I) Reactance and Resistance

Inductance in an AC Circuit

Introduction

In this experiment you will use an Oscilloscope to measure the voltages across a Resistor (R) and an inductor (L) in a simple series RL Alternating Current (AC) circuit. The Function (Signal) Generator produces the AC signal, which can be converted into a voltage vs. time graph by the Oscilloscope. You will investigate the dependence of the resistor and inductor voltages, as well as their phase difference, on the frequency of the AC signal and calculate the inductance of the coil.

Equipment

Variable Resistor, Inductor (coil), Function (signal) generator, Oscilloscope

Theoretical Background

i) Reactance and Resistance

A resistor provides resistance to the flow of current in a circuit and the relationship between the voltage across the resistor (V_R) and current (I) is given by Ohm’s law, which states that the resistance of many materials remains constant over a wide range of applied currents and voltages:

(1)

The resistor converts electrical energy into heat energy – it dissipates electrical energy. These basic principles apply whether the current in the circuit is constant, as in direct current (DC) circuits, or variable, as in AC circuits. In AC circuits, V_R can be expressed as:

(2)

where t is time, V _{0 R} is the amplitude or maximum resistor voltage, ω = 2π f is the angular frequency of the AC signal, and f is the AC frequency, which can be adjusted by using the Function Generator. Ohm’s law holds in AC circuits, as well, and the current and voltage across the resistor oscillate in phase at the same frequency:

(3)

where is the amplitude or maximum current through the resistor.

Now consider a coil of wire, also known as inductor, and suppose it has a negligible resistance. When the current through it changes with time, an electromotive force (emf) is induced in the coils of the wire to oppose the change in current. This is explained mainly by Faraday’s law of magnetic induction, a fundamental law of electromagnetism that predicts how magnetic fields interact in an electrical circuit to produce an emf. An emf produces energy to drive electrons around a circuit just like the voltage of a battery. This means that when an AC voltage is applied to a coil a periodic emf is induced in the coil that opposes the applied AC voltage, since the AC voltage and current are variable. This induced emf or back emf acts as an AC resistance. In AC circuits this resistance is called reactance and it is quite different from the normal resistance of a resistor. The reactance does not dissipate electrical energy as heat, and it is directly proportional to the AC frequency or the rate of change of the AC current and voltage. It can be shown that the reactance X_L of an inductor can be expressed as:

(4)

where ω and f have been defined above, and L is the inductance of the coil. L is constant for a given coil, and depends on the number of its turns of wire, geometry (e.g. cross-sectional area, length, and/or shape), as well as on the type of core material used. The SI unit of inductance is the Henry (H), which is equivalent to 1 V s/A. Applying Ohm’s law for an inductor in an AC circuit yields:

(5)

where V _{0 L} and I _{0 L} are the amplitudes of the inductor voltage and current, respectively. The voltage across the inductor is 90⁰ out of phase with the current through it, as the voltage reaches its peak value one-quarter of an oscillation period before the current reaches its maximum value.

Дата добавления: 2015-10-29