# Impedance matching without smith charts

1. Jul 1, 2012

### rppearso

Can anyone help me with impedance matching WITHOUT using smith charts. I am trying to develop a mathcad template to calculate imput impedances for normal power line systems as well as parallel and series impedance matching. I was not very good at using smith charts and never really got the hang of it and would like to build a tool I could use to be successful in graduate level antenna theory.

2. Jul 1, 2012

### yungman

Yes, it is absolutely possible. What do you mean by power line? Are you talking about real power line that is 60Hz and running miles long? Post your parameter and what are you looking for. I might have to take a little bit of time reviewing the formulas particular if you are talking about power lines which might be parallel line tx lines. My only doubt is finding the impedance of the line if it is just lines hanging on the post. If we can determine the impedance then, yes you can do the matching without Smith Chart.

3. Jul 2, 2012

### yungman

The basic input impedance formula is:

$$Z_{in}(z)= Z_0 \frac{Z_s-Z_0 \tanh (δz)}{Z_0-Z_s \tanh (δz)}\;\hbox { where }\; \delta =\alpha+j\beta \hbox { and } \beta =\frac {2\pi}{\lambda}$$

Where z is the distance from the generator ( power station). $Z_s\;$ is the output impedance of the power station ( source impedance). $Z_0\;$ is the characteristics of the transmission line. $\alpha\;$ is the attenuation constant that you don't want to deal with if possible.

But for 60Hz power line hanging in the air, dielectric loss can be assume zero or assume lossless dielectric. The only thing that contribute to loss is the ohmic loss due to the resistance of the wire. At 60Hz, I don't think you need to worry about skin effect and just use the resistance given by the data sheet.

On first pass, it you can assume ideal lossless transmission line, things get much easier as the input impedance:

$$Z_{in}(z)= Z_0 \frac{Z_s-jZ_0 \tan (\beta z)}{Z_0-jZ_s \tan (\beta z)}\;\hbox { where }\; \beta =\frac {2\pi}{\lambda}$$

Since dielectric is air in power line, so $\epsilon=\epsilon_0\;$ and speed of propagation is $3\times 10^8 m/sec\;$. With this, you find $\beta$.

The difficult parameters are the $Z_s$ which you have to find out. Also, $Z_0\;$ is the characteristic impedance of the transmission line and you need to provide the dimension and structure of the power lines to determine. I am not particularly good in doing this. Particular it is 3 phase and they do cross talk to each other.

If you can get through to this point, all you have to do is to terminate with complex conjugate and you get matching termination.

The maximum power transfer is if the $Z_s=Z_0$, then you just terminate with the characteristic impedance of the transmission line and you are done, no worry about the distance or Smith Chart. And sorry I don't know anything about power station. I am just doing the impedance match thingy for you. And I have a suspicion that's the easy part, your power station impedance and the transmission line impedance is the hard part. The calculations are simple enough I would just do it in Excel. I did a lot of the simulation in Excel.

Last edited: Jul 2, 2012
4. Jul 8, 2012

### rppearso

I have derived what I think should work as far as a program input.

Zd=-Zl/(Zl-1) Single stub tuner where Zl=Zo*(1+ $\Gamma$*e^-2*$\beta$*l)/(1-$\Gamma$*e^-2*$\beta$*l)

For l=0 to 1 step 0.0001

d=-1/2*$\beta$*ln((1/$\Gamma$)*((Zd/Zo)-1)/((Zd/Zo)+1))

If Im(d) < 0.0001 and display 1 = null THEN display 1 Re(d) and l
Else if Im(d) < 0.0001 then display 2 Re(d) and l
End if
Next l

The display was to represent a cell in excel however since these are all complex results I dont think I will be able to use VB unless I painstakingly break up the complex and real parts of the equation so that VB can treat them a simply different variables.

Otherwise I would wirte this in matlab which is the language of EE anyways.

Last edited: Jul 8, 2012