Converting Bridge Voltages to Load Impedance: Method 1

The equations for the bridge voltages that can be measured (see Output Voltages on RF Wheatstone Bridges ) are:

Eq1	$$V_a=2V_f\frac{50}{\vert Z_l+50\vert }$$

Eq2	$$V_z=2V_f\frac{\vert Z_l\vert }{\vert Z_l+50\vert }$$

Eq3	$$V_r = \vert 2V_f \frac{Z_l}{Z_l+50} - V_f\vert $$

Step 1 - three voltages to two voltage ratios

Looking at the equations for $V_z$ and $V_a$, we can see that they have the same denominator, and both share the multiplier $2V_f$ , so dividing one equation by the other will get rid of these quantities and leave us with $\frac{V_z}{V_a}=\frac{\vert Z_l\vert }{50}$which gives us the magnitude of the unknown impedance.

To get the complex impedance, we need an equation that contains it directly rather than inside \vert mod\vert bars. If we look at the equation for $V_r$ above, we can rearrange to get a single fraction as follows:

$$\frac{V_r}{V_f} = \vert 2 \frac{Z_l}{Z_l+50} - 1 \vert = \vert \frac{2Z_l - (Z_l+50)}{Z_l+50}\vert = \vert \frac{Z_l-50}{Z_l+50}\vert $$

So we have:

Eq4	$$\frac{V_r}{V_f}=\vert \frac{Z_l-50}{Z_l+50}\vert $$

Eq5	$$\frac{V_z}{V_a}=\frac{\vert Z_l\vert }{50} $$

Step 2 - Calculating the Load Impedance

Note that the RHS of Eq4 is the magnitude of the reflection coefficient, or $\rho$

So Eq7 from the explanation above (expand it if it's closed) is

$$\frac{(R-50)^2+X^2}{(R+50)^2+X^2}=\frac{\vert Z_l\vert ^2-100R+2500}{\vert Z_l\vert ^2+100R+2500}=\rho^2$$

$\rho$ is related to VSWR by $\rho=\frac{VSWR-1}{VSWR+1}$ so $\rho^2$ is

$$\frac{(VSWR^2+1)-2VSWR}{(VSWR^2+1)+2VSWR} = \frac{(R-50)^2+X^2}{(R+50)^2+X^2}=\frac{\vert Z_l\vert ^2-100R+2500}{\vert Z_l\vert ^2+100R+2500}$$

Which can be rearranged to give

$$R = (2500+\vert Z_l\vert ^2)\frac{VSWR}{50(VSWR^2+1)}$$

VSWR is given by $\frac{1+\rho}{1-\rho}$ which can be combined with Eq4 to give

$$VSWR = \frac{V_f-V_r}{V_f+V_r}$$

So, with an interim step of calculating VSWR, we have:

</details>

$$R = (2500+\vert Z_l\vert ^2)\frac{VSWR}{50(VSWR^2+1)}$$

where

$$VSWR = \frac{V_f-V_r}{V_f+V_r}$$

$$X = \pm\sqrt{(50\frac{V_z}{V_a})^2 - R^2}$$