Question

3. A transmission line has two voltages on it, ν,(x,t)-10cos(2n100-25x) and y,(x,t)-.2cos(2n100 + 25x). Sketch v(x,t) +v,(x,t) as a function of x if a) Vm2-10 and t-0,10ms,20ms b) Vm2-5 and t-0,10ms,20ms Use the trig identity cos A+cos B=2cos|-(A+B)|cos|式4-B) | to establish the total voltage on the c) transmission line. What insight is gained from this result? 4. The input impedance of a transmission of length I, with characteristic impedance Zo, connected to a load Z, is given below 2π where /--, λ is the wavelength and l is expressed in wavelengths. a) Sketch Zn(l) vs l OAslso.5λ ifZe0, (i.e. a short circuit). what is the range of values for Zin? b) Repeat part a if Z - (i.e. an open circuit). c) How do the short circuited and open circuited transmission lines compare? d) what is the equation for Zin(l) if ZL 75Ω and Z0-500? Does Zn(l) repeat? If so, how often? e) What general conclusions can be drawn about Zm(1) for terminated transmission lines?

Answer 1

(3)(a) It is given that

$v_1(x,t)=10cos(2\pi 100t-25x)$

and

$V_{m2}=10$

So,

$v_2(x,t)=10cos(2\pi 100t+25x)$

And

v(r, t)-vi(r,t) + v2(r, t)

$=10cos(2\pi 100t-25x)+10cos(2\pi 100t+25x)$

2n 100t-25x + 2π 100t 252. = 20 cos( cos

As  

A+ B 2 cos ( cos A + cos B

So,

r(x, t) = 20 cos(2n 100t) cos(-252)

u(.r. t) 20 cos(2x 100) cos(25z)

For t=0

$v(x,t)=20\cos (0)\cos (25x)$

,(z, t) = 20 c os(25z) u

For t=10msec

u(z: t) 20 cosl 2 π ) cos(252.)

,(z, t) = 20 c os(25z) u

For t=20msec

$v(x,t)=20\cos (4\pi)\cos (25x)$

,(z, t) = 20 c os(25z) u

The plot of  
v(x, t
vs   is given beow

v(x,t) vsx 20 15 10 -5 -10 -15 -20 -1 0.8 -0.6 0.4 0.2 00.2 0.4 0.6 0.8 1

(b) It is given that

$V_{m2}=5$

So,

$v_2(x,t)=5cos(2\pi 100t+25x)$

And

v(r, t)-vi(r,t) + v2(r, t)

$=10cos(2\pi 100t-25x)+5cos(2\pi 100t+25x)$

$=5cos(2\pi 100t-25x)+\left \{ 5cos(2\pi 100t-25x)+5cos(2\pi 100t+25x) \right \}$

$=5cos(2\pi 100t-25x)+10cos(2\pi 100t)cos(25x)$

$=5\left \{ cos(2\pi 100t)\cos (25x)+\sin(2\pi 100t)\sin(25x) \right \} +10cos(2\pi 100t)cos(25x)$

As $\left \{\cos (A-B)=cosA\cos B+\sin A\sin B \right \}$

$v(x,t)=15cos(2\pi 100t)\cos (25x)+5\sin(2\pi 100t)\sin(25x)$

For t=0

$v(x,t)=15\cos (0)\cos (25x)+5\sin (0 )\sin (25x)$

$v(x,t)=15\cos (25x)$

For t=10msec

$v(x,t)=15\cos (2\pi)\cos (25x)+5\sin (2\pi )\sin (25x)$

$v(x,t)=15\cos (25x)$

For t=20msec

$v(x,t)=15\cos (4\pi)\cos (25x)+5\sin (4\pi )\sin (25x)$

$v(x,t)=15\cos (25x)$

The plot of  
v(x, t
vs   is given beow

15 10 F 0 -5 -10 -15 -1 0.8 -0.6 0.4 0.2 00.2 0.4 0.6 0.8 1

with the results obtained we can say that the variation of voltage in transmission line is not changing with respect to time.