Answer:
Reflection Coefficient = [tex]0.57e^{-i79.8}[/tex]
SWR=3.65
Position of [tex]V_{max} =3.11cm[/tex]
position of [tex]i_{max} =1.11cm[/tex]
Explanation:
To determine the above answers, let outline the useful formulas
refection coefficient [tex]p=\frac{terminalimpednce -characteristics impedance }{terminalimpednce +characteristics impedance } \\[/tex].
where terminal impednce = (30-i50)Ω
characteristics impedance= 50Ω
Secondly, the Standing Wave Ratio,[tex]SWR=\frac{1+/p/}{1-/p/}[/tex]
Now let us substitute values and solve,
a. [tex]p=\frac{terminalimpednce -characteristics impedance }{terminalimpednce +characteristics impedance } \\[/tex]
[tex]p=\frac{(30-i50)-50}{(30-i50)-50} \\[/tex]
[tex]p=\frac{-20-i50}{80-i50} \\[/tex]
multiplying the numerator and denominator by the conjugate of the denominator. we have
[tex]p=\frac{-20-i50}{80-i50}*\frac{80+i50}{80+i50}\\[/tex]
by carrying out careful operation, we arrived at
[tex]p=\frac{900-i5000}{8900} \\p=0.1011-i0.56179\\[/tex]
To express in polar form i.e [tex]re^{i alpha}[/tex]
[tex]r=\sqrt{0.1011^{2}+0.56179^{2}} \\r=0.57\\[/tex]
to get the angle
alpha=[tex]tan^{-1} \frac{0.56179}{0.1011} \\alpha=-79.8\\[/tex]
hence the Reflection Coefficient,p = [tex]0.57e^{-i79.8}[/tex]
b. we now determine the Standing Wave Ratio,[tex]SWR=\frac{1+/p/}{1-/p/}[/tex]
[tex]swr=\frac{1+0.57}{1-0.57} =3.65\\[/tex]
c. to determine the position of the maximum voltage nearest to the load,
we use the equation
[tex]Position of V_{max}=\frac{\alpha λ}{4\pi}+\frac{λ}{2}\\[/tex]
were [tex]λ[/tex] is the wavelength of 8cm
lets convert α to rad by multiplying by π/180
[tex]Position of V_{max}=\frac{-79.8 *8cm*\pi}{4\pi*180 } +\frac{8cm}{2} \\[/tex]
[tex]Position of V_{max}=-0.89+4.0=3.11cm\\[/tex].
d. also were we have minimum voltage,there the maximum current will exist, to find this position nearest to the load
[tex]Position of v_{min}=Position of V_{max}-\frac{λ}{4} \\\\Position of v_{min}=3.11cm-\frac{8cm}{4}=1.11cm[/tex].
since the voltage minimum occure at 1.11cm. we can conclude that the current maximum also occur at this point i.e 1.11cm
