Lab 7 - LR Circuits

Introduction

The English physicist Michael Faraday found in 1831 that when the current through a coil changes, the coil produces a changing magnetic field (in addition to the field of the changing current), which induces an electromotive force ("emf") in the coil itself. In 1834, the German physicist Heinrich Lenz refined this further by showing that the induced current driven by this emf will be in the direction that opposes the change in the original current. We call this phenomenon self-induction, and the coils are called inductors. At the time that Faraday announced his discovery, he was asked what possible use such knowledge could be? His reply was: "Of what use is a newborn baby?" As happens with many seemingly arcane discoveries, Faraday's investigations of induction led to several common and useful modern electrical devices. Inductors, like capacitors, affect the time characteristics of an AC (alternating current) circuit and are, therefore, used to tune radio circuits, filter out unwanted noise, etc. The telephone receiver makes use of a type of inductor, as do stereo speaker systems and microphones. In this lab you will examine the effect of an inductor on the current and voltage in a simple circuit.

Discussion of Principles

The inductance of a circuit, usually symbolized by L, and measured in henry (H), is the tendency of a circuit to oppose any changes in the current. This opposition to a change in the current shows up as a slowing of the rise or fall of the current in circuits. Inductance is a property of electrical devices. Devices having this property are called inductors. The inductance of a device, like resistance and capacitance, depends on geometrical factors like the size of the device and on the material from which the device is made. It does not depend on the current in the device. Consider a simple circuit consisting of a switch, a resistor R, and a battery. When the switch is closed, the current I in the circuit will increase very quickly to a steady value given by Ohm's Law,

I = ΔV / R, 

where

ΔV 

is the voltage or emf of the battery. Consider the same circuit with the addition of an inductor, as shown in Fig. 1.

Figure 1: LR circuit

When the switch is in position 1 as shown in Fig. 1(a), the rising current produces a rising magnetic flux in the inductor. This induced magnetic flux produces an electromotive force (emf) that is of opposite polarity to that of the battery, which results in an induced current opposing the current from the battery. The current, therefore, rises more slowly than it would otherwise have done without the inductor. How much more slowly the current increases depends on the size of the inductance and the value of the resistance. Even though the time it takes the current to rise to its final value is longer, the actual final value is the same as that which would be achieved without the inductor. Consider the situation where the switch has been in position 1 for sometime and a steady current is flowing in the circuit. If the switch is now moved to position 2 as in Fig. 1(b) the battery is no longer part of the circuit. The inductor will prevent the current from instantly dropping to zero. Just as the inductor resisted a rapid increase in current, it also resists a rapid decrease in current. For the circuit shown in Fig. 1(a), Kirchhoff's loop equation can be written as The solution to Eq. (1)

ΔV − IR − L

dI

dt

= 0.

 

is where I_f represents the final steady current flowing in the circuit after an infinite length of time, R is the circuit resistance, and L is the inductance of the coil. From this expression you can see that current increases exponentially. This is shown in Fig. 2(a). When the switch is moved to position 2, for the circuit shown in Fig. 1(b), Kirchhoff's loop equation is now given by The solution to Eq. (3)

IR − L

dI

dt

= 0.

 

is where I₀ represents the initial maximum current in the circuit at t = 0. You can see from this expression that the current decreases exponentially. See Fig. 2(b).

Time Constant

The mathematical analysis of a simple LR circuit is similar to that of a simple RC circuit (a circuit consisting of a resistor and a capacitor in series). In an RL circuit, the time constant

τ 

is defined by The time constant is also defined as the amount of time it takes the current to reach 63% or

(1 − e⁻¹) 

of its final value. Note that Eq. (2)

I = I_f1 − e^(−R/L)t 

has the same form as the equation describing the charging of a capacitor. Since the current is changing with time, the potential difference across the resistor must also be changing with time. The equation for the potential difference across the resistor is obtained by using Ohm's Law,

ΔV = IR, 

where

ΔV_f 

is the final or maximum potential difference across the resistor and is equal to the emf of the battery. Note that at

t = 0, (1−e^−t/τ) = 0 

and the current through the circuit and the potential difference across the resistor are zero. Therefore, the potential drop is entirely across the inductor. At t = infinity (a long time after the switch has been in position 1),

(1 − e^−t/τ) = 1 

and the current through the circuit and the potential difference across the resistor are maximum at I_f and

ΔV_R. 

Therefore, the potential drop is entirely across the resistor and the potential drop across the inductor is zero. Without the inductor in the circuit the current in the resistor would drop very quickly to zero once the switch is moved to position 2. When the inductor is in the circuit, it opposes this change in current, and so the current drops more slowly. The current and voltage across the resistor t seconds after the battery is removed from the circuit by moving the switch to position 2 are given by the following. Here

τ 

is the time needed for the current to decrease to 33% of its original value at t = 0. Consider Eq. (7)

I = I₀e^(−t/τ) 

and Eq. (8)

ΔV_R = ΔV₀e^(−t/τ) 

. At

t = 0, e^−t/τ = 1, 

and the current through the circuit and the potential difference across the resistor are maximum. Therefore, the potential drop is entirely across the resistor and the potential drop across the inductor is zero. At t = infinity (a long time after the switch has been in position 2),

e^−t/τ = 0, 

and the current through the circuit and the potential difference across the resistor are zero. Therefore, the potential drop is entirely across the inductor and the potential drop across the resistor is zero. At any given time t the sum of the potential drops across the resistor and the inductor will be equal to the emf of the battery. The potential difference

ΔV_R 

across the resistor as a function of time is shown here in Fig. 3 and Fig. 4 shows the voltage drop across the inductor as a function of time. Note that when the voltage across the resistor is a maximum, the voltage across the inductor is zero and vice versa as discussed earlier. Eq. (6)

ΔV_R = ΔV_f 1 − e^(−t/τ) 

can be algebraically rearranged as: where

τ 

has been replaced with L/R. Taking the natural log of both sides of this equation and multiplying by –1, we get If you consider the left side of the equation to be a single variable, say y, then the equation becomes

y = (R/L)t, 

which is a linear equation of the form

y = mx. 

The inductance can be determined from the slope of this line. Similarly, Eq. (8)

ΔV_R = ΔV₀e^(−t/τ) 

can be written as A plot of

−ln

ΔVR

ΔV0

versus time t for a decreasing current (soon after the switch is opened) will give a straight line with a slope of R/L from which the inductance can be determined.

Using a Square Wave to Simulate the Role of a Switch

In this experiment, rather than using a switch, we will be using a signal generator that can generate periodic wave forms of varying shapes, like a sine wave, a triangular wave, and a square wave. Both the frequencies and amplitudes of the wave forms can also be adjusted. Here we will use the signal generator to produce a time-varying voltage, with a square wave form, across the inductor similar to the one shown in Fig. 5. The output voltage from the signal generator changes back and forth from a constant positive value to a constant zero volts in equal intervals of time t. The time

T = 2t 

is the period of the square wave. During the first half of the cycle, when the voltage is positive, it is similar to the switch being in position 1. During the second half of the cycle, when the voltage is zero, it is the same as the switch being in position 2. So the square wave, which is a DC voltage that is turned on and off periodically, serves as both battery and switch in the setup in Fig. 1. The signal generator allows this switching to be done repeatedly, and it is possible to optimize the data collection by adjusting the frequency of the repetition. This frequency will depend on the time constant of the RL circuit. When the time t is larger than the time constant

τ 

of the RL circuit, the current in the circuit will have enough time to reach the steady state and the voltage across the inductor will be as shown in Fig. 4.

Objective

The objective of this experiment is to examine the dynamic behavior of an LR circuit by using an oscilloscope to visualize the voltage across the resistor for both rising and decreasing current. You will also determine the time constant and inductance of the coil.

Equipment

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

Setting Up the LR Circuit

The RLC circuit board that you will be using consists of three resistors and one inductance coil among other elements. The value of the inductor can be changed by inserting an iron core into the coil. See Fig. 6 below. In theory you can, therefore, have different combinations of resistors and capacitors. In this experiment you will use the 10-Ω resistor and the inductor coil.

Procedure A: Determining L from Time Constant

The computer will function as the oscilloscope to record

ΔV_R 

and as the signal generator.

Procedure B: Measuring Voltage for Increasing Current

Procedure C: Measuring Voltage for Decreasing Current

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