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Simplify RLC Circuit Analysis with the RLC Transfer Function

Published Date
Author Cadence PCB Solutions

Key Takeaways

  • Gain an understanding of what a transfer function is. 

  • Learn how the transfer function helps RLC circuit analysis.

  • Derive the transfer function of an RLC circuit.

I’ve always thought I would be talented at getting out of a maze; after all, I am skilled at finding ways out of trouble. However, my confidence vanished when I finally attempted a real-life maze. Within 30 minutes, I lost all sense of direction and had to admit defeat. 

While walking through a maze is not as technical as using an RLC transfer function, they do share some similarities. Without a bird’s eye view, understanding an RLC transfer function can be just as difficult as escaping a maze. However, do not be intimidated—in this article, we will break down the RLC transfer function and its uses. 

What Is A Transfer Function?

Transfer function equation

Laplace transform is used in a transfer function

A transfer function is a mathematical model that represents the behavior of the output in accordance with every possible input value. This type of function is often expressed in a block diagram, where the block represents the transfer function and arrows indicate the input and output signals.

Laplace Transform

A discussion of the transfer function isn’t complete without mentioning Laplace transform. Laplace transform is an integral transformation that converts time-domain parameters into their frequency domain counterparts. 

Functions expressed in the time domain, F(t), are converted to the frequency domain, F(s), when Laplace transform is applied. Laplace transform is helpful in expressing transfer functions, as it enables parameters of different categories to be visualized in the frequency domain.

There are various benefits to converting parameters to the frequency domain. It makes analyzing circuits with multiple nodes easier and offers better prediction in the impulse response, which is not feasible in the time domain. 

How Transfer Functions Help RLC Circuit Analysis

Circuit analysis using transfer function

Transfer functions help when analyzing RLC circuits

The most basic form of an RLC circuit consists of a resistor, inductor, and capacitor. RLC circuits are often used in oscillator circuits, filters, and telecommunications. Because circuit analysis involves AC signals, it isn’t a simple process.

While the resistor exhibits consistent behaviors in both DC and AC analysis, capacitors and inductors are influenced by the frequency of the signal. Also, capacitors and inductors introduce phase differences to the voltage and current across them, although in different directions. 

It is impossible to visualize how the output corresponds to input in an RLC circuit in the time domain. For example, say you want to have a better understanding of the relationship of the output voltage against the input voltage of an RLC circuit. This relationship would be better expressed in the frequency domain as a mathematical model of the output in relation to the input. 

Therefore, the transfer function becomes a useful tool in analyzing RLC circuits, as it enables the study of a circuit’s behavior through the use of simpler equations instead of requiring complex differential formulas. 

Deriving the RLC Transfer Function

It is important to note that the RLC transfer function is a mathematical model and not a specific formula. Still, it involves a sequence of steps to obtain the numerical value of the transfer function:

1. Determine the output and input parameter.

2. Perform the Laplace transform of both output and input. 

3. Get the transfer function from the ratio of Laplace transformed from output to input. 

Here’s an example of how voltage across the capacitor (Vc) on the RLC circuit is expressed against the input voltage (Vin ):

Vc (s) / Vin (s) = 1 / (RCs + LCs2 + 1)

Transfer functions are not limited to a single type of parameter. For example, you can express the relation of current (I) to the input voltage (Vin ) with the following transfer response equation: 

I(s)/ Vin(s) = 1/ (R + Ls + 1/Cs)

You’ll find transfer functions especially helpful when they are plotted in a chart. For example, a transfer function plotted in the frequency domain produces a similar graph (see below) for a specific set of R, L, and C values. It is expressed in dB against frequency: 

transfer function chart

Transfer function chart for Vout/Vin of a series RLC circuit. 

This visualization helps engineers determine the characteristics of the circuit when operating at various frequencies. 

While the concept of an RLC transfer function is simple, solving the equation and plotting the chart is not. This calls for simulation-capable PCB design software. Additionally, the Allegro PSpice designer provides accurate frequency response analysis, which is helpful in an RLC circuit schematic design. You can also use InspectAR to accurately assess and improve PCBs using augmented reality and intuitive interaction. Inspecting, debugging, reworking, and assembling PCBs has never been faster or easier. 

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts. You can also visit our YouTube channel for videos about Schematic Capture as well as check out what’s new with our suite of design and analysis tools.