A circuit connected to a 110-V, 60 -Hz source contains a 50-Ωresistor and a coil with an inducta

A circuit connected to a 110-V, 60 -Hz source contains a...

A circuit connected to a $110-\mathrm{V}, 60$ -Hz source contains a $50-\Omega$ resistor and a coil with an inductance of $100 \mathrm{mH}$. Find (a) the reactance of the coil, (b) the impedance of the circuit, (c) the current in the circuit, and (d) the power dissipated by the coil, and (e) calculate the phase angle between the current and the applied voltage.

Key Concepts

Phase Angle in AC Circuits

The phase angle in an AC circuit indicates the difference in timing between the voltage and the current waveforms. This angle is determined by the ratio of the inductive reactance to the resistance, typically calculated using the arctan function. The phase angle is essential in understanding the power factor, which reflects how much of the electrical power is effectively used for work.

AC Power Dissipation

In AC circuits, real power is dissipated only by the resistive components, even though the inductor may also impact the phase relationship between the voltage and current. Power dissipation is commonly calculated using either the product of the square of the RMS current and resistance or the voltage, current, and the cosine of the phase angle, representing the power factor of the circuit.

AC Circuit Current Calculation

The current in an AC circuit is determined using Ohm's law for AC systems, where the current is equal to the source voltage divided by the circuit's impedance. Because impedance takes into account both resistance and reactance, this calculation provides the magnitude of the current flowing through the circuit under alternating current conditions.

AC Circuit Impedance

Impedance is the complex combination of resistance and reactance in an AC circuit, defining the total opposition to current flow. In circuits with resistors and inductors, impedance is calculated using the square root of the sum of the squares of the resistive and inductive reactance components. It serves as the AC analog of resistance in DC circuits and is used for determining the circuit current from the source voltage.

Inductive Reactance

Inductive reactance represents the opposition that an inductor offers to an alternating current, and it depends on both the frequency of the AC source and the inductance of the coil. It is calculated as the product of 2?, the frequency, and the inductance, which illustrates that higher frequencies or larger inductance values result in a higher reactance, thereby reducing the current flow through the inductor.

Transcript

00:01 Hi, in the given problem, rms voltage applied to the lr circuit is 110 volt frequency of this source is 60 hertz.

00:17 The value of the resistance in the circuit is 50 -oom and the value of inductance is l is equal to 100 millie henry.

00:31 In the first part of the problem we have to find reactance of this circuit as the circuit is having inductance only along with the resistance.

00:40 So there will be only inductive reactance in this circuit and that inductive reactance will be given as xl is equal to 2 pi fl.

00:55 So plugging in all known values, this is 2 into 3 .14 into for frequency this is 60 hertz into self inductance 100 millie henry or 100 into 10 dash bar minus 3 henry.

01:12 So finally this inductive reactance of the circuit comes out to be 37 .7 om or we can say approximately this is 38.

01:25 Which is the answer for the first part of this problem.

01:32 Now in the second part of the problem, we have to find impedance of this circuit.

01:38 So impedance here will be the vector sum of inductive reactants with the resistance, which will be given as square root of addition of a square of reactance with the resistance.

01:53 So now plugging in all known values for excel, this is, 37 .7 square plus for r this is 50 square.

02:05 Here we are using the value of this excel without approximating it means 37 .7 not 38 om so here these squares will be given as z is equal to square root of 1 ,421 .3 plus 2 ,500 which will be added to be 3 ,921 .29 .29.

02:38 Or making a square root, it will come out to be 62 .6 om...

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