7. Networks at variable frequency

First, a series connection of a resistor $R$ and an inductor $L$ shall be considered (see Abbildung 2). This structure is also called RL-element.
Here, $\underline{U}_I= \underline{X_I} \cdot \underline{I}_I$ with $\underline{X}_I = R + j\omega \cdot L$ and corresponding for $\underline{U}_O$: \begin{align*} \underline{A} = \frac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}} = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot e^{j\left(\frac{\pi}{2} - arctan \frac{\omega L}{R} \right)} \end{align*}

This results in the following for

• the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and
• the phase response: $\Delta\varphi_{u} = arctan \frac{R}{\omega L} = \frac{\pi}{2} - arctan \frac{\omega L}{R}$

The main focus should first be on the amplitude response. Its frequency response can be derived from the equation in various ways.

1. Extreme frequency consideration of this RL circuit (in the equation and in the system)
2. Plotting amplitude and frequency response
3. Determination of prominent frequencies

These three points are now to be gone through.

7.2.1 RL High Pass

For the first step we investigate the limit consideration: Wwe look at what happens, when the frequency $\omega$ runs to the definition range limits, i.e. $\omega \rightarrow 0$ and $\omega \rightarrow \infty$:

• For $\omega \rightarrow 0$, $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}} \rightarrow 0$ as the numerator approaches zero and the denominator remains greater than zero.
• For $\omega \rightarrow \infty$, $A \rightarrow1$, because in the root in the denominator $(\omega L)^2$ becomes larger and larger in the ratio $R^2$ to . So the root tends to $\omega L$ and thus to the numerator.

It can thus be seen that:

• at small frequencies there is no voltage $U_2$ at the output.
• at high frequencies $A = \frac {U_O}{U_I} = \rightarrow 1$, so the voltage at the output is equal to the voltage at the input.

Result:
The RL element shown here therefore only allows large frequencies to pass (= pass through) and small ones are filtered out.
The circuit corresponds to a high pass.

This can also be derived from understanding the components:

• At small frequencies, the current in the coil and thus the magnetic field changes only slowly. So only a negligibly small reverse voltage is induced. The coil acts like a short circuit at low frequencies.
• At higher frequencies, the current generated by $U_I$ through the coil changes faster, the induced voltage $U_i = - dI / dt$ becomes large.
  As a result, the coil inhibits the current flow and a voltage drops across the coil.
• If the frequency becomes very high, only a negligible current flows through the coil - and hence through the resistor. The voltage drop at $R$ thus approaches zero and the output voltage $U_O$ tends towards $U_I$.

The transfer function can also be decomposed into amplitude response and frequency response.
Often this plots are not given in with linear axis but:

• the amplitude response with a double logarithmic coordinate system and
• the phase response single logarithmic coordinate system.

By this, the course from low to high frequencies are easier to see. The following simulation in Abbildung 3 shows the amplitude response and frequency response in the lower left corner.

For further consideration, the equation of the transfer function $\underline{A} = \dfrac {\underline{U}_O^\phantom{O}}{\underline{U}_I^\phantom{O}}$ is to be rewritten so that it becomes independent of component values $R$ and $L$.
This allows for a generalized representation. This representation is called normalization:

This equation behaves quite the same as the one considered so far.

Abbildung 4 shows the two plots. On the x-axis, $x = \omega L / R$ has been plotted as the normalization variable. This represents a weighted frequency.

Here, too, the behavior determined in the limit value observation can be seen:

• at small frequencies $\omega$ (corresponds to small $x$), the amplitude response tends toward zero.
• At high frequencies, the ratio $U_O / U_I = 1 $ is established.

Interesting in the phase response is the point $x = 1$.

• Further to the left of this point (i.e. at smaller frequencies) a tenfold increase of the frequency $\omega$ produces a tenfold increase of $U_O / U_I$.
• Further to the right of this point (i.e. at higher frequencies) $U_O / U_I = 1$ remains.

So this point marks a limit. Far to the left, the ohmic resistance is significantly greater the amount of impedance of the coil: $R \gg \omega L$. far to the right is just the opposite.

The point $x=1$ just marks the cut-off frequency.
It holds

\begin{align*} \underline{A}_{norm} = \frac{x}{\sqrt{1 + x^2}} \cdot e^{j\left(\frac{\pi}{2} - arctan x \right)} = \frac{U_O}{U_I} \cdot e^{j\varphi} \quad \quad \left\{\begin{array}{l} x \ll 1 & \widehat{=}& \omega L \ll R &: \quad\quad \frac{U_O}{U_I}=x &, \varphi = \frac{\pi}{2} \, \widehat{=} \, 90° \\ x \gg 1 & \widehat{=}& \omega L \gg R &: \quad\quad \frac{U_O}{U_I}=1 &, \varphi = 0 \: \widehat{=} \, 0° \\ x = 1 & \widehat{=}& \omega L = R &: \quad\quad \frac{U_O}{U_I}=\frac{1}{\sqrt{2}} &, \varphi = \frac{\pi}{4} \, \widehat{=} \, 45° \end{array} \right. \end{align*}