# RLC Circuit Analysis: Phase Difference & Impedance Modulus

• B
• Titan97
In summary, the two-step process for finding the phase difference between voltage and current is to take the modulus of the imaginary part of the applied voltage, and then to take the phase difference between voltage and current.

#### Titan97

Gold Member
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)##?

Are you familiar with phasor notation?

Titan97 said:
Why are they considering a phase difference of ϕϕ\phi?
That phase difference is between voltage and current.

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.

Last edited:
Titan97 said:
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

## 1. What is an RLC circuit?

An RLC circuit is an electrical circuit that consists of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. These components are used to control the flow of electric current in a circuit.

## 2. What is phase difference in RLC circuit analysis?

Phase difference refers to the difference in phase between the voltage and current in an RLC circuit. It is measured in degrees and indicates how much the current and voltage are out of sync with each other.

## 3. How is the phase difference calculated in an RLC circuit?

The phase difference in an RLC circuit can be calculated using the formula tan θ = XC - XL / R, where XC is the capacitive reactance, XL is the inductive reactance, and R is the resistance. This formula is based on the trigonometric relationship between the three components.

## 4. What is impedance modulus in RLC circuit analysis?

Impedance modulus, also known as impedance magnitude, is a measure of the total opposition to current flow in an RLC circuit. It is represented by the absolute value of the impedance, which is the combination of the resistance, inductive reactance, and capacitive reactance.

## 5. How is impedance modulus used in RLC circuit analysis?

Impedance modulus is used to determine the overall behavior of an RLC circuit. It can help predict how the circuit will respond to different frequencies and can also be used to calculate the power and energy dissipated in the circuit.