Inhaltsverzeichnis
7. Networks at variable frequency
Further content can be found at this Tutorial or that Tutorial
Introduction
In the previous chapters, it was explained what the „influence of a sinusoidal current flow“ of capacitors and inductors looks like. To describe this, the impedance was introduced. This can be understood as a complex resistance for sinusoidal excitation.
It applies to the capacitor:
\begin{align*} \underline{U}_C = \frac{1}{{\rm j}\omega \cdot C} \cdot \underline{I}_C \quad \rightarrow \quad \underline{Z}_C = \frac{1}{{\rm j}\omega \cdot C} \end{align*}
and for the inductance
\begin{align*} \underline{U}_L = {\rm j}\omega \cdot L \cdot \underline{I}_L \quad \rightarrow \quad \underline{Z}_L = {\rm j}\omega \cdot L \end{align*}
Complex impedances can be dealt with in much the same way as ohmic resistances in Electrical Engineering 1 (see: simple DC Circuits, linear Sources and two-terminal network, Analysis of DC Networks). In these transformations, the fraction $ j\omega \cdot$ is preserved. Circuits with impedances such as inductors and capacitors will show a frequency dependence accordingly.
Targets
After this lesson, you should:
- know that …
- know that … is formed.
- be able to … can …
7.1 From Two-Terminal Network to Four-Terminal Network
Until now, components such as resistors, capacitors, and inductors have been understood as two-terminal. This is also obvious since there are only two connections. In the following however circuits are considered, which behave similarly to a voltage divider: On one side a voltage $U_\rm I$ is applied, and on the other side $U_\rm O$ is formed with it. This results in 4 terminals. The circuit can and will be considered as a four-terminal network in the following. However, the input and output values will be complex.
For a four-terminal network, the relation of „what goes out“ (e.g. $\underline{U}_\rm O$ or $\underline{U}_2$) to „what goes in“ (e.g. voltage $\underline{U}_\rm I$ or $\underline{U}_1$) is important. Thus, the output and input variables ($\underline{U}_\rm O$) and ($\underline{U}_\rm I$) give the quotient:
\begin{align*} \underline{A} & = {{\underline{U}_{\rm O}^\phantom{O}}\over{\underline{U}_{\rm I}^\phantom{O}}} \\ & \text{with} \; \underline{U}_{\rm O} = U_{\rm O} \cdot {\rm e}^{{\rm j} \varphi_{u\rm O}} \\ & \text{and} \; \underline{U}_{\rm I} = U_{\rm I} \cdot {\rm e}^{{\rm j} \varphi_{u\rm I}} \\ \\ \underline{A} & = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}} = \frac {U_{\rm O} \cdot {\rm e}^{{\rm j} \varphi_{u\rm O}}}{U_{\rm I}\cdot {\rm e}^{{\rm j} \varphi_{u\rm I}}} \\ & = \frac {U_{\rm O}}{U_{\rm I}}\cdot {\rm e}^{{\rm j} (\varphi_{u\rm O}-\varphi_{u\rm I})} \\ \end{align*}
\begin{align*} \boxed{\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}} = \frac {U_\rm O}{U_\rm I}\cdot {\rm e}^{{\rm j} \Delta\varphi_{u}}} \end{align*}
Reminder:
- The complex-valued quotient ${\underline{U}_{\rm O}}/{\underline{U}_{\rm I}}$ is called the transfer function.
- The frequency-dependent magnitude of the quotient $A(\omega)={U_{\rm O}}/{U_{\rm I}}$ is called amplitude response and the angular difference $\Delta\varphi_{u}(\omega)$ is called phase response.
The frequency behavior of the amplitude response and the frequency response is not only important in electrical engineering and electronics but will also play a central role in control engineering.
7.2 RL Series Circuit
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, $\underline{U}_{\rm I}= \underline{X_\rm I} \cdot \underline{I}_{\rm I}$ with $\underline{X}_{\rm I} = R + {\rm j}\omega \cdot L$ and corresponding for $\underline{U}_{\rm O}$:
\begin{align*}
\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}
= \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}\cdot {\rm e}^{{\rm 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.
- Extreme frequency consideration of this RL circuit (in the equation and the system)
- Plotting amplitude and frequency response
- 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: We 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 $\omega \rightarrow \infty$, $A \rightarrow 1$, 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 $A = \frac {U_{\rm O}}{U_{\rm I}} = \rightarrow 1$, 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 $U_{\rm i} = - {\rm d}I / {\rm d}t$ 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 $U_\rm O$ tends towards $U_\rm I$.
The transfer function can also be decomposed into amplitude response and frequency response.
Often these 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 is 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 $\underline{A} = \dfrac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}}$ 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:
\begin{align*} \large{\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}} = \frac {\omega L} {\sqrt{R^2 + (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}} \quad \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad \large{\underline{A}_{norm} = \frac {\omega L / R}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)} } \large{ = \frac {x} {\sqrt{1 + x^2 }}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan x \right)} } \end{align*}
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_{\rm O} / U_{\rm 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 $U_{\rm O} / U_{\rm I}$.
- Further to the right of this point (i.e. at higher frequencies) $U_{\rm O} / U_{\rm I} = 1$ remains.
So this point marks a limit. Far to the left, the ohmic resistance is significantly greater than 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*} \vphantom{\HUGE{I }} \\ \underline{A}_{\rm norm} = \frac{x}{\sqrt{1 + x^2}} \cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - arctan x \right)} = \frac{U_{\rm O}}{U_{\rm I}} \cdot {\rm e}^{{\rm j}\varphi} \end{align*}
\begin{align*} \left\{\begin{array}{l} x \ll 1 & \widehat{=}& \omega L \ll R \, : \quad & \frac{U_{\rm O}}{U_{\rm I}}=x &, \varphi = \frac{\pi}{2} \, \widehat{=} \, 90° \\ x \gg 1 & \widehat{=}& \omega L \gg R \, : \quad & \frac{U_{\rm O}}{U_{\rm I}}=1 &, \varphi = 0 \; \widehat{=} \, 0° \\ x = 1 & \widehat{=}& \omega L = R , : \quad & \frac{U_{\rm O}}{U_{\rm I}}=\frac{1}{\sqrt{2}} &, \varphi = \frac{\pi}{4} \, \widehat{=} \, 45° \end{array} \right. \end{align*}
Reminder:
- The cut-off frequency $f_\rm c$ for high-pass and low-pass filters is the frequency at which the ohmic resistance just equals the value of the impedance.
- The cut-off frequency separates a range in which the filter allows signals through from one in which they are suppressed (=blocked).
- At the cut-off frequency, the phase $\varphi = 45°$ and the amplitude $A = \frac{1}{\sqrt{2}}$.
- In German the cut-off Frequency is called Grenzfrequenz $f_{\rm Gr}$
These statements apply to single-stage passive filters, i.e. one RL or one RC element. Multistage filters are considered in circuit engineering.
The cut-off frequency, in this case, is given by:
\begin{align*} R &= \omega L \\ \omega _{\rm c} &= \frac{R}{L} \\ 2 \pi f_{\rm c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{R}{2 \pi \cdot L}} \end{align*}
7.2.2 RL Low Pass
So far, only one variant of the RL element has been considered, namely the one where the output voltage $\underline{U}_{\rm O}$ is tapped at the inductance.
Here we will briefly discuss what happens when the two components are swapped.
In this case, the normalized transfer function is given by:
\begin{align*} \underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan \frac{\omega L}{R} } \end{align*}
The cut-off frequency is again given by $f_{\rm c} = \frac{R}{2 \pi \cdot L}$.
7.3 RC Series Circuit
7.3.1 RC High Pass
Now a voltage divider is to be constructed by a resistor $R$ and a capacity $C$. Quite similar to the previous chapters, the transfer function can also be determined here.
Here results as normalized transfer function:
\begin{align*} \underline{A}_{\rm norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{\frac{\pi}{2}-{\rm j} \; \arctan (\omega RC) } \end{align*}
In this case, the normalization variable $x = \omega RC$. Again, the cut-off frequency is determined by equating $R$ and the magnitude of the impedance of the capacitance:
\begin{align*} R &= \frac{1}{\omega_{\rm c} C} \\ \omega_{\rm c} &= \frac{1}{RC} \\ 2 \pi f_{\rm c} &= \frac{1}{RC} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{1}{2 \pi\cdot RC} } \end{align*}
7.3.2 RC Low Pass
Again, the voltage at the impedance is to be used as the output voltage. This results in a low-pass filter.
Here results as normalized transfer function:
\begin{align*} \underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan (\omega RC) } \end{align*}
Also, the cut-off frequency is given by $f_{\rm c} =\frac{1}{2 \pi\cdot RC}$