Phasors for Inductors: Study with Video Lessons, Practice Problems & Examples

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

Hey guys. In this video, we're going to talk about phasers and how they apply to the voltages and currents through inductors. Alright, let's get to it. Remember guys, that there are 2 functions that are very important regarding inductors in AC circuits. The current through an inductor and the voltage across an inductor at any time t. These functions are ωt, and the voltage occurs at some different angle θ′, which is ωt+π2. Because both functions occur at different angles, they are said to be out of phase. Okay? The current actually lags the voltage in this case, or you could say the voltage leads the current. What this means is readily apparent in phaser diagrams. In the first diagram, I plot the current at its angle ωt. In the second diagram, I plot the voltage at its angle ωt+π2, plus that 90 degrees. Combining these 2, we have the voltage ahead of the current by 90 degrees. Alright? It's very, very important to remember that the voltage across an inductor leads the current. This is opposite to capacitors in AC circuits, where the voltage lags the current. They're opposites. The voltage leads the current for inductors. The voltage lags the current for capacitors. Let's do a quick example about this. An AC source is connected to an inductor. At a particular instant in time, the current in the circuit is negative and increasing in magnitude. Draw the phasers for the voltage and the current that correspond to this instant in time. So here's my phaser diagram. Remember, what does it take for a phaser to be negative? It has to be on the left side of the graph because its horizontal component has to be negative. What does it take for it to be increasing in magnitude? It has to be moving towards the horizontal axis. Since it rotates counterclockwise, it has to be in the second quadrant. This has to be here so that it's negative, and since it's rotating counterclockwise, it's moving towards the horizontal axis. This is for the current. Remember that the current through an inductor lags the voltage. If you look all the way above in the green box, the voltage leads the current by 90 degrees. So, I would need to draw another phaser 90 degrees ahead, and that would be the voltage across the inductor. This is our phaser diagram for an inductor in an AC circuit. Alright, guys, this wraps up our discussion on phasors and how they pertain to inductors in AC circuits. Thanks for watching.

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

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.