Calculating Impedance, verify.

[AntiStrange]

Homework Statement

Calculate the circuit's impedance Z between input and ground for an applied sinusoidal voltage with frequency f = 1000Hz, and express Z in the form $re^{i\theta}$

There is a picture that goes with it, it's just a function generator (makes a sinusoidal AC voltage) connected in series with a 0.1μF capacitor, and a 1kΩ resistor.

I think I solved it, but I would just like it if someone could verify that I did it correctly as it is going into a lab report.

Homework Equations

i.$$Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}$$
where X_L is for a transistor (none exists in the circuit so it's zero) and X_C is for the capacitor

ii. $$X_{C} = \frac{1}{\omega C}$$

iii.$$tan(\phi) = \frac{\omega L - 1/\omega C}{R}$$

All equations extracted from "University Physics by Young & Freedman 11th edition

The Attempt at a Solution

From Eq i, I just plugged in 1kΩ for the resistor, and X_L is zero, and I found X_C from Eq ii (where C is the 0.1μF capacitor and ω is 2π*1000Hz).
This gave me a value for Z of 1879.6Ω.

Then, to find it in the form $re^{i\theta}$, more specifically it should be:
$$re^{i(\omega t + \phi)}$$
I know everything but φ, so I used Eq iii.
Again, L was zero so the first term in the numerator canceled out, and I found φ to be:
- 1.010 (radians).

So my final answer is:

$$1879.6 e^{i(2000\pi t - 1.01)} \Omega$$

Is that all correct?
thanks.

 

[anathema]

Your calculations seem to be correct. The impedance Z is indeed 1879.6Ω and can be expressed in the form re^{i\theta} as you have shown. Great job!

 

[tomcue]

Your solution looks correct. Just make sure to double check your calculations and units. Also, it might be helpful to include a brief explanation of your steps and equations used in your lab report. Overall, great job on solving the problem!