Lumped Element Equivalent Circuit for Microstrip line

Question

I am breaking my head already for 2 days on this problem I have. I want to represent my microstrip line in lumped components. My microstrip line has the following characterstics: $$ Z_0 = 50 \text{ [$\Omega$]}, \\ \phi = 180 ^{\circ},\\ \gamma = \alpha +j\beta \text{ [$1/m$]} \\ \beta = \omega \sqrt{\mu_0 \epsilon_r \epsilon_0} \text{ [$1/m$]} $$ where \$\gamma\$ is the propagation constant. With these known values it should be possible to calculate a lumped element representation of the microstrip line. I tried the following equations: $$ \frac{R + j\omega L}{\gamma} = \frac{\gamma}{G + j \omega C} \text{ [$\Omega$]}\\ R = \alpha Z_0 \text{ [$\Omega$]}\\ L = \frac{\beta Z_0}{\omega} \text{ [H]}\\ G = \frac{\alpha}{Z_0} \text{ [S]}\\ C = \frac{\beta}{\omega Z_0} \text{ [F]} $$

The resulting values for the lumped components do not represent my microstrip at all. When running simulations using the model it does not end up with the same scatter matrix as the microstrip.

Could someone help me to figure out what goes wrong? Am I using the wrong model for a short lossy transmission line? I am trying to get the same characteristic impedance and propagation constant as the microstrip line.

Thank you so much in advance! :)

Answer 1

R-L-C-G are "values" per length unit. \$ Zo^2=L/C \$ for a lossless line.

See this in french :-), specially page 67 for use of chained matrix [ABCD].

What you are searching is probably here.

Unless "errors" ... Maple sheet (very very old one) ...

Comparison and formulas for line and equivalent T (2-port) ... This for very short segment of line.

NB: rr may be perhaps simplified (?)

Answer 2

Your units for \$R\$ should be \$\rm\left[\Omega/m\right]\$, not just \$\rm\left[\Omega\right]\$. Similarly \$L\$ should be \$\rm\left[H/m\right]\$, \$C\$ should be \$\rm\left[F/m\right]\$, and \$G\$ should be \$\rm\left[S/m\right]\$.

Now you can divide up your model into as many short segments as you like. A minimum of 10 per wavelength at your frequency of interest is recommended. Since you already know your electrical length is \$180^\circ\$, you want at least 5 RLGC elements in your model. The more elements you use, the more accurate your model will be.

That said, I do not think the formulas you are using are all meant to apply to the same physical line. For example, you have \$R=\alpha Z_0\$ and \$G = \alpha/Z_0\$. These imply that \$R\$ and \$G\$ aren't are independently determined by the line geometry and material properties. I suspect that one formula is meant for a line with losses dominated by series resistance (an RLC line) and the other is meant for a line with losses dominated by shunt conductance (an LGC line). They are not meant to both be applied to the same line.