixed series and parallel arrangements of components

• Cascaded groups of filtration or amplification stages

These possibilities make some RLC circuits difficult to analyze, and they may not have a single resonance. To examine more complex circuits, you should use a SPICE-based simulator. This type of simulator will let you examine a circuit in the time domain or frequency domain, and you can use features like parameter sweeps to optimize the design of more complex circuits.

When you’re ready to build circuits for your new design and examine how they behave electrically, use the front-end design features from Cadence to build and simulate your circuits. The complete set of simulation features in PSpice Simulator lets you create detailed simulation files, simulate nearly any aspect of electrical behavior, and optimize circuit behavior with parameter sweeps. Once you’re ready to create a PCB for your circuits, simply capture your schematics in a blank layout and use Cadence’s board layout utilities to finish your design.

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