- #1
Andrusko
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Homework Statement
A transmission line has the following properties:
[tex]L_{0} = 1 mHm^{-1}[/tex]
[tex]C_{0} = 10 \mu F m^{-1}[/tex]
[tex]R_{0} = 10 \Omega m^{-1}[/tex]
[tex]G_{0} = 0 \Omega^{-1} m^{-1} [/tex]
That is, inductance per meter, capacitance per metre etc. The line is 10m long.
The problem is to find the angular frequency ([tex]\omega[/tex]) such that the propagation constant ([tex]\gamma[/tex]) and characteristic impedance ([tex]Z_{0}[/tex]) have phase angle [tex]\frac{-\pi}{6}[/tex]
Homework Equations
[tex]\gamma^{2} = (R_{0} + i\omega L_{0})(G_{0} + i\omega C_{0}) \cdots (1)[/tex]
[tex]Z_{0} = \sqrt{\frac{R_{0} + i\omega L_{0}}{G_{0} + i\omega C_{0}}} \cdots (2)[/tex]
and geometric representation of complex numbers will come into it.
The Attempt at a Solution
I worked out that the ratio of the imaginary part of the propagation constant and the real part is -sqrt(3). Ie;
[tex] \gamma = \alpha + \beta i[/tex]
[tex] \frac{\beta}{\alpha} = -\sqrt{3}[/tex]
Then went to find an equation for omega from eqn (1) above by substituting in known values and taking the square root of both sides. However, you end up with a pretty hideous equation with omega distributed through it everywhere and no way to solve for it, because to take the square root you must put the complex number in polar form. I ran into the same problem with solving for characteristic impedance.
It's obvious I'm approaching this from the wrong way, any suggestions?
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