Phasors: Videos & Practice Problems

Phasors are rotating vectors that effectively represent oscillating values like voltage and current in AC circuits. The x component of a phasor indicates the voltage's value, while its direction shows whether it's positive or negative. As phasors rotate counterclockwise, they can be analyzed like vectors, allowing for addition and subtraction. Understanding phasors is crucial for solving AC circuit problems, as they simplify the representation of sinusoidal functions, making calculations more manageable. The angular frequency, denoted as ω, determines the rotation speed of the phasor.

Phasors

Phasors are essential tools in analyzing alternating current (AC) circuits, representing oscillating values like voltage and current through rotating vectors. A phasor, or phase vector, captures all relevant information in its horizontal (x) component, allowing for a simplified representation of sinusoidal functions. For example, the voltage as a function of time can be expressed as:

$$ V(t) = V_{max} \cos(\omega t) $$

where \( V_{max} \) is the maximum voltage and \( \omega \) is the angular frequency. Understanding phasors involves examining their positions at specific times during one cycle of oscillation. At time \( t_1 \), when voltage is at its maximum, the phasor aligns entirely along the positive x-axis. At time \( t_2 \), when voltage is zero, the phasor points upwards along the y-axis, indicating no x component. At time \( t_3 \), the voltage reaches its maximum negative value, and the phasor points to the left along the x-axis. Finally, at time \( t_4 \), the phasor again points downwards, reflecting zero voltage.

Phasors rotate counterclockwise at the same angular frequency \( \omega \), which can be quantified in rotations per second. For instance, if \( \omega = 2 \) radians per second, the phasor completes two full rotations each second. This rotational nature allows phasors to be treated like vectors at any instant, enabling operations such as addition and subtraction, as well as magnitude calculations using the Pythagorean theorem.

In practical applications, when analyzing multiple phasors, the net phasor can be determined by vector addition. For example, if two phasors point in the same direction and a third is perpendicular, the resultant phasor will still point in the direction of the longer phasor, ensuring that the net value remains positive unless the resultant points to the left. This foundational understanding of phasors is crucial for further exploration of their applications in voltage and current analysis within circuits.

The following phasor diagram shows an arbitrary phasor during its first rotation. Assuming that it begins with an angle of 0° , if the phasor took 0.027 s to get to its current position, what is the angular frequency of the phasor?

• A. 0.141 s⁻¹
• B. 19.4 s⁻¹
• C. 38.8 s⁻¹
• D. 194 s⁻¹

Answer: (D)

Converting Between a Function and a Phasor

In an alternating current (AC) circuit, the relationship between the current and its phasor representation is crucial for understanding the behavior of the system. To illustrate this, consider a current described by a phasor equation. At a specific time of 15 milliseconds, we need to determine the position of the phasor, which begins at 0 degrees.

The angle of the phasor at any time can be calculated using the formula:

θ = ω × t

where θ is the angle in radians, ω is the angular frequency, and t is the time in seconds.

In this case, the angular frequency ω is given as 377 radians per second. To find the angle at 15 milliseconds (or 15 × 10^-3 seconds), we calculate:

θ = 377 × (15 × 10^{-3}) = 5.66 \text{ radians}

To convert this angle from radians to degrees, we use the conversion factor:

Degrees = \frac{θ}{\pi} × 180

Thus, the angle in degrees is:

Degrees = \frac{5.66}{\pi} × 180 \approx 324 \text{ degrees}

With the phasor starting at 0 degrees and rotating counterclockwise, it reaches 324 degrees, which places it in the fourth quadrant of the unit circle. This angle can also be expressed in alternative forms: as 54 degrees from the negative y-axis or as 36 degrees from the positive x-axis. Regardless of the representation, the key takeaway is that the phasor has traveled 324 degrees from its initial position, illustrating the fundamental concept of phasor rotation in AC circuits.

An AC source oscillates with an angular frequency of 120 s^-1 . If the initial voltage phasor is shown in the following phasor diagram, draw the voltage phasor after 0.01 s. (Select the correct absolute angle below of the phasor's location below after you have drawn it.)

• A. 8.3×10⁻⁵ radians
• B. 1.2 radians
• C. 2π radians
• D. 15 radians

Answer: (B)

A phasor of length 4 begins at 0° . If it is rotating at ω = 250 s⁻¹ , what is the value of the phasor after 0.007 s?

• A. 0.2
• B. 0.7
• C. 3.1
• D. 3.9
• E. 4.0

Answer: (B)

Here’s what students ask on this topic:

What is a phasor in AC circuits?

A phasor in AC circuits is a rotating vector that represents oscillating values like voltage and current. It simplifies the analysis of sinusoidal functions by capturing all the necessary information in its x component. The phasor's direction indicates whether the value is positive or negative, and it rotates counterclockwise at an angular frequency denoted by ω. This rotation allows phasors to be treated like vectors, enabling addition, subtraction, and magnitude calculations using the Pythagorean theorem. Phasors are essential tools for solving AC circuit problems as they make complex calculations more manageable.

How do you represent voltage using a phasor?

Voltage can be represented using a phasor by considering its maximum value and its oscillation over time. For instance, if the voltage is given by V(t) = V_max cos(ωt), the phasor will have a length equal to V_max and will rotate counterclockwise with an angular frequency ω. At different times, the phasor's projection onto the x-axis will represent the instantaneous voltage value. When the voltage is at its maximum, the phasor points entirely along the positive x-axis. When the voltage is zero, the phasor points along the vertical axis, and when the voltage is at its negative maximum, the phasor points along the negative x-axis.

Why are phasors useful in analyzing AC circuits?

Phasors are useful in analyzing AC circuits because they simplify the representation of sinusoidal functions, making calculations more manageable. By converting oscillating values like voltage and current into rotating vectors, phasors allow for easy addition, subtraction, and magnitude calculations using vector algebra. This approach reduces the complexity of solving AC circuit problems, as it avoids dealing with trigonometric functions directly. Additionally, phasors provide a clear visual representation of the phase relationships between different circuit elements, aiding in the understanding and analysis of AC circuits.

How do you add and subtract phasors?

Adding and subtracting phasors is similar to vector addition and subtraction. To add phasors, you align them head-to-tail and draw the resultant vector from the tail of the first phasor to the head of the last phasor. For subtraction, you reverse the direction of the phasor being subtracted and then add it to the other phasor. The resulting phasor represents the net effect of the combined phasors. This method allows for straightforward calculations of the overall voltage or current in an AC circuit by considering the contributions of individual phasors.

What is the significance of the angular frequency ω in phasors?

The angular frequency ω in phasors is significant because it determines the rotation speed of the phasor. It is defined as the rate at which the phasor rotates counterclockwise and is measured in radians per second. The angular frequency is directly related to the frequency of the oscillating value it represents, such as voltage or current, in an AC circuit. A higher angular frequency means the phasor completes more rotations per second, corresponding to a higher frequency of the oscillating signal. Understanding ω is crucial for accurately analyzing and interpreting phasor diagrams in AC circuits.