refelction coefficient calculation

[fran1942]

Hello, I am looking at the following calculation for the reflection coefficient of a transmission line using the formula ((ZL - ZO) / (ZL +ZO)
I can understand it all apart from the last part (0.355e^j).
Can someone please tell me how that figure was gained from (-0.354 + 0.030j)

(216 + j15 - 452.38) / (216 + j15 + 452.38)

= (236.38 + j15) / (668.38 + j15)

= -0.354 + 0.030j

= 0.355e^j

Thanks kindly for any help.

 

[elchiquito]

hello

-0.3530 + 0.0304i which can also be written in this form mag*exp(j*theta)

here magnitude will be .355 theta =3.0557 which by any logic cant be rounded off to 1. i strongly believe there might have been misunderstanding or a typo.

hope that helps

Regards
Elchiquito

 

[enjunear]

Agree w/ Elchiquito... the answer is missing something.

-0.353 + j0.030 is in rectangular format. A quick punch of the calculator to polar mode gives it as 0.354 @ 175.1 degrees. This math is a little rusty, but IIRC, in \[{e}^{j\theta}\] form, the angle is shown in radians, so 175.1*pi/180 = 3.06 radians. The final answer should be \[0.354{e}^{j3.056}\], if that's the final format you're aiming for. I tend to find looking at reflection coefficients as mag/angle(degrees) is the most insightful, since you can quickly envision where the point appears on the Smith Chart.

Related awesome mathematical identity: Euler's Identity

 

[albbg]

Sorry, but starting from -0.3530 + 0.0304i = mag*exp(j*theta) the math gives to me:

mag = -sqrt(0.3530² + 0.0304²) = -0.3543
theta = arctg[0.0304/(-0.3530)] = -0.0859 rad

 

[elchiquito]

Albbg

theta=-.0859(rad) refers to the same angle as 3.056(rad) depends how you measure the angle? 3.056+.0859 is Pi radians.

which still doesn't solve the problem, reason for the post. like me and enjunear has mentioned probably a mistake or a typo

Regards
Elchiquito

 

[enjunear]

Sorry, but starting from -0.3530 + 0.0304i = mag*exp(j*theta) the math gives to me:

mag = -sqrt(0.3530² + 0.0304²) = -0.3543
theta = arctg[0.0304/(-0.3530)] = -0.0859 rad

I have one issue with your math, but its minor and it makes our answers identical.
Issue: A magnitude should never be negative value... magnitude is synonymous with absolute value in this case.

Now, if you flip the sign on your magnitude, your vector's angle changes by 180 degrees (pi radians). So your calculated angle of -0.0859 rad becomes -0.0859 + 3.14159 = 3.056 radians.

So, magnitude = 0.3543, angle = 3.056 radians... so, our answers agree.

Also, your theta equation was incorrect by pi; you need to measure theta starting from the postive real axis. Since the vector is in the upper-left quadrant (-R, +X), your equation for theta should be

.

 

[albbg]

elchiquito, enjunear,

you are right; it's my mistake.