If R is 85Ω and XL is +j90Ω, calculate total impedance, Z.

The total impedance, \( Z \), in a circuit is the combination of resistance (\( R \)) and reactance (\( X_L \)), which are combined in rectangular form as:

\[
Z = R + X_L
\]

Here:
- \( R = 85 \, \Omega \) (pure resistance, real component)
- \( X_L = +j90 \, \Omega \) (inductive reactance, imaginary component)

Thus:
\[
Z = 85 + j90 \, \Omega
\]

To express this in polar form (magnitude and phase angle):
1. **Calculate the magnitude**:
   \[
   |Z| = \sqrt{R^2 + X_L^2} = \sqrt{85^2 + 90^2} = \sqrt{7225 + 8100} = \sqrt{15325}
   \]
   \[
   |Z| \approx 123.84 \, \Omega
   \]

2. **Calculate the phase angle**:
   \[
   \theta = \arctan\left(\frac{X_L}{R}\right) = \arctan\left(\frac{90}{85}\right)
   \]
   \[
   \theta \approx 46.77^\circ
   \]

So, the total impedance in polar form is:
\[
Z \approx 123.84 \, \Omega \, \angle \, 46.77^\circ
\]

### Final Answer:
- In rectangular form: \( Z = 85 + j90 \, \Omega \)
- In polar form: \( Z \approx 123.84 \, \Omega \, \angle \, 46.77^\circ \)

---

Would you like additional details or clarifications?  
### Relative Questions:
1. How do you convert rectangular form to polar form for complex impedance?  
2. What is the physical significance of the phase angle in impedance?  
3. How does inductive reactance affect the impedance in an AC circuit?  
4. What would happen to \( Z \) if \( X_L \) were negative (capacitive reactance)?  
5. How can total impedance be used to calculate circuit current?  

**Tip:** Always double-check the units for resistance (\( \Omega \)), reactance (\( \Omega \)), and ensure they are added as complex numbers!

Learn how to calculate the total impedance (Z) in an AC circuit when R = 85Ω and XL = +j90Ω. This guide covers the calculation of magnitude and phase angle, along with the conversion of impedance from rectangular to polar form. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation