Question

Show that the total instantaneous power in a balanced three-phase circuit is constant and equal to \(1.5 V_{m} I_{m} \cos \theta_{\phi},\) where \(V_{m}\) and \(I_{m}\) represent the maximum amplitudes of the phase voltage and phase current, respectively.

Short answer

The total instantaneous power is constant and equals \(1.5 V_{m} I_{m} \cos \theta_{\phi}\).

Step-by-step solution

Understanding Three-Phase Power

In a balanced three-phase circuit, there are three voltages and currents, typically expressed for phases A, B, and C. These are time-dependent sinusoidal functions with a phase difference of 120° among them. Thus, the phase voltages can be expressed as: \[ v_a(t) = V_m \cos(\omega t) \]\[ v_b(t) = V_m \cos(\omega t - 120^\circ) \]\[ v_c(t) = V_m \cos(\omega t - 240^\circ) \]

Expressing Phase Currents

Similarly, the phase currents are given by:\[ i_a(t) = I_m \cos(\omega t - \theta_{\phi}) \]\[ i_b(t) = I_m \cos(\omega t - 120^\circ - \theta_{\phi}) \]\[ i_c(t) = I_m \cos(\omega t - 240^\circ - \theta_{\phi}) \]Where \(\theta_{\phi}\) is the phase angle between the voltage and current.

Calculating Instantaneous Power in Each Phase

The instantaneous power for each phase can be calculated using the product of the voltage and current for that phase: \[ p_a(t) = v_a(t) \cdot i_a(t) = V_m \cos(\omega t) \cdot I_m \cos(\omega t - \theta_{\phi}) \]Apply the trigonometric identity \( \cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)] \) to simplify:\[ p_a(t) = \frac{V_m I_m}{2} [\cos(2\omega t - \theta_{\phi}) + \cos(\theta_{\phi})] \]

Key concepts

Balanced Three-Phase Circuit

In electrical engineering, a balanced three-phase circuit is a system that consists of three circuits powered by three alternating currents (AC), all having equal amplitude and frequency but differing in phase angle by 120 degrees. This configuration is widely used in electric power distribution because it allows for efficient delivery of electricity.
Three main elements define the system:

• Three-phase voltages
• Three-phase currents
• Phase angle differences

The balance in a three-phase circuit means the impedances are equal in each phase. This balance ensures that the power is distributed evenly across all phases, minimizing losses and improving efficiency. A three-phase system can supply a balanced load with constant power, reducing mechanical vibration in equipment connected to the system.
The phase voltages in a balanced system are given by sinusoidal functions like:

• For Phase A: \( v_a(t) = V_m \cos(\omega t) \)
• For Phase B: \( v_b(t) = V_m \cos(\omega t - 120^\circ) \)
• For Phase C: \( v_c(t) = V_m \cos(\omega t - 240^\circ) \)

Each equation describes a unique voltage wave that is spatially and temporally displaced, contributing to the circuit's overall balance and functionality.

Instantaneous Power

Instantaneous power in a three-phase circuit is the power value at any specific point in time. It is calculated by multiplying the instantaneous voltage and current of each phase.
For each phase, the power function is expressed as the product of the instantaneous voltage and current:

• Phase A Power: \( p_a(t) = v_a(t) \cdot i_a(t) \)
• Phase B Power: \( p_b(t) = v_b(t) \cdot i_b(t) \)
• Phase C Power: \( p_c(t) = v_c(t) \cdot i_c(t) \)

Using trigonometric identities, such as \( \cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)] \), these power equations can be simplified to highlight how instantaneous power behaves over time.
A notable aspect of a balanced three-phase system is that the total instantaneous power remains constant. While the instantaneous power of each phase varies sinusoidally, their sum results in a steady output, absent fluctuations, making it very efficient for operating electrical equipment.

Phase Voltage

Phase voltage refers to the voltage measured across a single phase in a three-phase system. In balanced circuits, the phase voltages are equal in magnitude but out-of-phase by 120 degrees. These voltages can be derived from line-to-line voltages or directly measured in delta or star configurations.
For a clearer understanding, consider the phase voltage for Phase A:

• \( v_a(t) = V_m \cos(\omega t) \)

This expression implies that the voltage is sinusoidal with an amplitude of \( V_m \) and oscillates with angular frequency \( \omega \).
The symmetry in the phase voltages greatly assists in predicting the circuit's behavior, ensuring that even if individual voltages fluctuate, the total power remains stable due to its balanced nature. Thus, phase voltage is crucial in analyzing a system's efficiency and operation.

Phase Current

Phase current is the current flowing through one of the phases in a three-phase circuit. Like phase voltage, phase current is also out-of-phase by 120 degrees relative to the other phases.
In mathematical terms, the phase current can be expressed as:

• For Phase A: \( i_a(t) = I_m \cos(\omega t - \theta_{\phi}) \)

Here, \( I_m \) is the maximum current amplitude, and \( \theta_{\phi} \) is the phase angle between the voltage and current.
Understanding phase current involves analyzing its relationship with the alternating voltage. The concept of phase angle \( \theta_{\phi} \) is particularly important as it affects the power factor of the system. A proper understanding of phase current helps ensure that power is delivered efficiently and equipment connected within the circuit operates smoothly.