RLC Circuit Analysis: Phase Difference & Impedance Modulus

[Titan97]

In an RLC series circuit let applied EMF be given $V=V_0\sin\omega t$, $$Z=Z_C+Z_R+Z_L=R+i\left(\frac{1}{\omega C}-\omega L\right)$$
$$|Z|=\sqrt{R^2+\left(\frac{1}{\omega C}-\omega L\right)^2}$$

Then $$i(t)=\frac{V(t)}{Z}=\frac{V_0e^{i\omega t}}{R+i\left(\frac{1}{\omega C}-\omega L\right)}$$

Its given in my book that
$$i(t)=\frac{V_0(\sin\omega t+\phi)}{\sqrt{R^2+\left(\frac{1}{\omega C}-\omega L\right)^2}}$$

Why are they considering a phase difference of $\phi$?

Also, why are they taking modulus of $Z$ and only the imaginary part of applied voltage?

What is the difference between the first $i(t)$ and the second $i(t)$?

 

[f95toli]

Are you familiar with phasor notation?

 

[cnh1995]

Why are they considering a phase difference of ϕϕ\phi?

That phase difference is between voltage and current.

 

[Titan97]

I get it. I can write $$R+i\left(\omega L-\frac{1}{\omega C}\right)=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}e^{i\phi}$$
Hence,
$$i(t)=\frac{V(t)}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}e^{i\phi}}$$
$$i(t)=\frac{V(t)e^{-i\phi}}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}$$

Now it's in phasor notation.

 

[SammyS]

I get it. I can write $$R+i\left(\omega L-\frac{1}{\omega C}\right)=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}e^{i\phi}$$ Hence,$$
i(t)=\frac{V(t)}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}e^{i\phi}}$$$$
i(t)=\frac{V(t)e^{-i\phi}}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}$$ Now it's in phasor notation.

There's $\ i\$ and then there's $\ i\$ .

You probably should write $\ i(t)\$ for the current

 

[Titan97]

@SammyS edited