# Circuit Theory/Impedance

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

!the impedance symbol is .. a
box

## 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.

