Electrical – Scattering Parameters and different load

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:

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

If you connect an ideal matched load to port 2, that will set \$a_2\$ to zero.

However, it's not always possible, or convenient, to get a load that's sufficiently accurate for your purposes. It depends what accuracy you want to work to, and what frequency you're using.

At very high frequencies, GHz, we often use a 'sliding load', which is an approximately matched load, that can be slid along a precision air-spaced coax, to change its electrical length (reflection angle) without changing the absolute reflection. The measurement software finds the centre of the circle that results as the load angle is changed, and this point becomes the effective reflection from a load ideally matched to the air-line.

Another method uses shorted or open sections of coax of different length. With these it's a case of generate enough equations for the unknowns, and solve them simultaneously. Again, the impedance of the coax becomes the effective load impedance with this method.

If you have a load that's not matched, but accurately known, it's still possible to put its value into the equations, and get a good measurement of the S parameters, when used in conjunction with at least two other reflection standards.