| Element | Time domain | Frequency domain |
|---|---|---|
| Resistor \(R\) | \(\displaystyle v = Ri\) | \(\displaystyle \mathbf{V} = R\mathbf{I}\) |
| Inductor \(L\) | \(\displaystyle v = L\frac{di}{dt}\) | \(\displaystyle \mathbf{V} = j\omega L\mathbf{I}\) |
| Capacitor \(C\) | \(\displaystyle i = C\frac{dv}{dt}\) | \(\displaystyle \mathbf{V} = \frac{\mathbf{I}}{j\omega C}\) |
For the resistor \(R\)
\(\displaystyle i = I_m \cos\left(\omega t + \phi\right)\)
\(\displaystyle v = iR = R\,I_m \cos\left(\omega t + \phi\right)\)
In phasor form
\(\boxed{\mathbf{V} = R\,I_m\angle\phi = R\,\mathbf{I}}\)
where \(\boxed{\mathbf{I} = I_m\angle\phi}\)
The voltage and current of resistor are in phase.
For the inductor \(L\)
\(\displaystyle i = I_m \cos\left(\omega t + \phi\right)\)
\(\displaystyle v = L\frac{di}{dt} = -\omega L I_m \sin\left(\omega t + \phi\right)\)
\(\displaystyle v = \omega L I_m \cos\left(\omega t + \phi + 90^{\circ}\right)\)
In phasor form
\(\displaystyle \mathbf{V} = \omega L I_m\angle\left(\phi + 90^{\circ}\right) = \omega L I_m e^{\displaystyle\,j\left(\phi + 90^{\circ}\right)}\)
\(\displaystyle \mathbf{V} = \omega L I_m e^{\displaystyle\,j\phi} e^{\displaystyle\,j90^{\circ}} = j\omega L I_m e^{\displaystyle\,j\phi}\)
\(\boxed{\mathbf{V} = j\omega L I_m\angle\phi = j\omega L\mathbf{I}}\)
where \(\boxed{\mathbf{I} = I_m\angle\phi}\)
The voltage and current of inductor are \(90^{\circ}\) out of phase. Specifically, the current lags the voltage by \(90^{\circ}\).
For the capacitor \(C\)
\(\displaystyle v = V_m \cos\left(\omega t + \phi\right)\)
\(\displaystyle i = C\frac{dv}{dt} = -\omega C V_m \sin\left(\omega t + \phi\right)\)
\(\displaystyle i = \omega C V_m \cos\left(\omega t + \phi + 90^{\circ}\right)\)
In phasor form
\(\displaystyle \mathbf{I} = \omega C V_m\angle\left(\phi + 90^{\circ}\right) = \omega C V_m e^{\displaystyle\,j\left(\phi + 90^{\circ}\right)}\)
\(\displaystyle \mathbf{I} = \omega C V_m e^{\displaystyle\,j\phi} e^{\displaystyle\,j90^{\circ}} = j\omega C V_m e^{\displaystyle\,j\phi}\)
\(\boxed{\mathbf{I} = j\omega C V_m\angle\phi = j\omega C\mathbf{V} \Leftrightarrow \mathbf{V} = \frac{\mathbf{I}}{j\omega C}}\)
where \(\boxed{\mathbf{V} = V_m\angle\phi}\)
The voltage and current of capacitor are \(90^{\circ}\) out of phase. Specifically, the current leads the voltage by \(90^{\circ}\).
