# S21 parameters multiplication

I have a two port system which consists of two subsystems. Is the multiplication of S21 parameters of the subsystems equal to that of the whole system?

• Only if $S_{11}(L)$ and/or $S_{22}(K)$ are 0. Commented Aug 4, 2016 at 23:20
• Does it mean there must be no input reflection for L and no output reflection for K?
Commented Aug 4, 2016 at 23:32
• If one or the other is 0, I think it's enough. Commented Aug 5, 2016 at 0:33

It isn't that simple. There can also be an effect due to reflections back and forth between the input of L and the output of K.

If L has no input reflections ($S_{11}=0$) or K has no reverse reflections ($S_{22}=0$), then your formula should work. (Edit: As I think about it some more, you'd also need to have a perfectly matched load on the output of L)

But if that's not the case, you have to jump through more hoops. The usual approach is to transform the model to a different representation called "T parameters":

$$T_{11}=\frac{1}{S_{21}}$$ $$T_{12}=-\frac{S_{22}}{S_{21}}$$ $$T_{21}=\frac{S_{11}}{S_{21}}$$ $$T_{22}=\frac{-(S_{11}S_{22}-S_{12}S_{21})}{S_{21}}$$

The T parameters can then be cascaded:

$$T^{(KL)}=T^{(L)} T^{(K)}$$

And then the T parameters can be transformed back to S parameters:

$$S_{11}=\frac{T_{21}}{T_{11}}$$ $$S_{12}=\frac{(T_{11}T_{22}-T_{12}T_{21})}{T_{11}}$$ $$S_{21}=\frac{1}{T_{11}}$$ $$S_{22}=-\frac{T_{12}}{T_{11}}$$

• ABCD parameters also (same as T, but scaled by Zo) ;-) Commented Aug 5, 2016 at 13:01