The article A thinking exercise on Jacobi Maximum Power Transfer #3 discussed Kurokawa's power reflection coefficient as in indicator of mismatch at a system node.

Above is a demonstration circuit in Simsmith, a linear source with Thevenin equivalent impedance of 50-j5Ω. The equivalent voltage is specified by useZo, which like much of Simsmith is counter intuitive (as you are not actually directly specifying generator impedance):

Vthev and Zthev are chosen so that ‘useZo' will deliver 1 watt to a circuit impedance that equals the G.Zo. Zthev will be Zo*.

So, in a lossless conjugate matched circuit we would expect the load power to be 0dBW.

In the circuit above, the load is not the conjugate of the source impedance so there will be some mismatch loss, to mean the ratio of power in a matched load to the power in the mismatched load.

In this scenario, the calculated power in the mismatched load is -0.307dBW, so mismatch loss is 0 – -0.307=0.307dB.

This mismatch load is often calculated as \( ML=10\cdot log_{10}(\frac1{1-|Γ|^2}) \) where Γ is calculated wrt 50+j0Ω, and the value is shown above at ML as 0.3856dB, quite different to the actual mismatch loss.

KML is mismatch loss calculated using Kurokawa's power reflection coefficient, and at 0.3069dB (rounds to 0.307dB) it reconciles with the 0.307dB calculated from the displayed power levels.

The expressions used for the displayed G quantities are show above.

What if we could introduce a circuit element that hid the true nature of the source and made it look like an ideal 50+j0Ω source?

The scenario above inserts an ideal isolator between source and load (the N block as it is known in Simsmith).

Note that this is not your Hammy Sammy isolator. Isolator is one of the many terms with a well understood meaning in industry, appropriated by hams and give new / incompatible meaning just to confuse.

An isolator, this isolator, is a two port device that allows RF to flow in one direction and not the other for a specified Zo. So in this instance, Zo is specified as 50+j0Ω, and power can flow from source to load (with no loss), but no power can flow from load to source.

An isolator is characterised by \(S=\begin{bmatrix}0 & 0\\1 & 0\\\end{bmatrix}\) using conventionial notation, but Simsmith does not follow convention, you must transpose P1 and P2.

Above is Simsmith's “backwards circuit” display option, the departure from convention in port labelling is more obvious, so to understand things in the Simsmith world, \(S_{simsmith}=\begin{bmatrix}0 & 0\\0 & 1\\\end{bmatrix}\) but to program it, you can undo the port transposition in the wiring and use the conventional S matrix, see below.

//Isolator P1 w2 gnd; P2 w1 gnd; sprm1 {w1 gnd w2 gnd} {{0,0},{1,0}} {50};

A consequence of the isolator is that the output of the isolator (P1… yes, more backwards confusion from Simsmith) appears to have a source impedance of 50+j0Ω. The displayed quantity L_sZ is the calculated impedance looking from load L back towards the source, it is 50+j0.0001247Ω… slightly off, probably due to rounding errors in a fairly complicated calculation.

Note that the effect of the isolator is that there is no reflected energy (wrt isolator Zo), so the source sees a load of 50+j0Ω.

Power into the isolator is -0.0108dBW (mismatch loss at this node is 0.0108dB) and power into the load is 0.396dBW, mismatch loss at the load input is 0.385dB. Both calculated (traditional) ML and KML reconcile with the calculated power levels.

Is an isolator the magic component that can deliver a transmitter with Thevenin source impedance equal to 50+j0Ω? Well, low loss isolators are practical at microwaves, and devices can be made to cover HF, the do it with considerable loss. For an ordinary HF SSB telephony transmitter there is little benefit and severe cost and efficiency issues.

## Classic misuse

… or where a little knowledge is dangerous.

A common newby online question is “my 50Ω VSWR is 3, surely that is really bad?”

The expert answers tend to go along the line “Don't worry, reflected power is 25%, which means your transmitted power is 75%, that is just 1.3dB lower, a quarter of an S-point and no one will notice it.” Some may provide a link to a handy dandy table of these magic values.

The experts have assumed the transmitter is well represented by a Thevenin equivalent circuit, and that Zs is 50+j0Ω, an unwarranted assumption. Not only is the transmitter unlikely to be sufficiently linear to apply linear circuit theory in that way, practical transmitters often include protection systems that may reduce power output to limit ‘reflected power' to 10% of rated power or less.

The fact that in many transmitters, protection circuits would have kicked in is a salutary warning. Operation well outside of specified load range might well result in degraded distortion products.

## Conclusions

Calculation of mismatch loss requires an understanding of the characteristics of the source.

It is unsafe to assume that a transmitter that is designed to work into a 50Ω load is well represented as a Thevenin source with Zs=50Ω.

Traditional mismatch formula fails under some load and source impedance combinations.

Kurokawa's power reflection coefficient may be a usable metric for linear systems, provided that actual source and load impedance are known.

## References

- Calculate Kurokawa's power reflection coefficient from Zload (or Yload or S11) and Zsource
- Kurokawa,K. Mar 1965. Power waves and the scattering matrix In IEEE transactions on microwave theory and technique Mar 1965 p 194.