Phasors for Inductors: Videos & Practice Problems

In AC circuits, the voltage across an inductor leads the current by 90 degrees, meaning the current lags the voltage. This phase difference is crucial for understanding inductors, as opposed to capacitors where the voltage lags the current. For example, if the current is negative and increasing, its phasor is in the second quadrant, while the voltage phasor is positioned 90 degrees ahead. This relationship is essential for analyzing circuits involving alternating current (AC) and inductors.

Phasors for Inductors

In alternating current (AC) circuits, understanding the relationship between voltage and current through inductors is crucial. The current through an inductor and the voltage across it can be represented as functions of time, specifically as i(t) = I_0 \sin(\omega t) for current and v(t) = V_0 \sin(\omega t + \frac{\pi}{2}) for voltage. This indicates that the voltage leads the current by 90 degrees, meaning the current lags behind the voltage. This phase difference is a key characteristic of inductors in AC circuits.

To visualize this relationship, phasor diagrams are employed. In these diagrams, the current phasor is plotted at an angle of \omega t, while the voltage phasor is plotted at \omega t + \frac{\pi}{2}. The result is a clear representation of how the voltage is ahead of the current by 90 degrees. This contrasts with capacitors, where the voltage lags the current, highlighting the fundamental differences in behavior between these two components in AC circuits.

For example, if an AC source is connected to an inductor and at a specific moment the current is negative and increasing in magnitude, the corresponding phasor for the current would be positioned in the second quadrant of the diagram, indicating a negative value while moving towards the horizontal axis. Since the voltage leads the current by 90 degrees, the voltage phasor would be drawn 90 degrees ahead of the current phasor, accurately reflecting the phase relationship.

In summary, the understanding of phasors and their application to inductors in AC circuits is essential for analyzing and interpreting the behavior of electrical systems. The key takeaway is that in inductors, the voltage leads the current, which is a fundamental concept that differentiates them from capacitors.

An AC source operates at a maximum voltage of 75 V and is connected to a 0.4 H inductor. If the current across the inductor is i(t) = i_MAX cos[(450 s ⁻¹)t], 
a) What is i_MAX? 
b) Draw the phasors for voltage across the inductor and current in the circuit at t = 4.2 ms. Assume that the current phasor begins at 0°.

• A. a) i_max = 0.42A b) θ_iL = 108^o θ_VL = 198^o
• B. a) i_max = 0.42A b) θ_iL = 108^o θ_VL = 18^o
• C. a) i_max = 0.42A b) θ_iL = 198^o θ_VL = 108^o
• D. a) i_max = 0.42A b) θ_iL = 18^o θ_VL = 108^o

Answer: (A)

Here’s what students ask on this topic:

What is the phase relationship between voltage and current in an inductor in an AC circuit?

In an AC circuit, the voltage across an inductor leads the current by 90 degrees. This means that the current lags the voltage. Mathematically, if the voltage is represented as $V=V{m}_{sin}\left(\omega t\right)$, the current can be represented as $I=I{m}_{sin}(\omega t-\pi /2)$. This phase difference is crucial for understanding the behavior of inductors in AC circuits.

How do you draw phasor diagrams for inductors in AC circuits?

To draw phasor diagrams for inductors in AC circuits, follow these steps: First, represent the current phasor at an angle $\omega t$. Next, draw the voltage phasor 90 degrees ahead of the current phasor, at an angle $\omega t+\pi /2$. This shows that the voltage leads the current by 90 degrees. Ensure the phasors are drawn to scale and accurately represent the phase difference.

Why does the voltage lead the current in an inductor?

The voltage leads the current in an inductor due to the property of inductance, which opposes changes in current. When an AC voltage is applied, the inductor generates a back EMF (electromotive force) that resists the change in current. This causes the current to lag behind the voltage. Mathematically, this is represented by the phase difference of 90 degrees, where the voltage is ahead of the current.

What happens to the phasor diagram if the current is negative and increasing in magnitude?

If the current is negative and increasing in magnitude, its phasor is in the second quadrant of the phasor diagram. This is because a negative current corresponds to a phasor on the left side of the graph, and increasing magnitude means it is moving towards the horizontal axis. The voltage phasor, which leads the current by 90 degrees, will be positioned 90 degrees ahead of the current phasor.

How does the phase relationship in inductors differ from that in capacitors?

In inductors, the voltage leads the current by 90 degrees, meaning the current lags the voltage. In contrast, in capacitors, the voltage lags the current by 90 degrees. This means that in a capacitor, the current leads the voltage. These opposite phase relationships are crucial for analyzing AC circuits involving inductors and capacitors.