How Do You Calculate the Phase Angle in an LCR Circuit?

[Sleepy_time]

Homework Statement

A resistor (R), capacitor (C), and inductor (L) are connected in series. What is the complex impedance, Z,
of this LCR series combination?

An AC supply of voltage V(t)=V₀e^iωt is applied across an LCR series combination.
Derive an expression for the current I(t) in the circuit. From your expression for I(t)
write down expressions for both the magnitude of the current and the tangent of the
phase angle between current and applied voltage.

Homework Equations

Z_L=iωL

Z_C=-i/(ωC)

Z_R=R

I(t)=V(t)/Z.​

where; ω=Angular frequency of the voltage and i=√(-1).

The Attempt at a Solution

For the total impedance I got:

Z=R+i(ωL-\frac{1}{ωC}),​

and for the current:

I=(V₀e^iωt)/(R+i(ωL-\frac{1}{ωC})).​

For the tangent of the phase angle it would just be the tangent of the argument of the above equation for the current? If so, I can't get it into a "nice" equation without a trig function of another trig function. Thank you for any help.

 

[tiny-tim]

Welcome to PF!

Hi Sleepy_time! Welcome to PF!

For the tangent of the phase angle it would just be the tangent of the argument of the above equation for the current? If so, I can't get it into a "nice" equation without a trig function of another trig function.

tan of the phase angle of a + ib is simply b/a

for (a+ib)/(c+id), use the tan(θ-ψ) formula

 

[Sleepy_time]

Hi tiny-tim, thanks for the help. So what I got is shown in the attachment. Have I got it right?

 

[tiny-tim]

Hi Sleepy_time!

erm …

An AC supply of voltage V(t)=V₀e^iωt
…
the tangent of the phase angle between current and applied voltage.​

apart from that, fine!

(remember, impedance, complex current, etc are all constants! )