# What Is the Time Constant of an RLC Circuit?

December 23, 2020

### Key Takeaways

• RLC circuits have damping, so they will not instantly transition between two different states and will exhibit some transient behavior.

• The time constant of an RLC circuit tells you how long it will take to transition between two different driving states, similar to the case where a capacitor is charged to full capacity.

• RLC circuits can have different damping levels, which can complicate the determination of the time constant. Complex RLC circuits can exhibit a complex time-domain response.

Reactive circuits are fundamental in real systems, ranging from power systems to RF circuits. They are also important for modeling the behavior of complex electrical circuits without well-defined geometry. An important part of understanding reactive circuits is to model them using the language of RLC circuits. The way in which simple RLC circuits are built and combined can produce complex electrical behavior that is useful for modeling electrical responses in more complex systems.

As all RLC circuits are second-order linear systems, they have some limit cycle in their transient behavior, which determines how they reach a steady state when driven between two different states. The time constant of an RLC circuit describes how a system transitions between two driving states in the time domain, and it’s a fundamental quantity used to describe more complex systems with resonances and transient behavior. If you’re working with RLC circuits, here’s how to determine the time constant in the transient response.

## RLC Circuit Time Constant

First-order and second-order systems (such as RL, RC, LC, or RLC circuits) can have some time constant that describes how long the circuit takes to transition between two states. Such a transition can occur when the driving source amplitude changes (e.g., a stepped voltage/current source) when the driving source changes frequency or when the driving source switches on or off. Because of this transition between two different driving states, it is natural to think of an RLC circuit in terms of its time constant.

In reality, an RLC circuit does not have a time constant in the same way as a charging capacitor. Instead, we say that the system has a damping constant which defines how the system transitions between two states. Because we are considering a second-order linear system (or coupled an equivalent first-order linear system) the system has two important quantities:

• Damping constant (𝛽): This defines how energy initially given to the system is dissipated (normally as heat).

• Natural frequency (𝜔0): This defines how the system would oscillate if there were no damping in the system.

The time constant in an RLC circuit is basically equal to 𝛽, but the real transient response in these systems depends on the relationship between 𝛽 and 𝜔0. Second-order systems, like RLC circuits, are damped oscillators with well-defined limit cycles, so they exhibit damped oscillations in their transient response. The conditions for each type of transient response in a damped oscillator are summarized in the table below.

 Condition Type of oscillation Notes 𝜔0 > 𝛽 Underdamped The voltage/current exhibits an oscillation superimposed on top of an exponential rise. 𝜔0 = 𝛽 Critically damped The system will exhibit the fastest transition between two states without a superimposed oscillation. 𝜔0 < 𝛽 Overdamped The system does not exhibit any oscillation in its transient response. The transient response resembles that of a charging capacitor.

For simple underdamped RLC circuits, such as parallel or series RLC circuits,  the damping constant can be determined by hand. Otherwise, such as in complex circuits with complex transfer functions, the time constant should be extracted from measurements or simulation data.

## Extracting the Time Constant of an RLC Circuit from Measurements

If you have some measurements or simulation data from an RLC circuit, you can easily extract the time constant from an underdamped circuit using regression. Let’s look at a simple example for an underdamped RLC oscillator, followed by considerations for critically damped and overdamped RLC oscillators.

### Underdamped

The graph below shows how this can easily be done for an underdamped oscillator. The data shows the total current in a series RLC circuit as a function of time, revealing a strongly underdamped oscillation. The successive maxima in the time-domain response (left) are marked with red dots. These data are then plotted on a natural log scale as a function of time and fit to a linear function. The slope of the linear function is 0.76, which is equal to the damping constant and the time constant. As a check, the same data in the linear plot (left panel) were fit to an exponential curve; we also find that the time constant in this exponential curve is 0.76. Two ways to extract the damping time constant of an RLC circuit.

In the above example, the time constant for the underdamped RLC circuit is equal to the damping constant. This is not the case for a critically damped or overdamped RLC circuit, and regression should be performed in these other two cases.

### Critical Damping and Overdamping

In the case of critical damping, the time constant depends on the initial conditions in the system because one solution to the second-order system is a linear function of time. In an overdamped circuit, the time constant is no longer strictly equal to the damping constant. Instead, the time constant is equal to: Time constant of an overdamped RLC circuit.

Here, we have a time constant that is derived from the sum of two decaying exponentials. When 𝜔0 << 𝛽, the time constant converges to 𝛽. The relationships discussed here are valid for simple RLC circuits with a single RLC block. More complex circuits need a different approach to extract transient behavior and damping.

## Higher-Order RLC Circuits

Higher-order RLC circuits have multiple RLC blocks connected together in unique ways and they might not have a well-defined time constant that follows the simple equation shown above. While, in principle, you can calculate the response in the frequency domain by hand, circuits with a large number of RLC elements connected in a mix of series and parallel are very difficult to solve. The time constant you observe depends on several factors:

• Where the circuit’s output ports are located.

• How power sources and components are arranged into a larger topology.

• Which voltage source is used for comparison in the circuit’s transfer function.

An example of a higher-order RLC circuit is shown below. In this circuit, we have multiple RLC blocks, each with its own damping constant and natural frequency. A more complex RLC network.

This is basically a higher-order filter, i.e., it mixes multiple filter sections together into a large RLC network. This type of circuit can have multiple resonances/anti-resonances at different frequencies and the frequencies may not be equal to the natural frequency of each RLC section. This occurs due to coupling between different sections in the circuit, producing a complex set of resonances/anti-resonances in the frequency domain.

There are two ways to determine the transient response and time constant of an RLC circuit from simulations:

• Use a transient simulation, as was discussed above; simply fit the circuit’s time-domain response (natural log scale) and calculate the transfer function from the slope.

• If you like determining transient responses by hand, you can use a frequency sweep to determine the poles and zeros in the transfer function.

Both methods can rely on using a powerful SPICE simulator to calculate the current and voltage seen at each component in the circuit. For complex circuits with multiple RLC blocks, pole-zero analysis is the fastest way to extract all information about the transient behavior, any resonant frequencies, and any anti-resonant frequencies.

When you need to determine the overdamped time constant of an RLC circuit, you can use the front-end design software from Cadence to start creating your circuit schematics and access simulation tools. The PSpice Simulator application makes it easy to determine the damping constant in an RLC circuit in a transient simulation. You can also perform more advanced pole-zero simulations to determine all possible transient effects in a complex RLC network.

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