The Importance of Capacitor Impedance in AC Circuit Analysis and How to Calculate It

May 11, 2022

Cadence PCB Solutions

Key Takeaways

Learn about capacitor impedance.
Gain a greater understanding of the importance of capacitor impedance in AC circuit analysis.
Learn how to calculate capacitor impedance.

Capacitors are remarkably common components in just about every electronic circuit.

The electronics field contains various parameters that gauge, assist, and affect every electronic device's functionality and performance. Amongst the crucial parameters is the impedance in an AC circuit. During the design process, one must accurately assess the impedance caused by various components to make informed design decisions.

Furthermore, parameters like the capacitance and impedance must stay within the acceptable design limits, or even accurate designs can fail to provide the desired functionality. There are also instances where converting one parameter into another's equivalent is a requirement. Moreover, converting parameters like capacitance into impedance is necessary when performing a detailed AC circuit analysis. Therefore, it is critical that designers understand the relationship between capacitance and impedance in AC circuits.

What Is Capacitance?

Below is a helpful definition of capacitance:

We associate capacitance, of course, with capacitors. We call a system's ability to store an electric charge “capacitance.” However, in physics, we refer to it as the ratio of change in an electrical charge within a system regarding the change in its electrical potential. The farad is capacitance’s standard unit of measurement.

Now that we have defined capacitance, let’s take a look at the role of a capacitor in an AC circuit.

The Function of a Capacitor in an AC Circuit

Capacitors are passive electronic components that provide energy storage in the form of an electrostatic field. A capacitor charges up when the AC reaches its peak in an AC circuit and releases the charge when the AC decreases. This behavior allows the capacitor to act like temporary storage that causes the current to lead voltage by 90 degrees.

Electrical engineers use capacitors to improve the power factor in an AC circuit. For example, an AC circuit powering inductive loads like a motor results in a lagging current. Adding a capacitor helps compensate for the lagging current and brings the power factor closer to unity. Power factor correction improves the efficiency of energy usage and reduces electricity bills.

Capacitors in Low Pass Filters

In PCB design, capacitors are often used in low pass or high pass filters. This is because the capacitor's impedance is a factor of the AC signal frequency that passes through it. A capacitor typically blocks lower frequency signals while allowing higher frequency signals to pass through.

Capacitor in low pass filter

Low pass RC filter. Source

The above diagram shows a low pass RC filter. The capacitor is shunted to the ground. As such, high frequency signals are channeled to the ground. This prevents high frequency noise above the cut-off frequency from passing through to the load.

A low pass RC filter’s cutoff frequency is defined by the following equation.

You can understand the filter’s behavior better by its transfer function with the following Laplace transform.

Low pass RC filter transfer function

Capacitors in High Pass Filters

Meanwhile, the capacitor is connected in series with the incoming signal to form a high pass filter. Based on the below diagram, the capacitor blocks DC or low-frequency signals from passing through to the load. Only signals above the cut-off frequency are allowed to pass.

High pass RC filter. Source

The cutoff frequency of the RC high pass filter is as follows:

Cut off frequency high pass RC filter

The high pass RC filter’s transfer function is given by the following equation.

A Capacitor’s Capacitance

The anatomy of a capacitor.

A capacitor incorporates two conductive plates separated by an insulating material called the dielectric. A capacitor's capacitance is directly proportional to its plate's surface area and inversely proportional to the plate's separation. Capacitance is also dependent on the dielectric constant of the substance separating these plates.

Capacitance is expressed by the following equation:

Capacitance equation

Where C is capacitance, q is the electrical charge, and V is the differential potential across conductive plates.

Now that we have a better understanding of capacitance, let’s take a look at the impedance in an AC circuit.

Understanding Impedance in an AC Circuit

Impedance is the active resistance of an electrical circuit or component to AC current, evolving from the combined effects of reactance and ohmic resistance. In other words, impedance is merely the extension of the principles of resistance in AC circuits. We also define it as any obstruction or the measure of the opposition of an electric current to the energy flow when applying voltage.

The more technical definition is the opposition an electric circuit offers to the flow of AC of a single frequency. In summary, it is a combination of reactance and resistance that we measure in ohms, representing it with the symbol Z.

However, reactance (X) expresses a component's resistance to AC, whereas impedance (Z) is the sum of resistance and reactance. We show it as a complex number by using the following formula:

Z = R + jX

Here, the complex impedance is Z.

We represent resistance as R (real aspect).

We represent reactance as X (imaginary aspect).

Keep in mind that reactance can be either negative or positive, whereas resistance is always positive. Also, reactance stores energy in a magnetic or electric field and resistance within a circuit dissipates power as heat.

Now that we have explored the impedance in an AC circuit, let’s take a look at how to calculate a capacitor’s impedance.

How to Calculate a Capacitor's Impedance

A capacitor introduces a certain level of capacitance into a circuit. Functionally, a capacitor affords temporary electrical energy storage in the form of an electric potential where the capacitor's current leads its voltage by 90°. The formula for capacitor impedance is as follows:

Z_C= -jX_C

XC is the capacitive reactance that characterizes how much resistance a capacitor will have at a particular frequency.

The above equation can further be expanded as follow:

Z_C = 1/j_ωC

Here, the impedance of a capacitor is ZC.

The angular frequency is ω, which we calculate as:

ω = 2πf

We represent the frequency of a signal as f and the capacitance of the capacitor as C.

In terms of capacitor parameters, an ideal capacitor's resistance is zero. However, the reactance and the impedance of an actual capacitor are negative for all capacitance and frequency values. The effective impedance (absolute value) of a capacitor depends and decreases with frequency.

It is evident from the equations above that a capacitor's reactance is inversely proportional to both capacitance and frequency. Therefore, higher capacitance and higher frequency translate into lower reactance. This allows the use of capacitors with other components in low pass or high pass filter designs and blocks selective frequencies.

Although capacitance in an AC circuit is easily discernible, the parameter impedance in an AC circuit requires thorough circuit analysis. Keeping this in mind, obtaining a greater understanding of the relationship between capacitance and impedance is paramount.

Using capacitors correctly and being sure their effects on the impedance in an AC circuit are predictable and acceptable requires good PCB Design and Analysis software. If you’re looking to learn more about how Cadence has the solution for you, talk to our team of experts or subscribe to our YouTube channel.