The phasor representation of current and voltage in a pure inductive circuit shows that _______.

Explanation:

Phasor Representation in a Pure Inductive Circuit

Definition: In electrical engineering, a pure inductive circuit is one where the circuit contains only an inductor, and the resistance and capacitance are negligible or absent. The phasor representation of current and voltage in such a circuit is essential for analyzing the behavior of alternating current (AC) signals.

In AC circuits, the voltage and current are sinusoidal in nature, and their magnitudes and phase relationships can be represented using phasors. A phasor is a complex number that represents the amplitude and phase angle of sinusoidal signals.

Correct Option: Voltage leads current by 90°

In a pure inductive circuit, the voltage across the inductor leads the current passing through it by 90°. This phase difference occurs due to the fundamental nature of inductors and how they interact with changing currents.

Explanation:

When an alternating current passes through an inductor, the inductor opposes the change in current by inducing a voltage across itself, as described by Faraday’s Law of Electromagnetic Induction. The induced voltage is proportional to the rate of change of current:

V = L × (dI/dt)

Where:

• V = Voltage across the inductor
• L = Inductance of the inductor
• dI/dt = Rate of change of current

In a sinusoidal AC signal, the current waveform is represented as:

I = I_m × sin(ωt)

Where:

• I_m = Maximum current amplitude
• ω = Angular frequency
• t = Time

Taking the derivative of current with respect to time gives:

dI/dt = I_m × ω × cos(ωt)

From the above equation, the voltage across the inductor becomes:

V = L × I_m × ω × cos(ωt)

Since cos(ωt) can be expressed as sin(ωt + 90°), the voltage waveform leads the current waveform by 90°. This phase relationship is a defining characteristic of pure inductive circuits.

Phasor Representation:

Using phasors, the voltage and current can be represented as vectors in the complex plane:

• Current (I): A vector pointing along the reference axis (typically the positive real axis).
• Voltage (V): A vector leading the current by 90°, pointing along the positive imaginary axis.

Advantages of Phasor Representation:

• Simplifies the analysis of AC circuits by focusing on amplitude and phase relationships.
• Allows easy visualization of phase differences between voltage and current.

Applications:

• Used in the design and analysis of electrical circuits involving inductors.
• Helps in understanding the behavior of transformers, motors, and other inductive devices.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Current and voltage are in phase

This option is incorrect because, in a pure inductive circuit, the voltage and current are never in phase. The voltage always leads the current by 90° due to the inductive property of the circuit.

Option 3: Voltage leads current by 180°

This option is also incorrect. A phase difference of 180° would mean the voltage and current are completely out of phase, which occurs in certain resistive-capacitive or resistive-inductive circuits under specific conditions but not in a pure inductive circuit.

Option 4: Current leads voltage by 90°

This option is incorrect because, in a pure inductive circuit, the voltage leads the current by 90°, not the other way around. Current leading voltage by 90° is a characteristic of pure capacitive circuits.

Conclusion:

Understanding the phase relationship between voltage and current in various types of AC circuits is fundamental in electrical engineering. In a pure inductive circuit, the voltage leads the current by 90°, which is a direct consequence of the inductive property of the circuit. The phasor representation provides a valuable visualization tool for analyzing and designing such circuits effectively.