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In a three-phase system, the positive-sequence is a type of phase sequence where the three phase voltages are arranged in a particular order. This order is generally denoted as \( e \rightarrow b \rightarrow c \). The phase angle between each of these phases is typically \( 120^{\circ} \). This sequence allows for efficient energy transfer and is commonly used in power systems.

For example, considering the given voltage expression \( v_{en}(t) = 120 \cos(100\pi t + 75^{\circ}) \). The voltage for phase \( e \) leads phase \( b \) by \( 120^{\circ} \) and phase \( c \) by \( 240^{\circ} \). Such a sequence is crucial for maintaining the balance and stability of the electrical system. Without the positive-sequence, motors might not start properly, and transformers can experience uneven loading. Remember, in the positive-sequence, the phase sequence is natural and promotes clockwise rotation in electric motors.

The negative-sequence of a three-phase system is the reverse of the positive-sequence. Here, the order is \( e \rightarrow c \rightarrow b \). Like the positive, the negative-sequence voltages are also separated by \( 120^{\circ} \) between each phase.

In a balanced negative-sequence system, phase \( e \) still leads but this time instead, followed by phase \( c \) and finally phase \( b \). This sequence causes reverse rotation, which can be problematic for motors and generators, potentially causing mechanical stress and inefficient operation. For our specific example, reversing the sequence leads actually to phase \( b \rightarrow c \rightarrow e \) for practical implications in reverse. The phase angles for voltages would appear reversed considering their specific signs.

The negative-sequence is generally an undesirable condition in power systems, as it can disrupt the normal operation of equipment, leading to adverse effects like overheating and vibration in rotating machines.

Angular frequency, denoted as \( \omega \), is a fundamental concept in three-phase systems that relates the oscillation rate of a waveform. It indicates how quickly the voltage waveform completes its cycles. Angular frequency is given by the expression \( \omega = 2\pi f \), where \( f \) is the frequency in hertz.

In the given exercise, the angular frequency is derived from the term \( 100\pi \) present in the voltage equation \( v_{en}(t) = 120 \cos(100\pi t + 75^{\circ}) \). This implies our system has an angular frequency of \( 100\pi \), equating to a frequency \( f \) of 50 Hz.

Angular frequency is crucial for understanding the dynamics of three-phase systems. It has a direct influence on the operation of components within the system, such as transformers and motors, helping determine their speed and efficiency. A consistent angular frequency ensures that the system runs smoothly without variations in cycle speed, which is vital for maintaining synchronization across the network.

Phase voltages in a three-phase system refer to the voltages measured across each of the individual phases \( e \), \( b \), and \( c \) relative to the neutral point \( n \). Each of these voltages should ideally have the same magnitude and be equally spaced in angle around the phase circle.

Understanding the expressions for phase voltages is important, as they dictate how electricity flows through the system. For our case, we have the initial phase voltage for \( e \) as \( v_{en}(t) = 120 \cos(100\pi t + 75^{\circ}) \). The other phase voltages, \( v_{bn}(t) \) and \( v_{cn}(t) \), will depend on sequence types:

• **Positive-sequence:** \( v_{bn}(t) = 120 \cos(100\pi t - 45^{\circ}) \), \( v_{cn}(t) = 120 \cos(100\pi t - 165^{\circ}) \)
• **Negative-sequence:** \( v_{bn}(t) = 120 \cos(100\pi t - 165^{\circ}) \), \( v_{cn}(t) = 120 \cos(100\pi t - 45^{\circ}) \)

Properly calibrated phase voltages ensure stable loads, efficient power delivery, and prevent unexpected outages or equipment failures within the system.