Phasors for Capacitors: Videos & Practice Problems

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

In alternating current (AC) circuits, understanding the behavior of capacitors is crucial, particularly how voltage and current relate to each other through phasors. In these circuits, the current phasor is represented by the angle \( \theta = \omega t \), while the voltage phasor is described by \( \theta' = \omega t - \frac{\pi}{2} \). This indicates that the voltage lags the current by 90 degrees, a key distinction from resistors where both voltage and current are in phase.

When visualizing these relationships through phasor diagrams, the current phasor is positioned at \( \omega t \), and the voltage phasor, lagging behind, is at \( \omega t - \frac{\pi}{2} \). This separation of angles illustrates that the current leads the voltage in a capacitor circuit. It is essential to remember that for capacitors, the voltage always lags the current.

To illustrate this concept, consider an AC source connected to a capacitor. If at a specific moment the voltage across the capacitor is positive and increasing, the phasor for voltage will be directed to the right and moving closer to the horizontal axis, indicating an increase in magnitude. As phasors rotate counterclockwise, the current phasor will be positioned 90 degrees ahead of the voltage phasor. This configuration visually represents the relationship between voltage and current in a capacitor, reinforcing the understanding that the current leads the voltage.

In summary, the critical takeaway is that in a capacitor circuit, the voltage lags the current by 90 degrees, which is fundamental for analyzing AC circuits involving capacitors.

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

Answer: (C)

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.