Understand what the impedance of an RLC circuit is.
Learn how to calculate series and parallel RLC circuit impedance.
Learn how to analyze complex RLC circuits.
A series RLC circuit (Source)
RLC circuits are basic building blocks of more complex analog systems and they 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. In this article, we will explore how to determine this impedance for both series and parallel RLC circuits.
What Is the Impedance of an RLC Circuit?
First, let’s clarify a few definitions:
RLC circuits are often used as examples for basic impedance analysis. While the resistor is a pure DC component, inductors and capacitors have varying impedances according to the signal frequency. The total impedance of an RLC circuit is determined by the impedance of each component.
Here is how impedance for resistors, inductors, and capacitors is expressed in magnitude and phase angle:
The resulting total impedance exists in the complex plane. RLC impedance is expressed in complex numbers or represented with a magnitude and phasor angle. This is best visualized with the vector diagram that describes the relationship between the impedance of all three components.
Series RLC impedance phasor diagram. (Source)
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 impedance 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 a series RLC circuit, shown below, the impedance can be easily derived using Kirchoff’s voltage law.
Series RLC circuit impedance. Source
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:
To calculate the magnitude and phase angle of series RLC impedance, the above equation is solved as follows.
Note that this is the same impedance you would find if you used the series rule for calculating the equivalent 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. In this case, the impedance 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.
Parallel RLC circuit impedance. Source
The total impedance of the parallel RLC circuit is described by the following equation.
With some algebra, the above equation can be solved for its magnitude and phase angle as follows.
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 the value of the resonant frequency, we can see that the impedance of both circuits is equal to R at resonance.
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 the 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.
If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.