# Microwave Figures of Merit

New for January 2017: here's a page on quality factor.

Figures of merit are often used in this industry. Figures of merit should always have the property that maximizing them is a good thing; otherwise they would be figures of demerit.  We'll try to explain a few of them here, starting with switch device FOM.

## Figure of Merit for switches

When we talk about the switch device figure of merit, why is there usually a  2π term in the denominator?

Electrical engineers are familiar with the time constant RC for a circuit with resistance and capacitance, which comes out in units of time, in microseconds, nanoseconds, or whatever you prefer. If you took its reciprocal, you would get units of Hz, MHz, GHz, etc.  Yet the RC time constant doesn’t have a 2π term…

Switch FOM is not a time constant, as the resistor and capacitor we are focusing on are not in the circuit at the same time (they refer to the two switch states).  Let’s see where the true FOM comes from.

Resistance of a switch device in its on-state is:

Where RON is the resistance normalized to periphery (in ohm-mm for example, as in switch FET).  Obviously, it is best to minimize RSw for low insertion loss.

Capacitive reactance of a switch device in its off-state is:

Where Q the periphery of the device, COFF is the capacitance normalized to periphery (pF/mm for example).  Obviously, it is best to maximize XSw for high isolation. RSW and XSW work against each other: reduce RSw for low loss and you are giving up isolation due to XSw.

Let’s look at the ratio of switch device on-resistance to off-reactance. Conveniently, the “Q" periphery term drops out and the equation is true for all device sizes.  This term needs to be minimized, by selecting the most capable process (as opposed to choosing device periphery).

At what frequency are Ron and XCoff equal? This is called the switch figure of merit, in units of frequency.

Rearranging:

Or simply,

Which leads us back to where we started:

If you made a switch that you wanted to operate at the switch device's FOM frequency, it would not be very good.  It might have equal insertion loss and isolation (ignoring that there is a capacitive term in the on-state as well which makes matters worse).  In reality, you can make a decent switch up to about 20% of the device FOM frequency, discounting other performance-robbing parameters like the substrate being too tall to eliminate spurious modes  Switch FOM provides a convenient way to judge semiconductor switch processes. Figures of merit should always have the property that maximizing them is a good thing; otherwise they would be figures of demerit.  That is why we don’t use time as a switch FOM, we focus on frequency.

Why do some authors leave out the 2π term and come up with their own FOM?  No one has provided a documented analysis in an article that could be referenced.  So, let’s all use the correct term in IEEE and other journal papers from now on, and reference Microwaves101 for offering the reasoning behind it.

### References

Here are some references that previously attempted to derive switch device FOMs. All were provided by Rob from Baltimore, thanks!

M. Hines, "Fundamental Limitations in RF Switch and Phase Shifting Using Semiconductor Diodes", Proc. of IEEE, June 1964, pp. 697-708.

A. Gopinath at al, "GaAs FET RF Switches", IEEE Trans. on Elect Dev.,Vol. ED 32, No. 7, July 1985, pp.1272-1278.

R. Gutmann and D. Fryklund, "Characterization of Linear and Nonlionear Properties of GaAs MESFETs for Broad Band Control Applications", IEEE Trans. on Micr. Theory and Tech., Vol MTT-35 No. 5, May 1987, pp. 516-521.

Author : Unknown Editor