Variable Capacitor

Purpose

Capacitance controlled by signal

Library

Electrical / Passive Components

Description

This component models a variable capacitor. The capacitance is determined by the signal fed into the input of the component. The current through a variable capacitance is determined by the equation

\[i = \frac{\mathrm{d}}{\mathrm{d}t}C \cdot v + C \cdot \frac{\mathrm{d}}{\mathrm{d}t}v\]

Since \(v\) is the state variable the equation above must be solved for \(\frac{\mathrm{d}v}{\mathrm{d}t}\). The control signal must provide the values of both \(C\) and \(\frac{\mathrm{d}}{\mathrm{d}t}C\) in the following form: \(\left[ C_1\; C_2 \ldots C_n\; \frac{\mathrm{d}}{\mathrm{d}t}C_1\; \frac{\mathrm{d}}{\mathrm{d}t}C_2 \ldots \frac{\mathrm{d}}{\mathrm{d}t}C_n\right]\). It is the responsibility of the user to provide the appropriate signals for a particular purpose (see further below).

If the component has multiple phases, you can choose to include the capacitive coupling of the phases. In this case the control signal vector must contain the elements of the capacitance matrix (row by row) followed by their derivatives with respect to time, e.g. for two coupled phases: \(\left[ C_{11}\; C_{12}\; C_{21}\; C_{22}\; \frac{\mathrm{d}}{\mathrm{d}t}C_{11}\; \frac{\mathrm{d}}{\mathrm{d}t}C_{12}\; \frac{\mathrm{d}}{\mathrm{d}t}C_{21}\; \frac{\mathrm{d}}{\mathrm{d}t}C_{22}\right]\). The control signal thus has a width of \(2 \cdot n^2\), \(n\) being the number of phases.

Note

The momentary capacitance may not be set to zero. In case of coupled capacitors, the capacitance matrix may not be singular.

There are two common use cases for variable capacitors, which are described in detail below: saturable capacitors, in which the capacitance is a function of the voltage and electrostatic actuators, in which the capacitance is a function of an external quantity, such as a capacitor with movable plates.

Saturable Capacitor Modeling

When specifying the characteristic of a saturable capacitor, you need to distinguish carefully between the total capacitance \(C_{\mathrm{tot}}(v) = Q/v\) and the differential capacitance \(C_{\mathrm{diff}}(v) = \mathrm{d}Q/\mathrm{d}v\).

With the total capacitance \(C_{\mathrm{tot}}(v) = Q/v\) you have

\[\begin{split}i &= \frac{\mathrm{d}Q}{\mathrm{d}t}\\ &= \frac{\mathrm{d}}{\mathrm{d}t}\left(C_{\mathrm{tot}} \cdot v\right)\\ &= C_{\mathrm{tot}} \cdot \frac{\mathrm{d}v}{\mathrm{d}t} + \frac{\mathrm{d}C_{\mathrm{tot}}}{\mathrm{d}t} \cdot v \\ &= C_{\mathrm{tot}} \cdot \frac{\mathrm{d}v}{\mathrm{d}t} + \frac{\mathrm{d}C_{\mathrm{tot}}}{\mathrm{d}v} \cdot \frac{\mathrm{d}v}{\mathrm{d}t} \cdot v\\ &= \left(C_{\mathrm{tot}} + \frac{\mathrm{d}C_{\mathrm{tot}}}{\mathrm{d}v}\cdot v\right) \cdot \frac{\mathrm{d}v}{\mathrm{d}t}\end{split}\]

which can be implemented as in Fig. 202.

Fig. 202 Schematic for first equation

With the differential capacitance \(C_{\mathrm{diff}}(v) = \mathrm{d}Q / \mathrm{d}v\) you have

\[\begin{split}i &= \frac{\mathrm{d}Q}{\mathrm{d}t} \\ &= \frac{\mathrm{d}Q}{\mathrm{d}v} \cdot \frac{\mathrm{d}v}{\mathrm{d}t} \\ &= C_{\mathrm{diff}} \cdot \frac{\mathrm{d}v}{\mathrm{d}t}\end{split}\]

which can be implemented as in Fig. 203:

Fig. 203 Schematic for second equation

Note that in both cases the \(\frac{\mathrm{d}}{\mathrm{d}t}C\)-input of the Variable Capacitor is zero!

Actuator Modeling

In an electrostatic actuator the capacitance is determined by an external quantity such as the distance \(x\) between the movable plates of a capacitor: \(C = C(x)\). Therefore you have

\[\begin{split}i &= C \cdot \frac{\mathrm{d}v}{\mathrm{d}t} + \frac{\mathrm{d}C}{\mathrm{d}t} \cdot v \\ &= C \cdot \frac{\mathrm{d}v}{\mathrm{d}t} + \frac{\mathrm{d}C}{\mathrm{d}x} \cdot \frac{\mathrm{d}x}{\mathrm{d}t} \cdot v\end{split}\]

which can be implemented as in Fig. 204.

Fig. 204 Schematic for third equation

Note that \(x\) is preferably calculated as the integral of \(\mathrm{d}x/\mathrm{d}t\) rather than calculating \(\mathrm{d}x/\mathrm{d}t\) as the derivative of \(x\).

Parameters

Capacitive coupling

Specifies whether the phases should be coupled capacitively. This parameter determines how the elements of the control signal are interpreted. The default is off.

Initial voltage

The initial voltage of the capacitor at simulation start, in volts \((\mathrm{V})\). This parameter may either be a scalar or a vector corresponding to the implicit width of the component. The positive pole is marked with a “+”. The initial voltage default is 0.

Probe Signals

Capacitor voltage

The voltage measured across the capacitor, in volts \((\mathrm{V})\). A positive voltage is measured when the potential at the terminal marked with “+” is greater than the potential at the unmarked terminal.

Capacitor current

The current flowing through the capacitor, in amperes \((\mathrm{A})\).