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.