Si~ulation & ~o Farid Najm, Editor

Simulink Simulation of Transmission Line

- Karl E. Lonngren and Er-Wei Bai

3- oftware that is dedicated to one par- ticular application such as control systems, circuit analysis, and magnetic field calcula- tions is usually confined to the field where it originated, and its global applications usu- ally remain unexplored. One of these pro- grams, but with the capability to expand, is MATLAB’ [I], a product of The Math- Works, Inc. MATLAB can manipulate and invert large matrices and can be used in many numerical applications. MATLAB’s capabilities can be extended with a recent addition called SimulinkTM, a program which is normally used in the analysis and synthesis of modern control systems.

Simulink, now incorporated into MAT- LAB, can also be used to analyze distributed transmission lines. Hence, it can be directly

used in the analysis and design of micro- wave, electric power, and VLSI networks. In addition to demonstrating its wider appli- cability, we include several examples that are normally described in books or papers directed to the study of pulse propagation along transmission lines [Z].

Transmission Line Mode! Consider the standard model of a transmis- sion line (Fig. 1). Both the voltages and the currents can be separately analyzed using Kirchhoff‘s laws and put in terms that can be analyzed using Simulink. Let’s analyze the model, writing all time-based variables in the transmission line in terms of the Laplace transform variable, s. The spatial variation of the transmission line will be

1. Distributed transmission lines. The signal generator V s has an internal impedance Zs. The load impedance is ZL. The units of the series inductances L, and shunt capacitances C, are Henrieshnit length, and faradshnit length, respectively. (a) Voltage response of the transmission line. (b) Cur- rent response of the transmission line.

incorporated into the discrete section numbe:.

For the simulation of the voltage re- sponse of the transmission line, the voltage Vi aross the capacitor in the first loop (which includes the voltage source in Fig. la) can be written in terms of the voltage source, Vs, and the voltage in the second loop V2 as

S L v , = 3 2 v, s L C+S2LCZS +S2L+ZS SL + zs + v2 s3L2C -t S2LCZS + s2L + z,

(1) where Z, is the source impedance. In the transmission line, the elements L and C are the inductance per unit length, and the ca- pacitance per unit length, respectively. The voltage across the capacitor in an intermedi- ate loop, n, can be wntten in terms of the similar voltage V,.i in the previous loop (n-l), and the voltage Vn+i in the following loop @+I).

A load impedance, ZL, is in parallel with the capacitor in the final loop, k. The load impedance can be linear or nonlinear.

We define the current in the load imped- ance at the end node, k , via the relation

ik = m , ) (3) where g( v) is an arbitrary nonlinear function that has to be specified by the sirgulator. In the linear case, g(V) is equal to a constant multiplied by V(Fig. 2). FromFig. 2, we find the voltage v k to be

10 Circuits & Devices

1 ‘k =E&-vk-’

(4) For purposes of simulating the current

response of the transmission line, the current ii in the first loop (which includes the volt- age source in Fig. 1 b) can be written in terms of the voltage source V, and the current in the second loop i2.

sc i,= v, s LC+sCZ,+1

1 + (5)

S2LC+SCZS + l z2

The current in an intermediate loop, n, can be written in terms of the current in the previous loop (n-I) and the following loop (n+l):

For the case of linear load impedance, i = VEL, the load impedance ZL is in parallel with the capacitor in the final loop, k. The current in this loop is written in terms of the current in the previous loop (k-1) as

(7) 1 + SCZ, i k =

s LC2Z, +S2LC+2SCZ, + I L k - ’

Equations 5-7 detennine the elements of a second transmission line.

In Fig, 2, the critical SimuEnk elements are shown for the elements specified with Eqs. 1-4. A dialog menu with Simulink al- lows all parameters of the polynomial to be specified. We specify the voltage source, Vs, as a half sine wave generator, which acts as a pulse generator in the simulation. The am- plitude and width were controllable parame- ters. In our application, fifteen identical intermediate elements were used. Although we will use only a Limited number of sec- tions in our transmission line model, it can be generalized to include as many as desired. In addition, the user can specify numerical values for the circuit elements L, C, Z,, and ZL. For clarity of presentation, we include a sequentially increasing “dc offset” to each section. Both linear [iEg(Vk) = constant* V k ] and nonlinear [ik=g(Vk)] load imped- ances are described with this model.

Applications In order to demonstrate the features of the Simulink transmission line simulator, we now analyze several well known examples. These include wave propagation and veloc- ity dependence upon the numerical values of the elements. In addition, we will discuss reflection at a linear terminating impedance. Dispersion of a pulse will be shown. Finally, we address reflection at a nonlinear termi- nating impedance.

2. Simulink simulation of the voltage response of the transmission line. The dotted lines indicate con- nections to the next section. The dashed lines indicate connections between a node and the oicillo- scope.

The characteristic impedance, Zc, of a transmission line an the velocity of propa- gation c of a wave‘ are respective1 given by

Initially, the impedances that terminate the line will be set equal to the characteristic impedance of the transmission line. This means that ZL = Z, and Z, = Z,, where Z, is a real number. Because the source imped- ance is matched to the characteristic imped- ance of the transmission line, the amplitude of the propagating wave will only be equal to one-half of the signal generator voltage Vs. The result of this “matching” of the transmission line implies that there will be no reflected wave. Figure 3a illustrates the propagating wave with L,= C = 0.2 and the wave being detected at each section. In this case 2, = 1. The propagation is readily ob- served. The simulation was repeated with the values L= C= 0.4 with all other elements remaining the same. As noted from Eq. 7 , only the velocity of propagation c should decrease by a factor of two and the charac- teristic impedance remains at Zc = 1. The results of this simulation are shown in Fig. 3b, where the decreased velocity is readily noted.

The effect of changing the value of the load impedance ZL is now examined in order to demonstrate that waves can propagate in both directions using this technique. We use the values L = C = 0.4 with ZL=O [short circuit] and Zt=lOO [open circuit]. The re- sults shown in Figs. 4a and 4b clearly show the expected behavior for the voltage waves. In addition, the reflection of the current waves using the circuit determined by Eqs. 5-7 for the same load impedances are shown in Figs. 4c and 4d.

The effect of joining two transmission lines of different characteristic impedances together is shown in Figure 5. For this ex- ample, we increased the number of sections to twenty in order to make the transmitted and the reflected signals readily apparent; The values L = C = 0.4 for j 513. The propagation for the values L = 0.4 and C = 0.1 [Zc=2] for j > 13 is shown in Fig. 5a. The propagation for the values L = 0.1 and C = 0.4 [Zc = 0.51 forj >13 is shown in Fig. 5b. The load impedance, ZL, differs in the two cases in order to match the lines f6r j > 13. Both the reflected and transmitted waves at the interface located at j = 13 can be fol-

12 Circuits & Devices

3. Sirnulation of the propagation of a voltage pulse. (a) L = C = 0.2. (b) L = C = 0.4.

4. Simulation of reflection of incident pulses from a load impedance. The transmission line has the values L = C = 0.4. (a) Voltage pulse, ZL = 0. (b) Voltage pulse, ZL = 100. (e) Current pulse, ZL = 0 (d) Current pulse, ZL = 100.

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5. Simulation of the reflection and transmission of an incident voltage pulse*om a transmission line of one impedance joined to another. The line is twenty sections long. (a) Z, [ j C l 3 ] = I and ZC[j>I3] = 2. (b) Zc [ j s I 3 ] = I and& &-I31 = ID.

6. Simulation of the dispersion of a narrow voltage pulse.

lowed. Hence the reflection and t r ansmis - sion coefficients can be computed. The ve- locities of propagation on either side of the transition at j = 13 are different as can be computed from Eq. 8.

In the data shown in Figs. 3 and 4, there is a hint of higher frequency oscillations that trail behind the main pulse. This suggests that dispersion may be present determined by wavelengths of the propagating wave being of the same size as the section size. In order to confirm the interpretation of disper- sion, a very narrow pulse was used since the

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7. Simulation of reflection of incidentpulses from a nonlinear load imped- ance. The transmission line has the values L = C = 0.2.

response of the transmission line that we are examining to such a stimulus can be written in terms of Airy functions [3]. High fre- quency oscillations trail behind a low fre- quency pulse in the Airy function. The simulation that confirms this interpretation is shown in Fig. 6.

As an example for a nonlinear load con- ductance, we chose

(9) i = O.Oltanh(V) The results of the Simulink simulation

are shown in Fig. 7. Other choices would, of course, yield different results. In addition,

the source impedance could also be made nonlinear.

Conclusion Using the Simulink opbon of MATLAB, we have demonstrated that transmssion lines can be easily simulated The lines can be made to be of any length, inhomogeneities can be easily studied, and different linear or nonlinear terminating elements can be ex- amined. Effects of coupling between this transmission line and other lines can also be incorporated into the simulation. We believe

Circuits & Devices

that it has many design, research, and edu- cational applications.

with the Department of Electrical and Com- and K E Lonngren, “A Transmission Line Simu- puter Engineenng, The University of Iowa lator for High Speed Interconnects,” IEEE Trans

actions on Circuits an Systems-II Analog and

References Digital Signal Processing, Vol 39, 1992, pp 1 e g , R H Bishop, Modern Control Systems 201-21

Analysis and Design using MATLAB, Addison- This work was supported in part by the Wesley, Reading, Mass (1993) Landt, et a1 , “Properties of Plasma Waves De- National Science Foundation Grant #ECS e , Liboff Dalman, fined by the Dispersion Relation ,” IEEE Trans- 90-06921 m l ~ ~ i ~ n lines, Waveguides, a& smith Charts, On P1asma Science, vO1 pS-2, 1974, PP

e g IC E Hsuan, Ackno wledgrnent

MacMillan, New York (1985), pp 25-66, S Chowdhury, J S Barkatullah, D Zhou,E W B ~ I ,

93-108

Karl E Lonngren and Er-Wei Bai are

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