# 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:

${\displaystyle Z=R+X}$
Impedance = ${\displaystyle Z}$
Resistance = ${\displaystyle R}$
Reactance = ${\displaystyle X}$

## Reactance

Reactance comes from either inductors or capacitors:

${\displaystyle X_{L}}$
${\displaystyle X_{C}}$

Reactance comes from solving the terminal relations in the phasor domain/complex frequency domain as ratios of V/I:

${\displaystyle {\frac {V}{I}}=R}$
${\displaystyle {\frac {V}{I}}=X_{L}=j\omega L}$  or ${\displaystyle X_{L}=sL}$
${\displaystyle {\frac {V}{I}}=X_{C}={\frac {1}{j\omega C}}}$  or ${\displaystyle X_{C}={\frac {1}{sC}}}$

Because of Euler's equation and the assumption of exponential or sinusoidal driving functions, the operator ${\displaystyle {\frac {d}{dt}}}$  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.