MCS2_003 Floating Electrode Actuator

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Floating Electrode Actuator

In the previous notes we mostly treated voltage-controlled electrostatic actuators, where the electrical loading is prescribed directly by an external voltage source. This lecture introduces a different situation: a suspended conductor carries a fixed charge , but is also acted on by a voltage source through a nearby driving electrode.

The important point is that the suspended electrode is floating. It is conductive, so it has a single electric potential throughout its volume, but it is not directly wired to a voltage source. Its potential is determined by charge redistribution between the variable drive capacitance and the parasitic capacitance.

Figure 3.1: Floating electrode actuator.

The variable capacitance between the fixed driving electrode and the floating electrode is

where is positive toward the driving electrode. The parasitic capacitance is assumed constant.

Charge Loading Only

First suppose that both fixed electrodes are grounded and charge is placed on the floating conductor. The two grounded electrodes must carry charge in total. Since the two capacitances are connected to the same floating potential, the charge divides proportionally to capacitance:

The two equations used to derive (3.2) are:

The first equation is charge conservation: the floating electrode carries , so the two grounded electrodes together carry . The second equation is the floating-conductor constraint: both fixed electrodes are grounded, so the voltage of the floating conductor computed through the driving capacitor must equal the same voltage computed through the parasitic capacitor.

Solving (3.3):

The floating electrode voltage can be computed through either capacitor. Through the driving capacitor,

and through the parasitic capacitor,

This agreement is a useful sanity check: the floating conductor must have one potential.

Charge and Voltage Loading

Now the driving electrode is held at voltage , while the parasitic electrode remains grounded. The electrode charges become

The first term in each expression is the voltage-only contribution, i.e. the charge distribution that would appear if and the source voltage were applied. In that case the two capacitors form a series path from the driving electrode at voltage , through the floating conductor, to the grounded parasitic electrode. The equivalent series capacitance is

so the source-driven charge magnitude is . It appears with opposite signs on the two fixed electrodes: the driving electrode is connected to the positive terminal of the voltage source, so its voltage-induced charge is ; the grounded parasitic electrode carries the opposite charge, . The remaining terms in (3.7) are the charge-only contribution from the trapped floating-electrode charge .

Again, . The voltage of the floating electrode is

The first term is a capacitive divider contribution from the applied voltage. The second term is the contribution from the trapped charge.

Total Potential

The energy stored in the two capacitors is

The voltage source also contributes potential energy. When the voltage source moves charge through a prescribed potential difference, the source does work on the electrical subsystem; in the potential formulation this enters with the opposite sign:

Here is the initial charge of the voltage source. Equivalently, it fixes the reference for the source work: the source only does incremental work on the additional charge that passes through it while the voltage is held at . The final term is independent of for a prescribed voltage history, so it shifts the potential by a constant and does not affect force or stiffness.

The mechanical potential is

Therefore

Why the sign changes

For an isolated charged capacitor, the electrostatic energy is . For a voltage-driven capacitor, the voltage source supplies or removes charge while holding fixed, so the effective potential contains the familiar term . The floating electrode actuator contains both ingredients: trapped charge and prescribed voltage .

Normalized Potential

Define

Then

and the total potential can be written as

For force and stability, all -independent terms can be ignored. The derivative of (3.16) reduces to a particularly simple form:

The actuator therefore responds only to the effective electrostatic load

This is the central result of the lecture. The trapped charge shifts the voltage axis.

Static Response

At equilibrium , so

Equivalently,

The two signs correspond to two voltage directions. Because the system already carries charge, pull-in can occur under either positive or negative applied voltage.

Figure 3.2: Static response of the floating electrode actuator for an arbitrary normalized charge and parasitic capacitance. The charge shifts the standard pull-in curve upward by , producing positive and negative pull-in voltages.

In the plot, the green point is the initial charged state at . The applied voltage can move the system toward a positive pull-in point or toward a negative pull-in point. The dashed segments are unstable equilibria.

Stiffness and Pull-In

The normalized stiffness is

Substituting the equilibrium condition (3.19) into (3.21) gives the stiffness along the equilibrium branch:

Pull-in occurs when , hence

The critical effective voltage is

and therefore the two static pull-in voltages are

Figure 3.3: Equilibrium stiffness along the response curve. Stability is lost when .

The initial charge loading is possible only if the trapped charge is not already beyond pull-in:

This means the charge itself can destabilize the actuator, even before any external voltage is applied.

מסקנה:

A floating electrode actuator behaves like a voltage-driven parallel-plate actuator in the effective voltage . The trapped charge does not change the shape of the pull-in curve; it shifts the voltage axis and creates two possible pull-in voltages.

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