What options do I have to measure directly impedance change (ohm) or deduce it indirectly from S-Parameters?

Basically I want to measure how impedance (Ohm) relative to Z0 (50 Ohm) waveguide is changing by varying a parameter. I currently measure the reflection coefficient S11 in a 1-port setup. Our VNA can store complex S11 parameters, but also magnitude (dB or linear) and phase.

As far as I understand it's not so easy now to get from S11 to Ohm and Smith Chart was designed for that reason. A webpage says:

In order to convert S-parameters to impedances, you must specify Z0. Usually it's 50 ohms, sometimes 75 ohms.

The calculation to get from S-parameters to impedances is more complicated than, for example, VSWR. This is one of the reasons the Smith Chart was invented, you could enter coordinates either way and the graph would solve the equations for you. Here's one form of the equations, sent by an alert engineer named Steve:

REAL=(Z0*(1-(MAG*MAG)))/(1+(MAG*MAG)-
(2*MAG*COS((ANG/360)*2*PI())))

IMAGINARY=(2*MAG*SIN((ANG/360)*2*PI())*50)/
(1+(MAG*MAG)-(2*MAG*COS((ANG/360)*2*PI())))

He sure like brackets! Here's the input and output impedance, with real and imaginary parts plotted separately. Ideally the real part is 50 ohms, and the imaginary is zero.

So with REAL and IMAGINARY R and X in Z=R+jX are meant here or this again real and imaginary numbers of the S11 parameters?

As I recorded now some data where only the S11 real and imaginary pair numbers where stored, I'm wondering if I can deduce from S11 the magnitude to get with above formulas to Ohm, so I don't have to measure again? I'm using origin for post-processing the data, also matlab. Both have tools to plot smith charts and read VNA data to my knowledge.

Thanks for your kind help

• Transform your S-matrix to a Z-matrix $Z_{11}$ will be the input impedance. – Captainj2001 Jul 9 '16 at 12:01
• A link to some formulas would be helpful as I'm no elec. engineer and don't the math by heart. I found the formulas magnitude [dB] = 20 * Log(sqr(Re^2 + Im^2)) and phase = arctan(Im / Re)which allows me to use above formula. If it is just a matrix multiplication with the identity matrix as you pointed out, then root square of S11 will scale with the change of impedance in %? If I want absolute real Z in Ohm I need it to relate to Z0(50 or 75 ohm)? – Hauser Jul 9 '16 at 14:07

Since you're doing a 1-port measurement you can obtain the load impedance from $S_{11}$ which is equivalent in this case to the reflection coefficient $\Gamma$.

$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$$

Rearranging this equation gives:

$$Z_L = Z_0\frac{1 + \Gamma}{1 - \Gamma} = Z_0\frac{1 + S_{11}}{1 - S_{11}}$$

which will be your load impedance relative to $Z_0$.

Regarding the fact that $S_{11}$ will be a complex number, in general load impedances that are real at one frequency will be complex at most others. At RF frequencies parasitic capacitances and inductances of components (loads) cannot be ignored.

Waveguides are not the same as transmission lines however, for most transmission lines a Transverse ElectroMagnetic or TEM mode of wave propagation is assumed. However, in waveguides either Transverse Electric (TE) or Transverse Magnetic (TM) modes of propagation are selected. The impedances of these modes are defined differently than for TEM lines. If you would like to know more see:

• we are near to the solution and I can mark this as accepted... so reflection coefficient is synonym to S11, this is all very confusing because the gamma pops up on some websites but S11 not... So I enter in your equation the real number of the complex S11 (I get a pair of numbers for every frequency when I measure with the VNA the smith chart in a frequency span)? It's also confusing to call gamma reflection coefficient if the VNA always spits out two numbers for every frequency when choosing S11 instead of magn + phase – Hauser Jul 9 '16 at 14:31
• @Hauser For a 1-port device the reflection coefficient and $S_{11}$ are the same, this is not the case for multi-port networks however, UNLESS all of the other ports have no reflections. A good resource to look at is microwaves101.com or if you can get a copy Microwave Engineering by D.M. Pozar. – Captainj2001 Jul 9 '16 at 14:38
• So the numbers I get are always complex pairs as we measure a coplanlar waveguide via a microwave probe. What disturbs me a bit, is that the formulas of "Steve" in my question contains magnitude and phase to calculate real part of ZL and your formula not. Is steve formula only fitting 2-port networks and for 1-port the mag*cos term vanishes? My imaginary values look plotted against the parameter we change (a field) similar to real values, but above a distinct frequency this changes completely. This might not affect calculation of ZL by your fomula? – Hauser Jul 9 '16 at 15:00
• PS: I surely can get the Pozar bible and read it over some weeks, but this is not the point of this website ;) – Hauser Jul 9 '16 at 15:01
• @Hauser The reflection coefficient almost always be a complex number, so yes, your effective load impedance will also have some component to it that is either inductive (positive imaginary part) or capacitive (negative imaginary part) in addition to the real part of the impedance. For the formulas I gave you will use the R + jX form of the impedance measured from your VNA. There are many ways to manipulate the Mag/Phase or Real/Imaginary numbers to get the resistive part and the reactive part to the impedance. Steve's formula seems sound as well. – Captainj2001 Jul 9 '16 at 15:04