Coil Parameters Overview

The question presents a scenario involving an iron cored coil, specifying the energy stored within its magnetic field and the power dissipated as copper loss in its windings. Our objective is to calculate the time constant of this specific coil. Understanding the relationship between energy storage, power dissipation, and the time constant is fundamental in analyzing inductive circuits.

Energy Stored in an Inductor Coil

An inductor, such as an iron cored coil, stores energy in the magnetic field generated when current flows through its windings. The amount of energy stored (\(E\)) is dependent on the coil's inductance (\(L\)) and the current (\(I\)) passing through it. The formula that describes this relationship is:

\[E = \frac{1}{2} L I^2\]

From the problem statement, the energy stored \(E\) is given as \(1000 \, \text{J}\) (Joules).

Copper Loss in an Iron Cored Coil

Copper loss (\(P_{Cu}\)) refers to the power dissipated as heat due to the resistance of the coil's windings when current flows. This power loss is a form of energy conversion, where electrical energy is transformed into thermal energy. It is also commonly known as \(I^2R\) loss. The formula for copper loss is:

\[P_{Cu} = I^2 R\]

Here, \(I\) represents the current flowing through the coil, and \(R\) represents the resistance of the coil's copper windings. The problem states that the copper loss \(P_{Cu}\) is \(2000 \, \text{W}\) (Watts).

Time Constant Definition for an RL Circuit

For a circuit containing both resistance and inductance (an RL circuit), the time constant (\(\tau\)) is a critical parameter. It quantifies the rate at which the current or voltage in the circuit changes when a sudden change occurs, such as connecting to a voltage source. The time constant for an RL circuit is defined as the ratio of its inductance to its resistance:

\[\tau = \frac{L}{R}\]

The time constant provides insight into the transient response of the coil.

Coil Time Constant Solution Steps

To determine the time constant (\(\tau\)) of the iron cored coil, we need to utilize the given energy stored and copper loss values to find expressions for \(L\) and \(R\) that can then be substituted into the time constant formula.

We have the following two fundamental equations derived from the problem's information:

• Energy stored: \(E = \frac{1}{2} L I^2\)
• Copper loss: \(P_{Cu} = I^2 R\)

From the energy stored equation, we can rearrange it to isolate the inductance (\(L\)):

\[2E = L I^2\] \[L = \frac{2E}{I^2}\]

Similarly, from the copper loss equation, we can express the resistance (\(R\)):

\[R = \frac{P_{Cu}}{I^2}\]

Now, substitute these derived expressions for \(L\) and \(R\) into the time constant formula \(\tau = \frac{L}{R}\):

\[\tau = \frac{\left(\frac{2E}{I^2}\right)}{\left(\frac{P_{Cu}}{I^2}\right)}\]

Observe that the term \(I^2\) appears in both the numerator and the denominator, allowing it to cancel out. This simplification leads to a direct relationship between the time constant, energy stored, and copper loss:

\[\tau = \frac{2E}{P_{Cu}}\]

Finally, we can insert the given numerical values into this simplified formula:

• Energy stored (\(E\)) = \(1000 \, \text{J}\)
• Copper loss (\(P_{Cu}\)) = \(2000 \, \text{W}\)

Performing the calculation:

\[\tau = \frac{2 \times 1000 \, \text{J}}{2000 \, \text{W}}\] \[\tau = \frac{2000}{2000} \, \text{s}\] \[\tau = 1.0 \, \text{s}\]

Thus, the time constant for the given iron cored coil is \(1.0 \, \text{s}\). This value indicates how quickly the current in the coil would build up or decay in response to a change in the applied voltage.