Step Response In a Hybrid Lumped-Distributed Network

Introduction

Transmission line theory has been around for more than a century. However, the distributed nature of it has kept it interesting for all these years. One interesting subject is the step response in a hybrid lumped-distributed network. While for a purely lumped network it is quite easy to analyze and derive the response equation, for a distributed or a hybrid network, it may not be. This short post covers this topic briefly . It is assumed the reader has the fundamental knowledge of transmission lines.

A Review of Transmission Line

Transmission lines are mainly defined with the following factors:

  • Characteristic impedance Z_{o}=\sqrt{\frac{R+j\omega L}{G+j\omega C}} where R, L,G, C are the resistance, inductance, conductance and capacitance per unit length. In a lossless transmission line Z_{o}=\sqrt{\frac{L}{C}}.
  • Propagation constant \gamma=\alpha+j\beta where \alpha is the loss (this post considers a lossless transmission line in which \alpha=0)  per unit length and \beta is the wave number.
  • The wave velocity (phase velocity) \vartheta=\frac{{c}}{\sqrt{\epsilon_{r}}} where c is the speed of the light and \epsilon_{r} is he permitivity of the medium. For an IC designers \epsilon_{r}=4.2 as the signal travels in silicon-doixide. 
  • The signal wavelength \lambda=\frac{\vartheta}{f} where \vartheta is the wave velocity and f is the frequency of the signal.

In a circuit where the signal wavelength is comparable to the size of the elements, the lumped analysis of the voltage and current (Kirchhoff circuit laws) is not valid anymore and should be treated with wave equation. In
this case, the signal travels like a wave on a transmission line and could have reflection if not terminated with a matched impedance load. If a transmission line with characteristic impedance of Z_{o} is terminated with a load impedance of Z_{L}, then the reflection coefficient \Gamma_{L} is defined \frac{Z_{L}-Z_{o}}{Z_{L}+Z_{o}}.

Step Response Equation

We now can use the terminologies we defined earlier to derive the step response of a distributed network. Consider the circuit illustrated in the following figure.

Voltage of transmission line

Figure 1. Exciting a transmission line with a length of l. The signal travels with a velocity of \vartheta through the line. The reflection coefficients are \Gamma_{S} and \Gamma_{L} at the begining and end of the line respectively.

The source has an impedance of Z_{S} driving a transmission line with a length of l and characteristic impedance of Z_{o}. The transmission line is then terminated with Z_{L}\Gamma_{S} and \Gamma_{L} are the reflection coefficients at the source and load respectively. Assume at t=0 the source voltage V_{S} is launched. Then there is a voltage divide ratio between the source impedance, Z_{S} , and Z_{o} of the line. This defines the forward voltage travelling from source to the load at t=0. We call this V_{1}=V_{S}\frac{Z_{o}}{Z_{o}+Z_{S}} . It takes t_{0}=\frac{l}{\vartheta} for V_{1} to get to the other side of the line where it sees Z_{L}. The reflected wave is now V_{2}=\Gamma_{L}V_{1}. The same analysis is applied for the travelling signal at each end point of the transmission

To derive the equation of the voltage at each node one need to usethe superposition of each reflected waves. Let's start with V_{I}.Looking at Figure 1 it is shown this node will have transition at time 0,2t_{0},4t_{0},...,2nt_{0}. So:

\begin{align}
V_{I}(t)&=V_{1}u(t)+(V_{2}+V_{3})u(t-2t_{0})+(V_{4}+V_{5})u(t-4t_{0})\\ \nonumber
&=+...+(V_{2n}+V_{2n+1})u(t-2nt_{0})
\label{eq:VI}
\end{align}

Similarly, V_L is derived as:

\begin{align}
V_{L}(t)&=(V_{1}+V_{2})u(t-t_{0})+(V_{3}+V_{4})u(t-3t_{0})\\ \nonumber
&=+...+(V_{2n-1}+V_{2n})u(t-(2n-1)t_{0})
\label{eq:VL}
\end{align}

To complete the discussion in this section the dual circuit of Figure 1 is also analyzed. Figure 2 shows the Norton equivalent of V_S and Z_S. Since the reflection coefficient for current always has a negative sign, the sign of each reflected current is toggled every other time. This has been shown in Figure 2.

Transmission line current

Figure 2. Dual circuit of Figure 1. The reflection coefficient for current always keeps a negative sign.

Similar to equation 2 one can derive I_L(t) with the same exact format. For a purely resistive network, both \Gamma_S and \Gamma_L are real numbers. This makes ploting the step response quite easy. For a complex source and load impedance such as RC, RL and RLC networks, the reflection coefficients are not real anymore and further consideration is required to derive the step response.

Example: Purely Resistive Load

Consider a Z_S = 650 \Omega, a 2mm transmission line with Z_o = 90 \Omega that is terminated with a Z_L = 130 \Omega in silicon-dioxide medium. Plot I_I(t) and I_L(t).

  • \Gamma_S = \frac{650-90}{650+90} = 0.756
  • \Gamma_L = \frac{130-90}{130+90} = 0.181
  • I_1 = I_S\frac{650}{650+90} = 0.878I_S
  • t_0 = \frac{2x10^{-3}}{1.5x10^8} = 13.33ps

By plugging the above numbers into the equations in Figure 2 one can derive I_I(t) = 0.878u(t)-0.039u(t-26.6ps)-0.005u(t-53.33ps) and I_L(t) = 0.719u(t-13.33ps)+0.099u(t-39.99ps)+0.014u(t-66.66ps). Figure 3 shows the waveforms of I_I(t) and I_L(t).

Step response current

Figure 3. Step responses of I_I and I_L.

Reference:
[1]. Brian C. Wadell, Transmission Line Design Handbook, Artech House, 1991.
[2]. Stuart M. Wentworth, Applied Electromagnetics : Early Transmission Lines Approach, Wiley, 2007.

About the author

Nader KalantariNader Kalantari received the B.S. degree in electrical engineering from the Isfahan University of Technology, Isfahan, Iran, in 1997, the M.S. degree in computer engineering from Wright State University, Dayton, OH, in 2003, the M.S. degree in electrical engineering from the University of California at Irvine, in 2005, and the Ph.D degree in electrical engineering from the University of California at San Diego, in 2013.
He was an Embedded System Design Engineer with Signal Ltd., Tehran, Iran. From 2004 to 2012, he was an Analog/RFIC Design Engineer working involved with transimpedance amplifiers (TIAs), phase-locked loops (PLLs), dc–dc converters and clock and data recovery (CDRs) with Maxim-IC, Starport Systems, Microsemi and Mindspeed Technology respectively. He is currently a Sr. staff scientist at Broadcom Corp., Irvine CA, working on RFIC blocks for wireless applications.

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