How do I determine the current amplitude and phase angle in an RLC AC circuit?

[kate12]

Voltage V(t)=V1 Re{eiwt} produces a current I(t)=I1 Re{e(iwt+θ). Determine I₁ and θ.

The circuit is AC so do I treat it the same as an SHM question? Is π/2? And what do I need to do to find I₁? I don't know where to start.

 

[ehild]

What is the circuit? You need its complex impedance to find I₁ and θ.

ehild

 

[kate12]

Ohh. The circuit is like this one:

with no values given.

 

[ehild]

What is the impedance in terms of R, L, and C?

ehild

 

[kate12]

z = R + iωL +1/iωC ?

What is the impedance in terms of R, L, and C?

ehild

 

[ehild]

OK, collecting the imaginary terms, it is Z=R+i(ωL -1/ωC ). This is a complex number. You can write it in exponential form:

Z=|Z|e^iψ (How?).

The complex voltage is V=V₁e^iωt

Use the equation I=V/Z to find the complex current.

ehild

 

[kate12]

OK, collecting the imaginary terms, it is Z=R+i(ωL -1/ωC ). This is a complex number. You can write it in exponential form:

Z=|Z|e^iωt (How?).

The complex voltage is V=V₁e^iωt

Use the equation I=V/Z to find the complex current.

ehild

I=V₀/|Z| ?

But that gives me I₀= V₀/|Z|e^i(ωt+θ).

 

[ehild]

I=V₀/|Z| ?

But that gives me I₀= V₀/|Z|e^i(ωt+θ).

You used I₁ and V₁ for the amplitudes in the original post.

I=V/Z . Z=|Z|e^iψ

That means

I₀e^i(ωt+θ)=V₀e^i(ωt)/[|Z|e^iψ]

Simplify with e^i(ωt).

I₀e^iθ=V₀/[|Z|e^iψ]=[V₀/|Z|]e^-iψ

The magnitudes and phases have to be the same on both sides.

ehild