# Solution to Exercise 7.1-2

# Forces in Capacitive Structures

| Here is the drawing once more to avoid jumping back and forth:
| 1. Find the proper relations for the forces pulling at the moveable plates for all three configurations.
|  | The total energy E for all three configurations is simply given by
E = | ó õ | U · I · dt | + | ó õ | k · (x – x0) · dx
|  | With the simple relations C = Q/U, I = dQ/dt and therefore U · I · dt = (Q/C) · (dQ/dt) · dt = Q/C · dQ, we obtain
E | = | Q2 2C | + | k · (x – x0)2 2
--- | --- | --- | --- | ---
| = | C · U2 2 | + | k · (x – x0)2 2
F | = – | U2 2 | · | dC dx | – | k · (x – x0)
--- | --- | --- | --- | --- | --- | ---
|  | Without integration limits we cannot get proper signs (one energy term must decrease if the other one increases because we have energy conservation) - but that is not important here since we know that the spring force and the capacitive force must have opposite signs, and we are only interested in the capacitive force F C.
| For the capacity C and the force F C = ½U2 · (dC/dx) we obtain:

|  | 1. Configuration: |  |
C1 | = | e0 · A x
--- | --- | ---
F 1 | = | U2 2 | · | e0 · A x2
--- | --- | --- | --- | ---
|  | 2. Configuration: x* is the (easy to calculate) plate overlap for zero voltage. But since it disappears upon differentiation, we do not need to spell it out. |  |
C2 | = | e0 · (x* – x) · h y
--- | --- | ---
F2 | = | U2 2 | · | e0 · h y
--- | --- | --- | --- | ---
|  | 3. Configuration: C3 is simply given by 2 C2 in parallel |  |
C3 | = 2C2 = | 2e0 · (3l/2 – x) · h y
--- | --- | ---
F3 | = 2F3 = | U2 | · | e0 · h y
--- | --- | --- | --- | ---

| 2. Compare the relative strength of the first and third configuration.
|  | If we simply take the relation F1/F3 for equal distances between the plates (i.e. x = y), we obtain
F1 F3 | = | A y · h
--- | --- | ---
|  | Considering that y · h << A for typical structures, configuration 1 can transmit much more force than the other ones for about identical size.

| 3. Discuss the pro and cons of the two configurations for driving an actual actuator.
| In configuration 1 the force decreases with the square of the distance between the plates; in the extreme case of zero distance the plates would stick together forever (in reality a fuse will blow).
|  | The design rule is obvious: Use with extreme care!
| In configuration 3 the force is independent on the position, which makes the design reasy. However, the force is relatively small.
|  | The consequences are obvious too: This is the preferred configuration, but you need to employ many combs to achieve sufficient force.

With frame

Exercise Basic 7.1-2 Capacitors and Forces

Exercise Basic 7.1-1Class Exercises and Quick Questions to 7.1 MEMS - Products and Developments

Exercise Basic 7.1-1Class Exercises and Quick Questions to 7.1 MEMS - Products and Developments

© H. Föll (Semiconductor Technology - Script)