Multi-section transformers can be used where a wider bandwith of matching is required than can be achieved by a single transformer.
To derive such a transformer, we start with the theory of small reflections, applied to a single-section transformer. The incident wave will partially reflect and partially transmit at the interface, which will then reflect at the load, and then reflect again at the boundary, and so on.
Summing the reflections/transmissions, the total reflection seen by the feedline is:
This is a geometric series, which sums to:
Consider now a multisection transformer, which is just lots of small sections of transmission line of equal length
Making a few assumptions:
- Assume the differences between adjacent impedances are small
- Assume all increase or decrease monotonically
- Assume is real
- will be real and of the same sign
The total reflection coefficient is therefore:
Any desired value of can be synthesised by suitably choosing and .
We show how to realise such a transformer with a maximally flat total reflection coefficient, a binomial transformer.
For an -section transformer:
- Set the first derivatives of to 0 at the center frequency
- Provided by a reflection coefficient of the form
- Magnitude is then
- To determine , let
- Expression reduces to
- All sections are of 0 electrical length as
is expressed as a binomial series:
Because we assume are all small, we can approximate the characteristic impedances as:
To find the bandwith of the binomial transformer, let be the maximum tolerated reflection coefficient over the passband.
is the lower edge of the passband. Therefore:
And the fractional bandwith: