microwave software®
OptiMatch® User Manual
Impedance Match Network Optimization Utility
NETWORK OPTIMIZATION EXAMPLES
Optimization & Match Bandwidth Limitations
Constrained vs. Unconstrained Optimization
There are many possible constraints that may be programmed into an optimization algorithm. One very useful one is the option that all element values be positive.
A circuit element, like a 5 pF capacitor, a 10 nH inductor, etc. is just another number to a mathematical algorithm. But, while numbers can be either positive or negative, element values must be positive to be realizable. By introducing a positive bound constraint, we require OptiMatch to never let an element change sign, even if by doing so it reduces the value of the error function.
To accomplish this positive bound constraint, we work with the square root of a number inside the optimizer, and the square outside of it. Then, if the optimizer does make a variable negative, there will be no change in the error function, (which is outside the optimizer), and the negative value will be withdrawn.
A few extra milliseconds are required if you elect to use the positive bound constraint, because of the squaring and un-squaring required. Also, the gradients are computed differently, using the chain rule of calculus, and will be somewhat larger. Its hard to say what the net effect will be since every case is different. However, element values will be held positive. A particular circuit may optimize better one way than another. We suggest you try both ways and try to get a feel for the difference.
Constrained vs. Unconstrained Optimization
There are many possible constraints that may be programmed into an optimization algorithm. One very useful one is the option that all element values be positive.
A circuit element, like a 5 pF capacitor, a 10 nH inductor, etc. is just another number to a mathematical algorithm. But, while numbers can be either positive or negative, element values must be positive to be realizable. By introducing a positive bound constraint, we require OptiMatch to never let an element change sign, even if by doing so it reduces the value of the error function.
To accomplish this positive bound constraint, we work with the square root of a number inside the optimizer, and the square outside of it. Then, if the optimizer does make a variable negative, there will be no change in the error function, (which is outside the optimizer), and the negative value will be withdrawn.
A few extra milliseconds are required if you elect to use the positive bound constraint, because of the squaring and un-squaring required. Also, the gradients are computed differently, using the chain rule of calculus, and will be somewhat larger. Its hard to say what the net effect will be since every case is different. However, element values will be held positive. A particular circuit may optimize better one way than another. We suggest you try both ways and try to get a feel for the difference.
Match Bandwidth Limitations
Years ago, Fano published an important paper regarding the theoretical limitations on the broadband matching of reactive loads. His paper, a true classic, is unfamiliar to many young engineers.
What Fano did was to prove that, given a series or parallel RC or RL load, was that there exists a finite limit on the minimum attainable VSWR across a given band of frequencies, even when using a match network with an infinite number of elements in it.
This is profound, and a remarkable achievement. His work today is referred to as "The Fano Bandwidth Limit." Using the work of Fano, you can determine, up front, the best that can be done in any reactive match situation. Why spin your wheels endlessly trying to do better than the theoretical limit?
Keep in mind that, in practice, there is no way you can actually reach Fano bandwidth limit, but you can approach it. If you would like to see a reference to an article entitled "Calculator program finds Fano bandwidth," written by Microwave Software President, James J. Lev, see Appendix D.
Our Utilities+ program has, among twenty-six other useful design aids, a Fano program you can use.
Concluding Remarks
These five examples should give you a good feeling about OptiMatch, and what it can do for you. As you can imagine, this is a complex subject, and there is a lot of room for experimentation. Have fun!