Phasors for Resistors: Videos & Practice Problems

In AC circuits, the voltage and current across a resistor are in phase, meaning they reach their maximum and minimum values simultaneously. This is represented by the same angle, ωt, in their phasor diagrams. For example, with an angular frequency of 20 rad/s, after 0.2 seconds, both voltage and current phasors would be at 4 radians (or 229 degrees), indicating they are aligned. Understanding this relationship is crucial for analyzing resistive circuits in alternating current systems.

Phasors for Resistors

In alternating current (AC) circuits, understanding the relationship between voltage and current across resistors is crucial. The voltage and current can be represented as sinusoidal functions, both described by the equation:

V(t) = V_max cos(ωt) and I(t) = I_max cos(ωt)

Here, ω represents the angular frequency, and t is time. Since both functions share the same angle, ωt, they are said to be in phase. This means that the voltage and current reach their maximum and minimum values simultaneously, resulting in a consistent phase relationship.

When visualizing this relationship using phasors, the current and voltage phasors will align at the same angle, indicating that they are in phase. This alignment is a key characteristic of resistors in AC circuits, where the voltage across the resistor is always in phase with the current flowing through it.

To illustrate this concept, consider an AC source with an angular frequency of 20 radians per second connected to a resistor. If the circuit is closed at time t = 0.2 seconds, the angle for the phasors can be calculated as:

θ = ωt = 20 rad/s × 0.2 s = 4 radians

This angle can also be converted to degrees:

θ = 4 rad × (180°/π) ≈ 229°

At 229 degrees, the phasors for both the current and voltage will be positioned in the third quadrant of the unit circle, demonstrating their in-phase relationship. This means that regardless of the specific angle chosen, as long as both phasors are at 229 degrees, they will always align, reinforcing the concept that voltage and current through a resistor in an AC circuit are consistently in phase.

In summary, recognizing that voltage and current are in phase in resistive AC circuits is fundamental for analyzing and understanding circuit behavior.

A 12 Ω resistor is connected to an AC source. If the resistor's voltage phasor is initially at 0° , and the figure below shows the phasor after 0.04 s, answer the following: 
a) What is the angular frequency of the source? Assume the phasor is on its first rotation. 
b) What does the current phasor diagram look like? 
c) What is the current in the circuit at this point (t = 0.04 ?)?

• A. a) ω = 0.029 s^-1 b) θ = 132^o c) -0.31A
• B. a) ω = 0.029 s^-1 b) θ = 132^o c) 0.31A
• C. a) ω = 18.3 s^-1 b) θ = 42^o c) 0.31A
• D. a) ω = 18.3 s^-1 b) θ = 42^o c) 0.42A

Answer: (C)

Here’s what students ask on this topic:

What does it mean for voltage and current to be in phase in a resistor?

In an AC circuit, when voltage and current are said to be in phase in a resistor, it means that they reach their maximum and minimum values simultaneously. This is represented by the same angle, ωt, in their phasor diagrams. Essentially, the voltage and current waveforms align perfectly, with no phase difference between them. This characteristic is unique to purely resistive circuits and is crucial for analyzing and understanding the behavior of resistors in AC systems.

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

To draw phasor diagrams for resistors in AC circuits, follow these steps: First, determine the angular frequency (ω) and the time (t). Calculate the angle θ using θ = ωt. For example, if ω = 20 rad/s and t = 0.2 s, then θ = 4 radians (or 229 degrees). Next, draw the phasor for the current at this angle. Since voltage and current are in phase in a resistor, draw the voltage phasor at the same angle. Both phasors will align, indicating they are in phase.

Why is it important to understand phasors in AC circuits?

Understanding phasors in AC circuits is crucial because they simplify the analysis of AC signals. Phasors convert sinusoidal functions into rotating vectors, making it easier to visualize and calculate the relationships between voltage, current, and impedance. This is especially important in complex circuits with multiple components, as phasors allow for straightforward addition and subtraction of AC signals, aiding in the design and troubleshooting of electrical systems.

What is the relationship between angular frequency and phasor angle in AC circuits?

The relationship between angular frequency (ω) and phasor angle (θ) in AC circuits is given by the equation θ = ωt, where t is the time. Angular frequency, measured in radians per second, determines how quickly the phasor rotates. The angle θ represents the position of the phasor at a specific time. For example, if ω = 20 rad/s and t = 0.2 s, then θ = 4 radians (or 229 degrees). This relationship helps in determining the phase position of voltage and current in AC circuits.

How do you convert radians to degrees in phasor diagrams?

To convert radians to degrees in phasor diagrams, use the conversion factor: 1 radian = 57.2958 degrees. Multiply the angle in radians by this factor to get the angle in degrees. For example, if the angle is 4 radians, the conversion to degrees is 4 × 57.2958 ≈ 229 degrees. This conversion is useful for visualizing and drawing phasor diagrams, as degrees are often easier to interpret.