Electrical Engineering ETR › Complex AC Analysis

Integral and Derivative

Calculating with sine and cosine in the time domain is particularly unpleasant when signal curves are integrated or differentiated over time. This is always the case with energy stores in a circuit, see the chapter Calculating with energy stores.

Here, too, the Laplace transform simplifies the mathematics considerably: an integration over time in the time domain corresponds to a division by the term jω in the frequency domain. A derivative with respect to time in the time domain corresponds to a multiplication by the term jω in the frequency domain.

For voltage and current at the capacitor, the formula in the time domain (with u_C0 = 0 V) is

\[ \begin{gathered} u_C(t) = \frac{1}{C} \int i_C(t)\,dt \\[6pt] u_C(\omega) = \frac{1}{C} \cdot \frac{1}{j\omega} \cdot i_C(\omega) = \frac{1}{j\omega C} \cdot i_C(\omega) \end{gathered} \]

For the inductor, we can also simplify the relationship between voltage and current:

\[ \begin{gathered} u_L(t) = L \cdot \frac{di(t)}{dt} \\[6pt] u_L(\omega) = L \cdot j\omega \cdot i_L(\omega) = j\omega L \cdot i_L(\omega) \end{gathered} \]

We verify the validity of the simplification with a formula for which we already know the solution. For a capacitor with an alternating voltage, the following applies in the time domain:

\[ \begin{gathered} u(t) = \hat{u} \sin(\omega t) \\[4pt] i(t) = C \frac{du(t)}{dt} \\[4pt] i(t) = C \frac{du(t)}{dt} = C \frac{d(\hat{u} \sin(\omega t))}{dt} = \omega C \cdot \hat{u} \cdot \cos(\omega t) = \omega C \cdot \hat{u} \sin\left(\omega t + \frac{\pi}{2}\right) \end{gathered} \]

The derivative of sin(ax) is a ∙ cos(ax). For this, we need the chain rule with the inner derivative. The cosine can be expressed as a sine shifted by π/2. The shift by π/2 corresponds to a multiplication by j. So the following applies

\[ \begin{gathered} i(t) = \omega C \cdot \hat{u} \sin\left(\omega t + \frac{\pi}{2}\right) = \omega C \cdot \hat{u} \cdot j \cdot \sin(\omega t) = j\omega C \cdot u(t) \\[6pt] i(t) = C \frac{du}{dt} = C \cdot j\omega \cdot u(t) \end{gathered} \]

Using the time derivative, we see that the calculation assumed before is correct.

Caution: here I have mixed the time domain and the frequency domain. Strictly speaking, the j does not exist in the time domain. Nevertheless, this somewhat sloppy approach can show that the simplifications for differentiation and integration are correct. If you show this to a mathematician, they will scratch their eyes out, but it works. You can try the proof for integration yourself.