Circuit Theory/Impedance

The impedance concept has to be formally introduced in order to solve node and mesh problems.



Symbols & Definition
Impedance is a concept within the phasor domain / complex frequency domain.

Impedance is not a phasor although it is a complex number.

Impedance = Resistance + Reactance:
 * $$Z = R + X$$


 * Impedance = $$Z$$
 * Resistance = $$R$$
 * Reactance = $$X$$

Reactance
Reactance comes from either inductors or capacitors:
 * $$X_L$$
 * $$X_C$$

Reactance comes from solving the terminal relations in the phasor domain/complex frequency domain as ratios of V/I:
 * $$\frac{V}{I} = R$$
 * $$\frac{V}{I} = X_L = j\omega L$$ or $$X_L = sL$$
 * $$\frac{V}{I} = X_C = \frac{1}{j\omega C}$$ or $$X_C = \frac{1}{sC}$$

Because of Euler's equation and the assumption of exponential or sinusoidal driving functions, the operator $$\frac{d}{dt}$$ can be decoupled from the voltage and current and re-attached to the inductance or capacitance. At this point the inductive reactance and the capacitive reactance are conceptually imaginary resistance (not a phasor).

Reactance is measured in ohms like resistance.

Characteristics
Impedance has magnitude and angle like a phasor and is measured in ohms.

Impedance only exists in the phasor or complex frequency domain.

Impedance's angle indicates whether the inductor or capacitor is dominating. A positive angle means that inductive reactance is dominating. A negative angle means that capacitive reactance is dominating. An angle of zero means that the impedance is purely resistive.

Impedance has no meaning in the time domain.