An ac generator has a frequency of 1200 Hz and a constant rms voltage. When a 470-Ωresistor is c

step 1 Here in this given problem frequency of the AC generator that is given as 1200 Hz. The resistanc

An ac generator has a frequency of 1200 Hz and a constant...

An ac generator has a frequency of $1200 \mathrm{Hz}$ and a constant rms voltage. When a $470-\Omega$ resistor is connected between the terminals of the generator, an average power of $0.25 \mathrm{W}$ is consumed by the resistor. Then, a 0.080-H inductor is connected in series with the resistor, and the combination is connected between the generator terminals. Concepts: (i) In which case does the generator deliver a greater rms current? (ii) In which case is the greater average power consumed by the circuit? Calculations: What is the average power consumed in the inductor-resistor series circuit?

Key Concepts

Inductive Reactance

Inductive reactance is the opposition to the change of current by an inductor in an AC circuit and is given by the formula X_L = 2?fL, where f is the frequency and L is the inductance. This concept is critical for understanding how an inductor affects the overall impedance and phase relationship between voltage and current, especially as it increases with frequency.

Impedance in AC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current and combines both resistance and reactance into a single complex quantity. In circuits with inductors or capacitors, impedance is used in place of resistance, and its magnitude is determined by both the resistive and reactive components, typically calculated as Z = ?(R² + X²) for a series circuit.

Phase Angle in AC Circuits

The phase angle in an AC circuit arises due to the presence of reactive components like inductors. It represents the angular difference between the voltage and current waveforms. In a purely resistive circuit the phase angle is zero, but when inductive reactance is present, the current lags behind the voltage, affecting both the overall current and the calculation of real power delivered to the circuit.

RMS Voltage and Current

RMS (Root Mean Square) values represent the effective voltage or current in an AC circuit that produces the same heating effect as a corresponding DC circuit. This concept is fundamental in analyzing AC circuits, as it allows for the use of power formulas that are familiar from DC analysis, such as power being proportional to the square of the RMS current times the resistance.

Resistive Power Consumption in AC Circuits

In AC circuits, when only resistive elements are present, the average power consumed can be calculated using the formula P = I_rms² × R. This relationship highlights that the power consumption depends solely on the current and the resistance, assuming the voltage and current are in phase, with no reactive components affecting the energy transfer.

Calculation of Power in Series AC Circuits

When resistors and inductors are combined in a series circuit, the overall impedance is higher due to the added inductive reactance, which reduces the magnitude of the current for a given applied voltage. The average power consumed in such circuits is determined by the resistive element only, using the updated current calculated from the total impedance, reinforcing that power is dissipated only in the resistor while the inductor temporarily stores and releases energy.

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