Binomial Expansion: Calculating Constants \alpha and \beta

[mezarashi]

Hello, another dull question on binomial expansion (approximation). I cannot follow the derivation for the approximate values of the two constants $$\alpha$$ and $$\beta$$.

(Text on propagation coefficient of TEM waves in transmission lines - constants of attenuation and phase-shift)

Given
$$\gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)}$$

Through "binomial expansion", taking the expansion to the third term.

$$\alpha \approx \frac{1}{2} (R\sqrt{\frac{C}{L}} + G\sqrt{\frac{L}{C}})$$

$$\beta \approx \omega\sqrt{LC}(1 + \frac{1}{8\omega^2}(\frac{R}{L} - \frac{G}{C})^2)$$

I know this is a messy one, so just a clue on what this is about would be great =D

 

[dextercioby]

If i were you, i'd square both members of the equality and solve it algebraically.

Daniel.

 

[mezarashi]

If i were you, i'd square both members of the equality and solve it algebraically.
Daniel.

Yes, I've tried that on one occasion. It works if you want to express either alpha or beta in terms of the other. If you know either, then the problem becomes quite easily solvable. Apparently you aren't able to separate the variables. For example, from squaring both sides and equivalating the real portion, you would get.

$$\alpha = \sqrt{\frac{\alpha^2 + \beta^2 + (RG - \omega^2 LC)}{2}}$$

 

[mezarashi]

Update: I've been able to solve for alpha and beta simultaneously using algebra. I'll be verifying them with the binomial approximation. In anycase, I'd still appreciate information on the approximation method. Thanks.