Question

What is the phase sequence of each of the following sets of voltages? a)$$\begin{array}{l} v_{\mathrm{a}}=120 \cos \left(\omega t+54^{\circ}\right) \mathrm{V} \\ v_{\mathrm{b}}=120 \cos \left(\omega t-66^{\circ}\right) \mathrm{V} \\ v_{\mathrm{c}}=120 \cos \left(\omega t+174^{\circ}\right) \mathrm{V} \end{array}$$ b)$$\begin{aligned} &v_{\mathrm{a}}=3240 \cos \left(\omega t-26^{\circ}\right) \mathrm{V}\\\ &v_{b}=3240 \cos \left(\omega t+94^{\circ}\right) \mathrm{V}\\\ &v_{\mathrm{c}}=3240 \cos \left(\omega t-146^{\circ}\right) \mathrm{V} \end{aligned}$$

Short answer

a) The phase sequence is \( v_b \), \( v_a \), \( v_c \). b) The phase sequence is \( v_c \), \( v_a \), \( v_b \).

Step-by-step solution

Extract Angles from Voltages (Set a)

Identify the phase angles from each voltage equation in set (a): - For \( v_{a} \), the angle is \( +54^{\circ} \).- For \( v_{b} \), the angle is \( -66^{\circ} \).- For \( v_{c} \), the angle is \( +174^{\circ} \).

Compare Angles (Set a)

Compare the angles: \( -66^{\circ} \), \( +54^{\circ} \), and \( +174^{\circ} \).- The smallest angle is \( -66^{\circ} \).- Next is \( +54^{\circ} \).- Largest is \( +174^{\circ} \).Thus, the phase sequence is \( v_b \), \( v_a \), \( v_c \).

Extract Angles from Voltages (Set b)

Identify the phase angles from each voltage equation in set (b): - For \( v_{a} \), the angle is \( -26^{\circ} \).- For \( v_{b} \), the angle is \( +94^{\circ} \).- For \( v_{c} \), the angle is \( -146^{\circ} \).

Compare Angles (Set b)

Compare the angles: \( -146^{\circ} \), \( -26^{\circ} \), and \( +94^{\circ} \).- The smallest angle is \( -146^{\circ} \).- Next is \( -26^{\circ} \).- Largest is \( +94^{\circ} \).Thus, the phase sequence is \( v_c \), \( v_a \), \( v_b \).

Key concepts

Alternating Current (AC) Circuits

Alternating Current (AC) circuits are fundamental in modern electrical systems. Unlike direct current (DC), where the flow of electric charge is unidirectional, AC circuits utilize electricity that changes its direction periodically. This is more efficient for transferring electricity over large distances. Here's why:

• **Efficiency in Power Transmission:** AC circuits can be easily transformed to higher or lower voltages using transformers. This ability minimizes energy loss over long distances, making it ideal for power distribution.
• **Variable Voltage:** The ability to change voltage levels is particularly useful in residential and industrial applications, where different appliances require varied voltage levels.

When discussing AC circuits, we often come across terms like phase and frequency. These describe the oscillation characteristics. For instance, the **frequency** (measured in Hertz) indicates how often the direction of current changes per second. In most household systems, the frequency is standardized, like 60 Hz in North America. Understanding these properties helps in analyzing circuit behaviors, especially in multi-phase systems where phase sequence is crucial for synchronous operations.

Phase Angles

Phase angles in AC circuits determine a wave's position at any given point in time. They are crucial because they help us understand how different AC quantities relate to each other. A typical example would be voltage waves in a three-phase system.

• **Understanding Phase Angle:** The phase angle is indicated in degrees or radians and represents where exactly the wave is in its cycle when compared to a reference point, often referred to as the zero-position or starting point.
• **Determining Order:** When analyzing sets of voltages, like those in the original exercise, determining the correct phase sequence can prevent potential issues in equipment connected to these circuits, like motors which could rotate in the wrong direction if phased incorrectly.

By extracting the phase angle from voltage equations, as solved previously, one can ascertain the correct sequence by simply arranging them from the smallest to the largest angle. This sequence is critical in systems that employ synchronized multi-phase AC inputs.

Voltage Equations

Voltage equations are invaluable for understanding and predicting the behavior of electrical systems. These equations provide the mathematical expressions that describe voltages as they vary with time in AC circuits. Commonly expressed in cosine or sine form, these equations illustrate several key aspects:

• **Amplitude:** This is the peak value of the voltage. In the equations given, for example, 120 V and 3240 V represent the maximum magnitude of the voltage.
• **Phase Shift:** Noted in the form of a plus or minus degree notation, this tells us about the shift of the wave compared to a standard reference.
• **Angular Frequency:** Represented by the Greek letter omega (\( \omega \)), it correlates with the frequency of the AC supply and is crucial for calculating time-based behaviors.

When handling such equations, understanding how these components play together helps in predicting performances and troubleshooting AC circuits, which is essential for efficient design and operation of electrical setups.