Solving Complex Impedance Calculations: Step-by-Step Guide

In summary, the conversation discusses finding the total value of Z in different forms and understanding how the numbers were determined. The steps involve finding the modulus and phase of Z using the rectangular and polar forms, and using the inverse tangent function to find the angle. The conversation also mentions needing a step by step explanation and not understanding the concept of tangent.
  • #1
maddyfan811
5
0
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.
 
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  • #2
maddyfan811 said:
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 [tex]\sqrt{R^2 + X_L^2}[/tex] and the phase is the angle between the hypotenuse and the side R: [tex]tan^{-1}\frac{X_L}{R}[/tex]
 
  • #3
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?
 
  • #4
maddyfan811 said:
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. [tex]tan^{-1}[/tex] is the trigonometric function inverse of the tangent. It means the arc whose tangent is...
 
  • #5
I'm sorry I don't understand. Can you give a step by step example?
 
  • #6
Have you ever studied trigonometry? Are you familiar with the functions sine, cosine and tangent?
 

1. What is impedance?

Impedance is a measure of the opposition to the flow of electric current in a circuit. It is represented by the symbol Z and is measured in ohms.

2. Why are complex impedance calculations necessary?

Complex impedance calculations are necessary because they take into account both the resistance and reactance of a circuit, which allows for a more accurate representation of the behavior of the circuit.

3. What is the formula for calculating complex impedance?

The formula for calculating complex impedance is Z = R + jX, where R is the resistance and X is the reactance. The j represents the imaginary unit, which is used to represent the reactance.

4. How do I solve complex impedance calculations step-by-step?

To solve complex impedance calculations, follow these steps:

  1. Separate the circuit into its individual components (resistors, capacitors, and inductors).
  2. Calculate the individual impedances for each component using the appropriate formulas.
  3. Combine the individual impedances using the rules for series and parallel circuits.
  4. Convert the final impedance back into polar form (Z = |Z|∠θ).

5. What are some common mistakes to avoid when solving complex impedance calculations?

Some common mistakes to avoid when solving complex impedance calculations include forgetting to convert between polar and rectangular forms, using incorrect formulas for specific components, and miscalculating the total impedance by not considering the phase differences between components.

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