Calculating Impedance of a RLC Circuit Connected to a 60 Hz AC Source

[ayajek]

Homework Statement

A series RLC circuit is connected to a 60 Hz AC source which produces an amplitude of εmax=20 V. The circuit element values are R= 20 Ω, L= 20 mH, C= 150 μF.

Calculate the total impedance of the circuit.

Homework Equations

Xc=1/ωC
XL=ωL
Z=√(R^2 +(XL-Xc)^2)
tan φ = (XL-Xc)/R
ω=1/√LC
εmax=Imax*Z

The Attempt at a Solution

I started out with Z=√(R^2 +(XL-Xc)^2) , and to get XL and Xc I used ω=2*pi*f .
I got my XL and Xc marked wrong. It wasn't a calculation mistake, so am I approaching the answer incorrectly?

 

[Hesch]

Calculate the total impedance of the circuit.

am I approaching the answer incorrectly?

It seems that your approach is correct, though I don't like your sign conventions. Example:

tan φ = (XL-Xc)/R.

I'd rather write:

tan φ = (XL+Xc)/R , but the value of Xc is negative.

I don't know if you are familiar with complex calculations where Z could be calculated as

Z = R + jωL + 1/( jωC ) = R + jωL + ( -j / ( ωC ) ).

 

[William White]

Inductive reactance

Xl = 2pi f L

Capacitive reactance

Xc = 1 / (2pi f c)

If you can use complex numbers (makes this much simpler)

z = R + j(Xl - Xc)
So, for example if
Xl = 12 ohms
Xc = 17 ohms
R = 25 ohms

you would have

z = 25 - j5 ohms

(because 12-17 = -5)and you can happily use ohms law to get the current

i = v / z