An industrial plant consumes 160 kW at 0.9 PF lagging. The line voltage at the plant is 480 ∠0^

step 1 In this problem I can write the value of p line is equal to PS minus PL which is equal to 100k m

An industrial plant consumes 160 kW at 0.9 PF lagging. The...

An industrial plant consumes $160 \mathrm{~kW}$ at 0.9 PF lagging. The line voltage at the plant is $480 \angle 0^{\circ} \mathrm{V}$ rms. The transmission line feeding the plant is 2 miles long. If the impedance of the line is $0.1+j 0.2 \Omega$, find the power factor at the sending end of the line.

Key Concepts

Phasor Analysis

Phasor analysis is a technique used to simplify the study of AC circuits by representing sinusoidal voltages and currents as rotating vectors (phasors) in the complex plane. This approach enables the simultaneous handling of magnitude and phase relationships, which is particularly important when dealing with power factors and impedance. By using phasor analysis, engineers can more easily calculate the effects of voltage drops, current flows, and phase shifts in a power system.

Transmission Line Impedance

Transmission line impedance models the non-ideal characteristics of the conductors used to transmit electrical power. It is represented by a complex number that includes both resistance (the real part) and reactance (the imaginary part). This impedance causes voltage drops along the line and affects the amount of power that ultimately reaches the load. Accurate knowledge of the line impedance is crucial for designing efficient power systems and for correctly calculating voltage drops and power losses.

Voltage Drop in AC Circuits

Voltage drop in AC circuits is the reduction in voltage as electrical current travels through the impedance of a transmission line. When current flows through a line with both resistance and reactance, a complex voltage drop occurs, which alters the magnitude and phase of the voltage at the load compared to the voltage at the source. Evaluating the voltage drop is essential for ensuring that sufficient voltage is delivered to devices and for determining the overall power factor of the system at the source.

Complex Power

Complex power represents both the real and reactive components of power in an AC circuit and is typically expressed as S = P + jQ, where P is the real power measured in kilowatts (kW) and Q is the reactive power measured in kilovolt-amperes reactive (kVAR). This formulation provides a complete picture of power consumption, allowing engineers to analyze and manage both the energy used for work and the energy stored and released by inductive or capacitive elements in the system.

Power Factor

Power factor is a measure of how effectively electrical power is being used in an AC circuit. It is defined as the ratio of real power, which performs useful work, to apparent power, which is the product of the circuit’s voltage and current. A lagging power factor indicates that the current lags behind the voltage, a common scenario in inductive loads. Understanding power factor is essential because it affects the efficiency of power transmission and the performance of electrical equipment.

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