# Taking e to a complex power (telecommunication formula)

1. Feb 28, 2012

### fran1942

Hello, I am have this telecommunications formula which calculates the magnitude of a wave on a transmission line:
VO = VI e^(-yi)

where:
VO = voltage out.
VI = voltage in (5)
y = propagation coefficient (0.745 + j1.279)
i = length of line (2500)

I am trying to use it to calculate the magnitude of a wave on a transmission line that is:
- 2.5km from the source
- whose input signal is 5V

My attempt is to write it out as below:
VO = 5*e^((-0.745 + j1.279)2500)

But I am unsure if I am have written this out correctly and if so, how to take 'e' to the power of a complex number.

Any help regarding how to finish this would be greatly appreciated.

2. Feb 28, 2012

### jbunniii

Here's the definition of the exponential of a complex number:

e^(a + bi) = (e^a)(cos b + i*sin b)

where a,b are the real and imaginary parts of the complex number.

3. Feb 28, 2012

### fran1942

thanks for that.
My attempt is below but I doubt I have implemented the "negative" y exponent correctly. Could someone please tell me how I have gone wrong.

Original formula:
VO = VI e^(-yi)
= VO = 5*e^((-0.745 + j1.279)2500)

My rewrite using:
e^(a + bi) = (e^a)(cos b + i*sin b)

is:

5 * e^(-0.745)(-cos 1.279 - sin 1.279)(2500)

= 0.4747(-1.022)(2500)

= -1212.94

4. Feb 28, 2012

### jbunniii

No, you dropped the "i" in the term i*sin b. Your result should be a complex number.

Also, isn't the 2500 part of the exponent?

5. Feb 28, 2012

### fran1942

Thanks for your patience. I have tried again. Is this an improvement ?

Original formula:
VO = VI e^(-yi)
= VO = 5*e^((-0.745 + j1.279)2500)

My rewrite using:
e^(a + bi) = (e^a)(cos b + i*sin b)

is:

5 * e^(-0.745*2500)(-cos 1.279 - isin 1.279)

= -1862.5(-.9998 - j.0223)

6. Feb 28, 2012

### jbunniii

No, this still isn't right. Multiply out the complex number first:

(-0.745 + j1.279)2500 = -1862.5 + j*3197.5

Then a = -1862.5, b = 3197.5