Antenna Resistance: Complex Implications in Real Life

[Seanskahn]

Hello community

Sorry if this is a repost, but I don't find what i am looking for.

I am reading the book "anntenas", by Kraus and Marhefka

In page 30, says the antenna impedances are complex. Z_{$$\tau$$} = +
R_{$$\tau$$} + jX_{$$\tau$$} ..

Could anyone please tell me what does the complex part imply in real life, and why is it regarded as complex.
Thank you

edit: the tau is supposed to be subscripted, but it is appearing as superscript :s

 

[K^2]

Impedance is used with complex notation for oscillation current. Suppose you have current oscillation at frequency ω and amplitude I₀. Then the current is given by:

$$I = Re(I_0 e^{i \omega t}) = I_0 cos(\omega t)$$

The complex voltage is related to complex current using analog of Ohm's law.

$$V = Re(Z I_0 e^{i \omega t})$$

If Z is real, this is no different from Ohm's Law. But let's substitute your complex form in.

$$V = Re((R + iX) I_0 e^{i \omega t})$$

$$V = I_0 Re(R e^{i \omega t} + X e^{i (\omega t + \frac{\pi}{2})})$$

$$V = I_0 R cos(\omega t) - I_0 X sin(\omega t)$$

The first term is still just IR, but second term is phase-shifted by 90°. This expression can also be re-written using a single cosine function.

$$V = I_0 \sqrt{R^2 + X^2} cos(\omega t + \tan^{-1}(\frac{X}{R}))$$

In other words, the norm of Z gives you the ratio of peak voltage to peak current, and so plays a role of effective resistance, while the angle of Z in complex plane gives you the phase shift between voltage and current.

 

[Seanskahn]

hi K²

Thanks you for the reply

then let me ask the next obvious question.
why don't we then just use any 2 dimensional plane for the Z vector?

e^theta is still sin(theta) + cos(theta), albeit the sin and cos part is summed, and not distinguishable

 

[K^2]

Because then you can't write V = ZI. Multiplication is not defined in vector spaces, but is defined in complex space.

It's just a convenience, though. You can solve any circuit using ordinary voltage and current by breaking up each into sin and cos components. It just gets a little messy. Once you know a few tricks, working with impedances is very fast.

 

[Seanskahn]

hi K²

thank you very much.