Impedance

In phasor analysis, impedance is a complex number. The reciprocal of impedance is admittance. In standard form, it is:

$$Z=R+jX$$

$R$ is called the resistance and the imaginary part $X$ is called the reactance. Both are in units of ohms. But note that $R$ is not referring to an actual resistor (necessarily), it is just the real part of the complex number. Impedance isn’t a phasor.

In standard form, for a resistor, $Z=R$, where $R$ is positive.

$$Z_R=R$$

For an inductor, both $R$ and $X$ are positive.

$$Z_L=j\omega L$$

For a capacitor, $R$ is positive and $X$ is negative.

$$Z_C=\frac{1}{j\omega C}=-\frac{j}{\omega C}$$

For impedances in series, we have:

$$Z_{eq}=\sum _{i=1}^n{Z}_k$$

For impedances in parallel, we have:

$$\frac{1}{Z_{eq}}=\sum _{k=1}^n\frac{1}{Z_k}$$

Addendums

The magnitude of the impedance depends on the frequency for both the inductor and capacitor. Frequency plotted against impedance is: We can derive a few key implications from this. For IC design at high-frequencies, this should be noted.

• As frequency approaches 0 (DC state):
  • The inductor impedance approaches 0 (short circuit).
  • The capacitor impedance approaches infinity (open circuit).
• As frequency approaches infinity:
  • The inductor impedance approaches 0 and becomes infinity (OC).
  • The capacitor impedance approaches 0 (SC).

Standard impedance values in electronics and PCB design is $Z=50\Omega$.