How to Model RLGC Transmission Line

This article covers:

• Transmission line (TL) equation and its solution
• Input impedance of a terminated TL
• Conversion between RLGC and γZ parameters
• S, Z, Y and ABCD Parameters of TL
• Examples of obtaining parameters from simulation

Transmission Line

Transmission Line Equation

We have

$$v(z,t)-R\Delta z i(z,t)-L\Delta z\frac{\partial i(z,t)}{\partial t}-v(z+\Delta z)=0\\ i(z,t)-G\Delta zv(z+\Delta z,t)-C\Delta z\frac{\partial v(z+\Delta z,t)}{\partial t}-i(z+\Delta z,t)=0$$

Divide them by $\Delta z$ and set $\Delta z\to0$

$$\frac{\partial v(z,t)}{\partial t}=-Ri(z,t)-L\frac{\partial i(z,t)}{\partial t}\\ \frac{\partial i(z,t)}{\partial t}=-Gv(z,t)-C\frac{\partial v(z,t)}{\partial t}$$

For the sinusoidal steady-state condition

$$\frac{d V(z)}{d t}=-(R+j\omega L)I(z)\\ \frac{d I(z)}{d t}=-(G+j\omega C)V(z)\\$$

With manipulation, it turns into

$$\frac{d^2 V(z)}{d t^2}-\gamma^2V(z)=0\\ \frac{d^2 I(z)}{d t^2}-\gamma^2I(z)=0\\$$

where

$$\gamma=\sqrt{(R+j\omega L)(G+j\omega C)}=\alpha+j\beta$$

The solution is

$$V(z)=V^+e^{-\gamma z}+V^-e^{\gamma z}\\ I(z)=I^+e^{-\gamma z}-I^-e^{\gamma z}$$

$V(z)$ can be used to derive $I(z)$ as

$$I(z)=\frac{\gamma}{R+j\omega L}(V^+e^{-\gamma z}-V^-e^{\gamma z})$$

we have

$$\frac{V^+}{I^+}=\frac{V^-}{I^-}=Z_0=\frac{R+j\omega L}{\gamma}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}$$

The solution of $V(z)$ can be written in the time domain as $\mathrm{Re}[V(z)e^{j\omega t}]$

$$v(z,t)=|V^+|\cos(\omega t-\beta z+\phi^+)e^{-\alpha z}\\ |V^-|\cos(\omega t-\beta z+\phi^-)e^{\alpha z}$$

where we have

$$\lambda=\frac{2\pi}{\beta}\\ v_p=\frac{\omega}{\beta}=\lambda f$$

For lossless lines $\beta=\omega\sqrt{LC}$

$$\lambda=\frac{2\pi}{\omega \sqrt{LC}}\\ v_p=\frac{1}{\sqrt{LC}}$$

Terminated Transmission Line

Since

$$Z_L=\frac{V(0)}{I(0)}=\frac{V^++V^-}{V^+-V^-}Z_0$$

The ratio of the reflected voltage wave amplitude to the incident voltage wave amplitude is defined as the voltage reflection coefficient:

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

Using reflection coefficient, we may rewrite the transmission line solution as

$$V(z)=V^+(e^{-\gamma z}+\Gamma e^{\gamma z})\\ I(z)=\frac{V^+}{Z_0}(e^{-\gamma z}-\Gamma e^{\gamma z})$$

The maximal voltage amplitude to the minimal voltage amplitude is defined as standing wave ratio (SWR):

$$\operatorname{SWR}=\frac{1+|\Gamma|}{1-|\Gamma|}$$

At arbitrary point, the reflection coefficient is

$$\Gamma(-l)=\frac{V^-e^{-\gamma l}}{V^+e^{\gamma l}}=\Gamma(0)e^{-2\gamma l}$$

and input impedance is

$$\begin{aligned} Z_{in}=Z(-l)&=Z_0\frac{1+\Gamma e^{-2\gamma l}}{1-\Gamma e^{-2\gamma l}}\\ &=Z_0\frac{Z_L+Z_0\tanh{\gamma l}}{Z_0+Z_L\tanh{\gamma l}} \end{aligned}$$

For lossless lines

$$\begin{aligned} Z_{in}=Z(-l)&=Z_0\frac{1+\Gamma e^{-2j\beta l}}{1-\Gamma e^{-2j\beta l}}\\ &=Z_0\frac{Z_L+jZ_0\tan{\beta l}}{Z_0+jZ_L\tan{\beta l}} \end{aligned}$$

Thus

• For a quarter-wavelength transmission line where $\beta l=\pi/2$, we have

  $$Z_{in}Z_L=Z_0^2$$

• An open stub where $Z_L=\infty$

  $$Z_{in}=-jZ_0\cot{\beta l}$$

• An short stub where $Z_l=0$

  $$Z_{in}=jZ_0\tan{\beta l}$$

Parameter Conversion

γZ to RLCG

$$R={\rm Re}\{\gamma Z\}$$ $$L={\rm Im}\{\gamma Z\}/\omega$$ $$G={\rm Re}\{\gamma/Z\}$$ $$C={\rm Im}\{\gamma/Z\}/\omega$$

S, Z, Y and ABCD Parameters

$$S=\frac{1}{D_s} \left[ \begin{matrix} (Z^2-Z_0^2)\sinh\gamma l & 2ZZ_0\\ 2ZZ_0 & (Z^2-Z_0^2)\sinh\gamma \end{matrix} \right]\\ D_s=2ZZ_0\cosh\gamma l+(Z^2+Z_0^2)\sinh\gamma l\\ ABCD= \left[ \begin{matrix} \cosh\gamma l & Z\sinh\gamma l\\ \frac{\sinh\gamma l}{Z} & \cosh\gamma l \end{matrix} \right]\\ Z= \left[ \begin{matrix} \frac{Z}{\tanh\gamma l} & \frac{Z}{\sinh\gamma l}\\ \frac{Z}{\sinh\gamma l} & \frac{Z}{\tanh\gamma l} \end{matrix} \right]\\ Y= \left[ \begin{matrix} \frac{1}{Z\tanh\gamma l} & -\frac{1}{Z\sinh\gamma l}\\ -\frac{1}{Z\sinh\gamma l} & \frac{1}{Z\tanh\gamma l} \end{matrix} \right]$$

Note that $Z$ is the characteristic impedance of the transmission line, and $Z_0$ is the characteristic impedance of the measurement system.

Distortionless Line

In a lossy line, the phase term $\beta$ is generally a complicated function of frequency $\omega$. If $\beta$ is not a linear function of frequency, the phase velocity can vary with frequency, causing signal dispersion. However, for the lossy lines, when

$$\frac{R}{L}=\frac{G}{C}$$

it is still distortionless. In this case

$$\begin{aligned} \gamma&=\sqrt{(R+j\omega L)(G+j\omega C)}\\ &=\sqrt{(j\omega L)(j\omega C)\left(1+\frac{R}{j\omega L}\right)\left(1+\frac{G}{j\omega C}\right)}\\ &=j\omega\sqrt{LC}\left(1+\frac{R}{j\omega L}\right)\\ &=R\sqrt{\frac{C}{L}}+j\omega\sqrt{LC} \end{aligned}$$

Parameters Extraction

Direct Simulation

The characteristic impedance can be obtained from direct simulation. First, two ports of identical impedance are connected to the sides of the TL. Then, a parameter sweep of the port impedance is conducted at the intended frequency. The impedance that have the lowest reflection can be used as characteristic impedance.

Real part of characteristic impedance is given by

Also, we note that when the characteristic impedance matches, the gain of the TL can be written as

$$\begin{aligned} S_{21}\biggl|_{Z=Z_0}&=\frac{1}{\sinh\gamma l+\cosh\gamma l}\\ &=e^{-\gamma l}\\ &=e^{-\alpha l}e^{-j\beta l} \end{aligned}$$

Hence, we have

$$S_{21}[dB]=20\log_{10}e^{-\alpha l}=-20\alpha l\log_{10}{e}\\ S_{21}[rad]=-\beta l$$

The above equations can be used to solve for γ. In this case, we have a TL length of $400\times10^{-6} \mathrm{m}$ and $S_{21}|_{Z=Z_0}=-0.22 dB/-28.95^\circ$. We can write

$$\alpha=\frac{S_{21}[dB]}{-8.686[dB/m]\cdot l[m]}=63.3\\ \beta=\frac{S_{21}[rad]}{-l[m]}=1263[rad/m]$$

Calculation

Alternatively, the parameters of a TL can be calculation directly from S parameters. This function is supported in Matlab. Calculation process is as follows:

PolartoComplex = @(len, the) len*cos(the*pi/180) + 1i*len*sin(the*pi/180);

S11 = PolartoComplex(0.019, 9.658);
S12 = PolartoComplex(0.975, -28.942);
S21 = PolartoComplex(0.975, -28.942);
S22 = PolartoComplex(0.019, 9.658);
length = 400e-6;	% length of transmission line
freq = 28e9;		% frequency of simulation
Z0 = 50;
rlgc_params = s2rlgc([S11, S12; S21, S22], length, freq, Z0)

Corresponding output is

    R: 5.1717e+03
    L: 3.6799e-07
    G: 0.4737
    C: 1.4002e-10
alpha: 62.5539
 beta: 1.2634e+03
   Zc: 51.3183 - 1.5526i

Verification of the Model

8 RLGC segments are used for verification

Detailed schematic of each one

Results

Appendix

Field Solution

$$\nabla\times E=-j\omega\mu H\\ \nabla\times H=j\omega\epsilon E+\sigma E$$

which can be transformed into

$$\nabla^2 E+\omega^2\mu\epsilon\left(1-j\frac{\sigma}{\omega\epsilon}\right)E=0$$

For that matter

$$\gamma=\alpha+j\beta=j\omega\sqrt{\mu\epsilon}\sqrt{1-j\frac{\sigma}{\omega\epsilon}}\\$$

We can use a complex permittivity to represent the loss and set $\sigma=0$

$$\epsilon=\epsilon'-j\epsilon''=\epsilon'(1-j\tan\delta)$$

The intrinsic impedance is complex

$$Z=\frac{j\omega\mu}{\gamma}$$

Low-Loss Approximation

For a low-loss line both conductor and dielectric loss will be small

$$\begin{aligned} \gamma&=\sqrt{(R+j\omega L)(G+j\omega C)}\\ &=\sqrt{(j\omega L)(j\omega C)\left(1+\frac{R}{j\omega L}\right)\left(1+\frac{G}{j\omega C}\right)}\\ &\approx j\omega\sqrt{LC}\sqrt{1-j\left(\frac{R}{\omega L}+\frac{G}{\omega C}\right)}\\ &\approx j\omega\sqrt{LC}\left[1-j\left(\frac{R}{\omega L}+\frac{G}{\omega C}\right)\right] \end{aligned}$$

we have

$$\alpha\approx\frac{1}{2}\left(\frac{R}{Z_0}+GZ_0\right)\\ \beta\approx\omega\sqrt{LC}\\ Z_0\approx\sqrt{\frac{L}{C}}$$

How to convert $\gamma,Z$ parameters to $A, \epsilon_{eff},Z$ parameters?

$$\alpha=\frac{A}{20\log_{10}{e}}\approx\frac{A}{8.686}\\ \beta=\omega\sqrt{\mu\epsilon}=\omega\sqrt{\mu_0\epsilon_0\epsilon_{eff}}$$

where $\epsilon'=\epsilon_0\epsilon_{eff}$ and $Z_0=\sqrt{\mu_0/\epsilon_0}=377 \Omega$. These translates to

$$A=20\alpha\log_{10}{e}\\ \epsilon_{eff}=\left(\frac{c\beta}{\omega}\right)^2$$

Reference

1. W. R. Eisenstadt and Y. Eo, "S-parameter-based IC interconnect transmission line characterization," in IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol. 15, no. 4, pp. 483-490, Aug. 1992. DOI: 10.1109/33.159877
2. David M. Pozar. Microwave engineering.