Write down equation of induced current at any instant in L-C-R- circuit when ( XL >XC) or circuit is inductive.

In an L-C-R circuit where the inductive reactance (XL) is greater than the capacitive reactance (XC), the circuit behaves inductively. This condition affects how we calculate the induced current at any instant. The equation for the induced current can be derived from the principles of alternating current (AC) circuits, particularly using Kirchhoff's voltage law and the concept of impedance.

Understanding the Circuit Components

In an L-C-R circuit, we have three main components:

• L: Inductance (measured in henries, H)
• C: Capacitance (measured in farads, F)
• R: Resistance (measured in ohms, Ω)

When the circuit is inductive (XL > XC), the total impedance (Z) of the circuit can be expressed as:

Calculating Impedance

The impedance in an L-C-R circuit is given by the formula:

Z = √(R² + (XL - XC)²)

Where:

• XL = ωL (inductive reactance)
• XC = 1/(ωC) (capacitive reactance)
• ω = 2πf (angular frequency, where f is the frequency of the AC source)

Induced Current Formula

The induced current (I) at any instant can be expressed using Ohm's law, which relates voltage (V), current (I), and impedance (Z):

I = V/Z

In an AC circuit, the voltage can be represented as:

V(t) = V₀ sin(ωt)

Where V₀ is the maximum voltage. Substituting this into the current equation gives:

Final Expression for Induced Current

Thus, the induced current at any instant can be expressed as:

I(t) = (V₀/Z) sin(ωt)

Since we know Z, we can substitute it into the equation:

I(t) = (V₀/√(R² + (XL - XC)²)) sin(ωt)

Example Calculation

Let’s say we have the following values:

• V₀ = 10 V
• R = 5 Ω
• L = 0.1 H
• C = 100 μF
• f = 50 Hz

First, calculate XL and XC:

• ω = 2π(50) ≈ 314.16 rad/s
• XL = ωL = 314.16 * 0.1 ≈ 31.42 Ω
• XC = 1/(ωC) = 1/(314.16 * 100 * 10^-6) ≈ 31.83 Ω

Now, since XL > XC, we can find Z:

Z = √(5² + (31.42 - 31.83)²) ≈ √(25 + 0.16) ≈ 5.00 Ω

Finally, substituting into the current equation:

I(t) = (10/5) sin(314.16t) = 2 sin(314.16t) A

This means the induced current oscillates with a maximum value of 2 A, following the sinusoidal pattern dictated by the AC source. Understanding these relationships helps in analyzing the behavior of inductive circuits effectively.