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

November 19, 2020 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.

 Two cylindrical capacitors sitting on top of a blue PCBA

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. During the design process, one must accurately assess these parameters to inform initial and final design decisions.

Furthermore, parameters like 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 the conversion of one parameter into another's equivalent is a requirement. Moreover, when performing a detailed AC circuit analysis, converting parameters like capacitance into impedance is necessary.

Capacitance in Capacitors

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 in regards to the change in its electrical potential. Furthermore, the farad is its standard unit of measurement.

A graphic showing the anatomy of a capacitor.

The anatomy of a capacitor. 

Capacitors are passive electronic components that provide energy storage in the form of an electrostatic field. 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 theses plates.

When calculating the impedance of a capacitor, we utilize the following formula:

Z = -jX

As I am sure you are aware, reactance is a more definite parameter and it characterizes how much resistance a capacitor will have at a particular frequency.

Impedance in an Electrical 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 offered by an electric circuit to the flow of AC of a single frequency. In summary, it is a combination of reactance and resistance that we measure in ohms, and we represent it with the symbol, Z.

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

Z = R + jX

Ideally, the impedance of a resistor is the equivalent of its resistance. Under these circumstances, the impedance's real or actual aspect is the resistance and the imaginary element is zero or nil.

Reactance, Resistance, and Impedance

In a DC circuit, we can describe a component or element using only its resistance. However, we interpret a capacitor's resistance in a DC circuit as an open due to its infinite resistance. Whereas, we consider an inductor's resistance in a DC circuit as a short (i.e., zero resistance). Ideally, utilizing inductors or capacitors in a DC circuit is an inefficient use of components. Although we continue to see their use in circuits because, functionally, they do not operate with a constant current or voltages.

As stated earlier, the impedance in an AC circuit is a measure of the opposition to the flow of current in a circuit when applying an AC voltage. It is essentially a ratio of voltage to current, and we express it in the frequency domain. Moreover, impedance is a complex number that incorporates real and imaginary aspects:

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 the form of a magnetic or electric field, and resistance within a circuit dissipates power as heat.

A Capacitor's Impedance

A capacitor introduces a certain level of capacitance into a circuit. Functionally, a capacitor affords temporary storage of electrical energy in the form of an electric field. This defines a capacitor's functionality, but in actuality, a capacitor lags the voltage by 90 degrees in comparison to its current, in the time domain.

Depending on your perspective, one could also say a capacitor's current leads its voltage by 900. In either case, we represent this operational fact by utilizing complex numbers. The formula for capacitor impedance is as follows:

ZC = -j

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

π = 3.14

The capacitance of the capacitor is C

In terms of capacitor parameters, an ideal capacitor's resistance is zero. However, the reactance and the impedance of an ideal capacitor are negative for all capacitance and frequency values. The effective impedance (absolute value) of a capacitor depends on the frequency, and typically, for an ideal capacitor, it will always decrease 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 a higher frequency both translate into a lower reactance. In summary, this inverse relationship between frequency and reactance facilitates capacitors' use to block low-frequency components of a signal while simultaneously allowing high-frequency components to pass.

Although capacitance in an AC circuit is easily discernible, it is the parameter impedance that requires thorough AC circuit analysis. Keeping this in mind, obtaining a greater understanding of the importance of these two parameters' interaction is paramount.

Using capacitors correctly and being sure their effects on your circuit are predictable and acceptable requires the use of a good PCB Design and Analysis software. Allegro PCB Editor, by Cadence, is one such software with a suite of robust tools to help you get the job done right the first time. 

If you're looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

 

About the Author

Cadence PCB solutions is a complete front to back design tool to enable fast and efficient product creation. Cadence enables users accurately shorten design cycles to hand off to manufacturing through modern, IPC-2581 industry standard.

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