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

© H. Föll (Semiconductor Technology - Script)