Capacitor Impedance

The capacitor is a reactive component and this mean its impedance is a complex number. Ideal capacitors impedance is purely reactive impedance. The impedance of a capacitor decrease with increasing frequency as shown below by the impedance formula for a capacitor. At low frequencies, the capacitor has a high impedance and its acts similar to an open circuit. In high frequencies, the impedance of the capacitor decrease and it acts similar to a close circuit and current will flow through it.

Z_C (\Omega )=\cfrac{1}{\mathrm{j}\omega C}

\omega =2\pi f

where :

f is the frequency in Hertz, (Hz)

C is the capacitance in Farads, (F)

The 1/j (or -j) phase indicates that the current leads the voltage by 90°. Figure 1 shows a visual representation of an AC voltage and current at the terminals of a capacitor:

Figure 1 : Visual representation of voltage lags in a capacitor. Voltage lags current by 90°

In figure 2, we have a Cartesian representation of impedance. In Cartesian form, the impedance is defined as:

Z (\Omega )= R + \mathrm{j}X

The real part (x-axis) of impedance is the resistance (R) and the imaginary part (y-axis) is the reactance (X).

Figure 2 : Cartesian representation of the complex impedance

For an ideal capacitor :

R (\Omega )= 0

and

X (\Omega )=-\cfrac{1}{\omega C}=-\cfrac{1}{2\pi f C}

Capacitor are not ideal component in real-life and the impedance formula above won’t give accurate result when working with high frequency. Most capacitors manufacturer will provide an impedance curve for their capacitor. In figure 3, you have the real-life model for a capacitor. As you can see in figure 3, the capacitor is far from ideal. One of the component that has the most impacts on the impedance at high frequency is the equivalent series inductance. The impedance of an inductance increase with frequency. The equivalent series inductance generally have a very small value and is negligeable in the lower frequency range but it won’t be negligeable in the high frequency range. The equivalent series resistance will also have an impact on the impedance of the capacitor.

Figure 3 : Real model of a capacitor
   C = Capacitance\\   C_{DA} = Dielectric\ Absorption\ Capacitance\\   R_{DA} = Dielectric\ Absorption\ Resistance\\   R_{L} = Leakage\ Resistance\\   L_{ESL} = Equivalent\ Series\ Inductance\\   R_{ESR} = Equivalent\ Series\ Resistance

In figure 4, you have the impedance curve for a random ceramic capacitor of 1uF. Above 10MHz, the impedance of the capacitor starts to increase because the impedance is now determnied by the equivalent series inductance. The ideal capacitor would have an infinitely decreasing impedance. When designing circuits in the high frequency range, the impedance curve of your actual capacitor needs to be considerated to avoid any issues. Also, it is important to note that different types and models of capacitors will not have the same impedance curve even if they have the same capacitance.

Figure 4 : Impedance of a random 1uF ceramic capacitor