An inductor behaves as a short circuit in the case of DC and behaves as an open circuit in AC. In other words, at a frequency of 0 [Hz], an inductor corresponds to a 0 [Ω] resistor, and at a high frequency it is a high value resistor, or simply an infinite resistance. The capacitor is the opposite, a short circuit at high frequency and an open circuit in DC.
It can therefore be said that both components have resistive behaviour and their resistance depends on the frequency in the circuit. Reactance is the AC resistance of an inductor or capacitor, measured in Ohms.
$ X_L = \omega ⋅ L $
$ X_C = \dfrac{1}{\omega ⋅ C}$
If the resistances of all components are known in an RLC circuit, then the impedance (AC resistance) can be calculated. The impedance is a complex number that sums the effects of the R, L and C components. Using impedance, power and other calculations are easily performed on the circuit.
$ Z = R + j ⋅ (X_L - X_C) $
$ Z = R + j ⋅ \left(\omega ⋅ L - \dfrac{1}{\omega ⋅ C} \right) $
Impedance as a complex number is represented as a vector, where the coordinates of the vector are the real and imaginary parts of the number. However, the same vector can also be given by its length and its angle.
$ Z = 8 + j ⋅ 3 $
$ Z = 8.544 \, \angle \, 20.56° $
$ |Z| = \displaystyle\sqrt{ R^2 + (X_L - X_C)^2 } $
$ \varphi = arctan \left( \dfrac{X_L - X_C}{R} \right) $
Polar coordinates (vector length, angle) can be used to describe the voltage and current of an RLC circuit in complex form. In this case, the vector length is the peak value and the angle of the vector is the initial phase angle. Knowing the impedance of the circuit elements and the supply voltage, the peak and the angle of the current can be calculated, which means the current function can be determined.
$ V(t) = V_{p} ⋅ sin(\omega ⋅ t + \varphi_V) = V_{p} \, \angle \, \varphi_V $
$ I(t) = \dfrac{V(t)}{Z} $
$ I_{p} \, \angle \, \varphi_I = \dfrac{V_{p} \, \angle \, \varphi_V}{Z} $
$ I(t) = I_{p} ⋅ sin(\omega ⋅ t + \varphi_I) $
The impedance as a complex vector length can be calculated from the ratio of the peak values or RMS values, and the impedance of a parallel RLC circuit can be determined by reciprocals, similar to parallel resistors.
$ |Z| = \dfrac{V_{p}}{I_{p}} = \dfrac{V_{RMS}}{I_{RMS}} $
$ \dfrac{1}{Z} = \dfrac{1}{R} + j ⋅ \left( \dfrac{1}{X_C} - \dfrac{1}{X_L} \right) $