Transmission Line Connected with a Nonlinear Load

                                                                                                                                                                     

Editors:  方紀穎  蕭名傑  蔡宜蒨

　

Advisor: 江簡富教授

　

　

 Load-line technique is a graphical technique frequently used to analyze the voltage and current along a transmission line connected with a nonlinear load.

            

               

　

Fig.1: Transmission line terminated by a passive nonlinear element and driven by a constant voltage source in series with a resistance.

　

Fig.1 shows a transmission line terminated by a passive nonlinear element having the I-V relationship of , and the source end is consisted of a resistor of 200Ω and a dc voltage of 50V. The characteristic impedance of the transmission line is Z0=50Ω, and the transverse time along the line is T=1μs. The switch S is closed at t=0, and the load-line technique is used to obtain the time variation of the voltages VS and VL at the source and at the load ends, respectively.

 

The Kirchhoff viltage law at z=0 implies

At t=0+,

            V+ and I+ are the voltage and current, respectively, of the (+) wave propagating in +z direction immediately after closure of the switch. We can solve VS and IS graphically by drawing two straight lines to represent (1) and (2) as shown in Fig.2. The point of intersection is denoted by A, which gives the values of VS and IS.

 

            When the (+) wave reaches the load end z=l at t=T, a (-) wave is induced, the voltage and current at t=T+ at z=l can be expressed a

                    

            where V- and I- are the (-) wave voltage and current, respectively. The solution of VL and IL is given by the intersection of the curve representing (3) and the straight line representing (4), the intersection is denoted by B.

 

            When the (-) wave reaches the source end z=0 at t=2T, another reflection is induced, denoted as (-+) wave.         

            The voltage and current at t=2T+ and z=0 satisfies the same KVL:

　

           

          

            The voltage and current are consisted of 

　

           

          

            where V-+ and I-+ are the (-+) wave voltage and current, respectively. Note that VS=V++V- and IS=(V+-V-)/50 is a solution to (5), implying that the straight line representing (6) passes through B. Thus, the solution of (5) and (6) is given by point C in Fig.2. 

　

           

Fig.2: Graphical representation of voltage and current waves.

 

            Continuing this process, we observe that the solutions at t = nT+ are the points of intersection of either the source or the load V-I curve with the straight lines of slope 1/Z0 or -1/Z0, respectively. The first straight line launches at the origin and the following straight line originates at the previous point of intersection. To be more specific, solutions of VS and IS are obtained by drawing the intersection of the straight lines of slope 1/Z0 and (1), whereas that of VL and IL are obtained by drawing the intersection of the straight lines of slope -1/ Z0 and (3), originating from A, and so on.

 

            Once we have the time variations of VS, IS, VL, and IL, the voltage and current at an arbitrary point on the transmission line at time t can be obtained. Consider the following example. When t=1.5T at z=0.6L, the voltage and current should be equal to that at the load end when t=1T+. On the other hand, at the same time, at z=0.4L, the (-) wave has not yet traveled to this point, hence the voltage and current is equal to that at the source end when t=0+.