Converting Capacitance to Impedance and the Relationship Between the Two

August 11, 2020 Cadence PCB Solutions

Key Takeaways

●     Learn about the formulas for converting capacitance to impedance.

●     Gain a greater understanding of how capacitance and impedance interact.

●     Learn more about the importance of impedance in AC circuit analysis.

 Electronic capacitor on a blue printed circuit board

Capacitance and its relationship to impedance

The field of electronics contains various parameters that gauge, assist, and affect the functionality, as well as the performance, of every electronic device. These parameters affect initial and final design decisions.

Parameters like capacitance and impedance must remain within the acceptable design confines or even the most accurate design will fail to provide the desired functional outcome. Furthermore, there are instances where the conversion of one parameter into the equivalent of another is a requirement. One such conversion, like capacitance to impedance, is a requirement of detailed AC circuit analysis.

Capacitors and Capacitance

The component associated with capacitance is, of course, a capacitor, and the ability of a system to store an electric charge is called capacitance. In physics, it is the ratio of change in an electrical charge in a system with regard to a change in its electric potential. In either case, the standard unit of capacitance is the farad.

These passive electronic components store energy in the form of an electrostatic field. In its purest form, a capacitor incorporates two conductive plates separated by an insulating material called the dielectric. A capacitor's capacitance is directly proportional to the surface area of its plates and inversely proportional to the separation between these plates. However, net capacitance also depends on the dielectric constant of the substance separating the plates.

When converting the impedance of a capacitor, we use the formula Z = -jX. Keep in mind that reactance is a more unambiguous parameter, and it defines how much resistance a capacitor will have at a specific frequency. As stated previously, knowing impedance is necessary for a detailed AC circuit analysis.

Inductor copper coil on the circuit board

Inductor and impedance

Understanding Impedance in an AC Circuit

Impedance is the active resistance of an electrical circuit or component to AC from the combined effects of reactance and ohmic resistance. In other words, impedance is the extension of the principles of resistance in AC circuits. We also define impedance 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 total 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.

Reactance (X) expresses a component's resistance to AC, whereas impedance (Z) indicates a component's resistance to both AC and DC. We show it as a complex number through the use of the following formula: Z = R + jX. Ideally, the impedance of a resistor is the equivalent of its resistance. Under these circumstances, the real or actual part of the impedance is the resistance, and the imaginary part is nil or zero.

Converting Capacitance to Impedance

There are capacitive reactance calculators that allow you to determine the impedance of a capacitor, provided you have its capacitance value (C) and the frequency of the signal passing through it (f). You input the capacitance in farads, picofarads, microfarads, or nanofarads and the frequency in units of GHz, MHz, kHz, or Hz. For example, a capacitance of 2 farads at a frequency of 100 hertz will yield an impedance of 0.0008 ohms.

The following is the formula necessary to calculate the above values:

XC = represents the capacitor's reactance in ohms

ω = expresses the angular frequency in rad/s

ƒ = the frequency in hertz

c = the capacitance in farads

We now understand the parameters of an ideal resistor where its impedance equals its resistance. However, a perfect capacitor's impedance is equal to the magnitude of its reactance, although these two parameters are not identical. We express reactance as an ordinary number in ohms, and the impedance of the capacitor is the reactance multiplied by -j. This correlates to the following formula: Z = -jX. In this context, the -j term represents the 90-degree phase shift that occurs between current and voltage in a purely capacitive circuit.

Using the Conversion Equation

Upon using the equation above (XC = 1/ωC = 1/2πƒC), you can obtain the reactance of a capacitor, and to convert this into the impedance of a capacitor you can utilize the formula Z = -jX. Remember that reactance is a more apparent parameter value, and it discerns how much resistance a capacitor has at a specific frequency.

It is apparent 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 a lower reactance. This inverse relationship between frequency and reactance facilitates the use of capacitors to block low-frequency components of a signal while simultaneously allowing high-frequency components to pass.

Capacitance in an AC circuit is easily discernable, but it is impedance that is a requirement of thorough AC circuit analysis. Logically, this means that gaining a greater understanding of the importance of how these two parameters interact is paramount.

Toroidal and cylindrical induction coils

Inductor copper wire winding

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