# Coupled Mode Theory

1. Aug 16, 2004

### Claude Bile

I need information on coupled mode theory, specifically with reference to cylindrical waveguides. The more info the better. I have some references, but they are quite scattered, I am yet to find a reference that really encapsulates the 'heart' of coupled mode theory.

If anyone has a good reference (web based or otherwise), on coupled mode theory, I would be greatful.

If somebody that is familiar with Coupled Mode Theory has some time on their hands, you are welcome to post info on the theory itself, however I gather that the theory is rather complex.

Regards, Claude.

2. Aug 16, 2004

### pervect

Staff Emeritus
I never heard of coupled mode theory. I can talk a little bit about the behavior of weakly and strongly coupled resonant transformers, which are coupled resonators, though they are lumped systems. This makes them easier to analyze, too.

For the strongly coupled resonant transformer, the voltage ratio depends on the turns ratio of the primary and secondary. For a weakly coupled case (like a typicial Tesla coil), the voltage ratio depends much more critically on the tuning of the primary & secondary than on the turns ratio.

More insight into the resonant transformer by writing down the differential equations

For the coupled inductors let the voltages across the primary and secondary be V1 and V2, the currents into the primary and secondary be I1 and I2. Orient the currents as follows.

V1-----I1----> <-----I2------V2

(I tried to draw a better pciture, couldnt' figure out how to turn off formatting)

For the coupled inductors we can write

$$\begin{array}{l} V_1 = L_1 \frac {dI_1}{dt} + M \frac {dI_2}{dt}\\ V_2 = M \frac {dI_1}{dt} + L_2 \frac {dI_2}{dt} \end{array}$$

To complete the circuit, you need to add a capacitor C1 across the primary, and a capacitor C2 across the secondary, and some series resistances R1 and R2 in series with the primary and secondary inductors

You can write down the diffeq's with the lapalace transform in terms of the current into the primary and secondary

$$\begin{array}{l} -I_1 / s C1 = L_1 s I_1 + R_1 I_1 + M s I_2 \\ -I_2/ s C2 = M s I_2 + L_2 s I_1 + R_2 I_2 \end{array}$$

The rest is a matter of analyzing the behavior of these diffeq's - you can substitute s=jw in the usual way to get the behavior vs frequency.

Well, there's one more piece of info you need to know

The strongly coupled case is represented by $$M = k \sqrt{L_1 L_2}$$ with k approximately equal to unity. The weakly coupled case is where k is less than unity. Qualitiatively, you should find that for the strongly coupled case, the bandwidth will be very wide, and the voltage will be a function of $$L_2 / L_1$$, which is equivalent to the turns ratio.

The weakly coupled case is much more similar to your coupled resonators. The bandwidth will be much narrower. I don't recall the expression for the voltage ratio offhand though.

I hope this is of some use, and not too far afield.

3. Aug 16, 2004