# SPICE Simulations for Time Domain Analysis of Simple RLC Circuits

May 20, 2019

I’ve always been a math lover, especially when it comes to circuit analysis. I’ve always found the elegance of mathematics satisfying, but sometimes things are easier if you work with the right simulation package.

If you are not the type that enjoys solving differential equations or you just enjoy writing code, you can gain some insight into the behavior of an RLC network in the time domain when you work with a SPICE simulator. You can also analyze this behavior in the frequency domain, either by iterating through multiple frequencies or by working in the frequency domain directly. Let’s take a look at the basic RLC networks and see how we can reproduce the behavior of these circuits in a SPICE package.

## Time Domain Analysis of Simple RLC Circuits

RLC circuits contain some arrangement of a resistor, capacitor, and inductor, commonly called an RLC circuit. As the capacitor and resistor have some frequency-dependent reactance, the behavior of this system will produce some interesting effects when driven with AC signals. When dealing with harmonic signals, pulses, or chirped signals, voltage sourcing will be vital to keep regulated. When working in the frequency domain, you can get a clear view of how the output from an RLC circuit and current in various portions of an RLC network are affected by the frequency of your voltage/current source.

Results from the frequency domain can be converted back into the time domain with an inverse Fourier transform (and vice versa). However, some effects like transient response are easier to calculate in the time domain. SPICE simulations lend themselves naturally to a time domain analysis thanks to their relatively simple matrix formulation.

RL and RC circuits are also interesting and are related to RLC circuits. If you are working with a larger RLC network that goes beyond a simple series RLC circuit, it helps to know how each of these circuits behaves in order to explain your time domain simulation results. When working with the analytical solution for an RLC circuit, the behavior of an RC or RL circuit can be found by taking L = 0 or C = 0 respectively in the solution for the relevant RLC circuit.

Note that an inductor in parallel with a resistor (RL circuit) will essentially form a short circuit when used with a DC source. Because the impedance of an inductor is a linear function of frequency, the impedance of the inductor will be zero with a DC source. This type of circuit is of less interest for DC circuits, but it is used to isolate amplifiers from capacitive loading effects at high frequencies.

With RC circuits, you can use a time domain simulation to analyze how an AC voltage couples into a circuit when a capacitor is placed in series with a resistor. Alternatively, when you have a resistor and capacitor in parallel, you can examine how an AC signal is bypassed around the resistor. This is particularly important in power integrity analysis, where you might need to check that any AC noise components are filtered from a DC power supply. Both of these analyses should be conducted as a function of frequency in order to see how an RC circuit acts as a filter.

## Transient Analysis With RLC Networks

RLC networks, as well as an RC or RL network within a larger RLC network, will have a specific response in time, depending on whether you are driving the circuit with a harmonic source, an arbitrary waveform, DC source, or any other source that can be easily defined as a function of time. This is what makes time-domain simulations so useful for RLC networks, allowing you to examine the response of your circuit to a pulsed or chirped (or both) voltage source.

Chirped pulses are important in radar and optical applications

When working with an AC source, most SPICE packages with a GUI will allow you to sweep through a range of AC frequencies and analyze the behavior of your device. However, you can also conduct transient analysis and examine how your circuit responds in time to AC sources with various frequencies. Here, you can examine the output from the circuit in the time domain at different frequencies and compare different qualities of the signal.

Transient analysis is also useful for examining how the network responds to DC inputs or impulses. With a DC input, you can examine the rate at which different portions of the circuit power up to different voltages and currents due to changes in the output from a DC voltage source. These curves appear as exponentials and are called transients. They have important effects in PCBs as they determine how quickly a circuit will respond to a definite change in the driving voltage, for example, due to noise in your power delivery network or due to a switching digital signal.

## Going Further With RLC Circuits

There are plenty of other aspects of an RLC network that can be analyzed with a SPICE-based simulator, but two very useful points will be discussed here. First, you can easily build higher order filters by daisy-chaining multiple RLC networks together. You can then simulate the transient response and voltage output from these higher order filters. If you also use a frequency sweep to analyze your circuit, you can determine the transfer function for your network.

Photosensitive sensor on a PCB

An excellent SPICE package will contain tools that allow you to examine how the output from your RLC network is affected by changing the operating temperature. This is particularly important for PCBs as your PCB will likely run above room temperature unless you design your board with strict thermal management.

Time-domain analysis of simple RLC circuits and more complex circuits doesn’t have to be complicated when you work with OrCAD PSpice Designer from Cadence. This unique package is adapted to complex PCB designs, and you can build models to simulate and analyze the behavior of circuits in your schematic and/or PCB.

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