Solving Complex Impedance Calculations: Step-by-Step Guide

[maddyfan811]

I have trouble figuring out how my textbook came up with the totals and am looking for step by step help. Here is what the text shows.

Z = R + j0 = R = 56 Ohm (in rectangular form [XL = 0])
Z = R < 0 degrees = 56 < 0 degrees Ohm (in polar form)

Z = 0 + jXL = j100 Ohm (in rectangular form [R = 0])
Z = XL < 90 degrees = 100 < 90 degrees Ohm (in polar form)

Z = R + jXL = 56 Ohm + j100 Ohm

Z = square root(R^2 + X^2L)<tan^-1(100 Ohm/56 Ohm) = 115<60.8 degrees Ohm

I think I figured out how to get the first number 115 but I'm having trouble on how the 60.8 degrees was determined. But a step by step explanation on how to get both numbers would be really helpful.

 

[CEL]

I have trouble figuring out how my textbook came up with the totals and am looking for step by step help. Here is what the text shows.

Z = R + j0 = R = 56 Ohm (in rectangular form [XL = 0])
Z = R < 0 degrees = 56 < 0 degrees Ohm (in polar form)

Z = 0 + jXL = j100 Ohm (in rectangular form [R = 0])
Z = XL < 90 degrees = 100 < 90 degrees Ohm (in polar form)

Z = R + jXL = 56 Ohm + j100 Ohm

Z = square root(R^2 + X^2L)<tan^-1(100 Ohm/56 Ohm) = 115<60.8 degrees Ohm

I think I figured out how to get the first number 115 but I'm having trouble on how the 60.8 degrees was determined. But a step by step explanation on how to get both numbers would be really helpful.

R and jXL form the orthogonal sides of a rectangle triangle. Z is the hypotenuse. It's modulus is $$\sqrt{R^2 + X_L^2}$$ and the phase is the angle between the hypotenuse and the side R: $$tan^{-1}\frac{X_L}{R}$$

 

[maddyfan811]

I think I figured out where I went wrong. I know I need to divide XL/R and then multiply it by tan^-1. Only problem is I don't know what tan^-1 is. What does tan^-1 equal?

 

[CEL]

I think I figured out where I went wrong. I know I need to divide XL/R and then multiply it by tan^-1. Only problem is I don't know what tan^-1 is. What does tan^-1 equal?

You don't have to multiply for anything. $$tan^{-1}$$ is the trigonometric function inverse of the tangent. It means the arc whose tangent is...

 

[maddyfan811]

I'm sorry I don't understand. Can you give a step by step example?

 

[CEL]

Have you ever studied trigonometry? Are you familiar with the functions sine, cosine and tangent?