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

In AC circuits, the current phasor leads the voltage phasor by 90 degrees for capacitors, meaning the voltage lags the current. This phase difference is crucial for understanding capacitor behavior. When the voltage across a capacitor is positive and increasing, its phasor points to the right and approaches the horizontal axis, while the current phasor is positioned 90 degrees ahead. This relationship is essential for analyzing circuits involving capacitors, emphasizing the importance of phase in alternating current (AC) systems.

Phasors for Capacitors

Hey, guys. In this video, we're going to talk about phasers for capacitors in AC circuits. So the current phaser and the functions that describe the voltage and the current across a capacitor at any time t in an AC circuit. These are given by these equations. Okay. Because the angle for each of them is different. Right? We have θ=ωt here and we have some other angle which I'll call θ'=ωt−π2 because the cosines each have a different angle we are saying that they are out of phase okay and in fact the voltage lags the current. This is very different than the phasers we saw for resistors which were in phase because both functions had the same angle ωt. Alright if we look at the 3 phasor diagrams that I drew, the first one, we have the current at an angle ωT, right? That is the angle of the current. In the second one, we have ωt−π2 as the angle for the voltage. That takes us all the way down into the negatives right because ωt itself as we can see is less than π2. So that takes us into the negatives and that also means that these have to have angles, right? This is just ωt which is the angle for the current minus degrees so they're separated by 90 degrees. So whenever you draw the phasor diagram which includes both the current and the voltage phases for a capacitor, okay, you have to draw it with the voltage lagging the current, okay, or the current leading the voltage. Okay? It's very important that you guys memorize that. The current leads the voltage or the voltage lags the current for a capacitor, alright? And that is the thing to take away from this. That the voltage across the capacitor always lags the current in the capacitor circuit. Okay? Let's do a quick example. An AC source is connected to a capacitor. At a particular instant in time, the voltage across the capacitor is positive and increasing in magnitude. Draw the phasors for the voltage and the current that correspond to this time. Okay? Now whenever a phaser is increasing in magnitude, it's because as it rotates it gets closer to a horizontal axis. Okay. As it gets closer and closer to that horizontal axis, its projection onto that axis gets larger and larger. And remember, the projection onto the horizontal axis tells us the value of that phaser. Okay. Now if this phaser is going to be positive it needs to be pointing to the right. And if it's going to be increasing in magnitude it it needs to be pointed to the right and moving towards the x-axis. Since phasers always rotate counterclockwise that means that the voltage phaser has to be here. Okay, and it's rotating like this. So it's positive because it points to the right and it's increasing in magnitude because as it gets closer to that horizontal axis more and more of it points horizontally until it's on the horizontal axis. Now it's at a maximum and then as it moves away from the horizontal axis it decreases and decreases in value until it's straight up and it's zero. Okay now if the voltage is here the current is 90 degrees ahead of it. So the current is going to be here. This is a current through a capacitor. This is the voltage of a capacitor. Okay? And this is a 90 degree angle. So this is what the phasor diagram looks like if the voltage is positive and increasing. Alright guys, that wraps up our discussion on phasers with capacitors. Thanks for watching.

An AC source operates at a maximum voltage of 60 V and is connected to a 0.7 mF capacitor. If the current across the capacitor is i(t) = i_MAX cos[(100 s⁻¹ )t], 

a) What is i_MAX? 

b) Draw the phasors for voltage across the capacitor and current in the circuit at t = 0.02 s. Assume that the current phasor begins at 0°.

• A. a) i_max = 4.2A b) θ_iC = 115^o θ_VC = 185^o
• B. a) i_max = 4.2A b) θ_iC = 185^o θ_VC = 115^o
• C. a) i_max = 4.2A b) θ_iC = 115^o θ_VC = 25^o
• D. a) i_max = 4.2A b) θ_iC = 25^o θ_VC = 115^o

Here’s what students ask on this topic:

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

In an AC circuit, the current phasor leads the voltage phasor by 90 degrees for a capacitor. This means that the voltage lags the current. Mathematically, if the current is represented as $I=I{0}_{cos}$, the voltage across the capacitor can be represented as $V=V{0}_{cos}$. This phase difference is crucial for understanding the behavior of capacitors in AC circuits.

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

To draw phasor diagrams for capacitors in AC circuits, start by representing the current phasor at an angle $\omega t$. The voltage phasor will be 90 degrees behind the current phasor, at an angle $\omega t-\pi /2$. If the voltage is positive and increasing, its phasor points to the right and approaches the horizontal axis. The current phasor, being 90 degrees ahead, will be positioned vertically above the voltage phasor. This relationship helps visualize the phase difference between voltage and current in capacitors.

Why does the voltage lag the current in a capacitor?

The voltage lags the current in a capacitor because of the nature of capacitive reactance. In a capacitor, the current is proportional to the rate of change of voltage. Mathematically, $I=C\delta V/\delta t$. This means that the current reaches its maximum value before the voltage does, resulting in the current leading the voltage by 90 degrees. This phase difference is a fundamental characteristic of capacitors in AC circuits.

What happens to the phasor diagram when the voltage across a capacitor is positive and increasing?

When the voltage across a capacitor is positive and increasing, its phasor points to the right and approaches the horizontal axis. As the phasor rotates counterclockwise, its projection onto the horizontal axis increases, indicating an increasing voltage. The current phasor, which leads the voltage by 90 degrees, will be positioned vertically above the voltage phasor. This configuration shows the phase relationship where the current is ahead of the voltage by 90 degrees.

How does the phase difference between voltage and current affect the analysis of AC circuits with capacitors?

The phase difference between voltage and current in capacitors is crucial for analyzing AC circuits. This 90-degree phase shift affects the impedance and power calculations. The impedance of a capacitor is given by $Z=-j/\omega C$, where $\omega$ is the angular frequency and $C$ is the capacitance. This phase difference also impacts the power factor, which is essential for determining the real and reactive power in the circuit. Understanding this relationship helps in designing and analyzing AC circuits involving capacitors.