Brown University
Physics Department
PHYS 0060
LAB B - 180
Circuits with Inductors
References:
Edward M. Purcell, Electricity and Magnetism 2nd ed, Ch.. 7,8 (McGraw Hill, 1985)
R.P. Feynman, Lectures on Physics, Vol. 2, Ch. 17,22 (Addison –Wesley, 1963).
Equipment: Pasco Digital Function Generator, Tektronix Digital Oscilloscope, Resistor Boxes,
Inductor Box.
Introduction:
The inductor is described, and inductance as the property of a circuit element is defined. We make frequent use of the comparisons one can make between inductive behavior and the capacitive behavior studied in the previous lab period. In particular, we find again a circuit situation in which steady, unidirectional current is not of interest, and only two situations need to be considered: (1) When the current in a circuit containing an inductor is suddenly turned on or off, there is a “transient behavior” that can be studied by applying square waves to the circuit. This is known as RL behavior, and a time constant similar in function to the RC time constant enters the picture here. (2) When the driving voltage is sinusoidal, there is a phase difference between the voltage dropped across and inductor and the current through it. It is most profitable to consider the RC circuit and the RL circuit in a comparative fashion in the laboratory, and so the former, whose behavior under an AC driving voltage was only looked at briefly in the last experiment, is here analyzed in detail along with the RL circuit. In both cases, the mathematical description of the circuit equations is considerably simplified by treating the capacitive and inductive “reactances” using elements of complex number theory. In order to provide a convenient reference for the aspects of complex numbers that are useful in describing AC behavior, such as appear in this write-­‐up, we will make use of the concepts and equations summarized in the note “Analyzing AC Circuits with Complex Numbers” (MGM44). A copy of those notes should be available for you before this and the following laboratory experiments are performed. In keeping with the notation used in the analysis note, we wil1 also use here the expression 140625 1 Brown University
Physics Department
PHYS 0060
LAB B - 180
exp(x) = ex Inductance and the Inductor.
When a current flows is in a wire, a magnetic field surrounds the wire. The simplest configuration that shows this is a long (in the limit, infinitely long) straight wire carrying a constant current I. If we depict the wire as perpendicular to the plane of the paper (see Fig. 1), and take the conventional positive current as coming out of the page1, the magnetic field produced by I is mapped by circles in the plane of the paper, concentric with the wire. The magnetic field itself has a direction, as shown in Fig. 1(a), that is given by a “right hand rule”: When the thumb of the right hand points in the direction of the current that produces a magnetic field, the curled fingers of the hand point in the direction of the magnetic field surrounding the wire. 1
Directed quantities, such as the current, coming out of the page depicted by a dot, representing
the head of an arrow. Quantities directed into the page represented in figures by an “x”
representing the tail of an arrow.
140625 2 Brown University
Physics Department
PHYS 0060
LAB B - 180
If the current reverses, as shown in Fig. 1 (b), the field also reverses. Some misconceptions that may arise with such a sketch should be prevented. The magnetic field is not uniform in strength as we move out radially from the wire. Each circle represents a region where the field is uniform, but it decreases the further we get from the wire. The field is proportional to the current everywhere, but the constant of proportionality decreases as a function of distance from the wire. Further, the magnetic field vector is not the direction of a force exerted by the magnetic force — the relation between those two vectors is considerably more complicated. Since we are not directly interested in that relation, we will not pursue the matter here. The important law of induction that we are concerned with describes what happens if a change in current occurs in the wire. The current of Fig. 1(a) will serve as an example. As that current decreases, the magnetic field decreases everywhere. The most graphic way to illustrate the induction effect is to consider the action in stages: (1) As the current changes. The magnetic field also changes proportionally everywhere. (2) As the magnetic field changes, a voltage is induced across each unit length of the wire, proportional to the time rate of change of the field (and so also proportional to the time rate of change of the current itself). (3) The direction of the induced voltage is such as would oppose the change taking place in the current. That is, it would be directed in such a way as to cause a “counter current” to be created that would seek to add to a current that was decreasing, and subtract from a current that was increasing. In Fig. 2, we sketch the inductive effect outlined by the three-­‐step process above. We have taken advantage in this example of the rather simple geometry to skip over details that are important when one seeks to find a more general description of the induction process. The text and lectures will cover those finer points. 140625 3 Brown University
Physics Department
PHYS 0060
LAB B - 180
However, the simple “infinite straight wire” would not give a very strong inductive effect. After all, every circuit you have built so far has, had more or less straight leads connecting different components without introducing noticeable magnetic effects. A more realistic configuration is one in which the conducting wire is wound in a coil, so the magnetic field close to the region of the coil is greatly strengthened by having contributions from many more elements of current-­‐carrying wire than the straight wire configuration gives. Fig. 3 sketches the configuration, which is sufficiently close to the actual nature of the circuit elements we have been describing, and which is called an “inductor” for obvious reasons (or “choke” for less obvious reasons), that it has become the electrical symbol for the device. A voltage difference appears across the ends of a change in current passing through the coil. For completeness, the sketch includes the symbol for a closely related device in which the magnetic field from one coil encompasses the wires of another coil. In this case, a changing current in the first coil includes a voltage difference across the ends of the second coil. The device is called a transformer – we will not consider it further here. The circuit behavior of an inductor is parametrized by the statement that the voltage drop across it is proportional to the time rate of change of the current through it. This can be written as the equation VL = L di/dt Eq. (1) Note that the voltage drop across an inductor is proportional to the rate of change of the current through it – not on the amount of current itself. In contrast to the capacitor, whose effect on a circuit is strongest at low frequency, the inductor will show its strongest effects at high frequency. The unit of inductance is the Henry, which corresponds in magnitude to a one-­‐volt difference between two circuit points for each ampere per second that the current is changing. The inductance unit is slightly large for practical work, but not (as is the case with the Farad capacitance unit) remarkably so. Inductances in the millihenry or microhenry range are more common, but a one-­‐
Henry inductance is easily obtainable 140625 4 Brown University
Physics Department
PHYS 0060
LAB B - 180
The Transient Behavior of an R-­‐L Circuit. In Fig. 4, we show the schematic of a circuit that includes a resistor H and an inductor L in series with a voltage source and a switch. Obviously no current is flowing, since the switch is open. At every instant, the voltage drop across the resistor, the inductor, and the switch must sum to the voltage rise V across the battery as we traverse the circuit in a clockwise direction. With no current flow, and also no change in current occurring, the voltage drop is of course entirely across the switch. Suppose the switch is closed at time T = 0. At every instant the voltage dropped across H and L must equal the applied voltage. V = i(t)R + L di(t)/dt Eq. (2) where the current is written explicitly as a function of time, and all other quantities are constants. At t = 0 there is no current, hence no drop across the resistor. But at that instant the time rate of change of the current must then be at its maximum absolute value and decreasing, so that Eq. (2) is satisfied. After a long time, with “long” again to be defined as we did in the corresponding RC circuit once we have the form of the voltage law at hand, the current will reach a stable, constant value. Since there is no voltage drop across a coil if the current through it is constant, the drop will be entirely across the resistor. In Appendix A, we show that Eq. (2) is satisfied by the time dependent current i(t) = (V/R) [1 -­‐ exp (-­‐Rt/L) ] Eq. (3) flowing in the circuit. Again we have characteristic time appearing in the exponential and playing a role analogous to the RC time constant of the previous experiment. Here the characteristic time is the ratio L/R, which you can easily show from the defining equation for L does have the dimensions of time. Now a “long time” can be defined as a time that consists of several time constants, because the same exponential behavior seen in the RC case operates here, with a different combination of circuit elements making up the time constant that characterizes the voltage and current changes. The argument would be very similar if we now asked what would happen it we again opened the switch. While there was no voltage drop across the inductor before we did so, there was a magnetic 140625 5 Brown University
Physics Department
PHYS 0060
LAB B - 180
field in the region of the coil. Opening the switch causes the current to stop immediately, but the voltage across the inductor, caused by the collapsing magnetic field, then simultaneously attains a maximum value, and decreases with the same characteristic time, to zero. Again it is across the switch that the voltage will be seen to rise, after it is reopened, to the value V that is had before it was first closed. Expt. 1 – Observing the Rise and Fall of Current (Transients) We will again use the square waves generated by the Pasco signal generator to set up an oscilloscope display equivalent to opening and closing the switch at a selectable frequency. In order to see the full rise and fall of the transient voltage across the inductor, the period of the square wave, which is 1/2f, with f the frequency of the generator, should be large compared to the characteristic time, L/R, of the RL circuit shown in Fig. 5. Recall that the signal generator puts out bipolar square waves whose average value is zero, the AC ground found not only on a correspondingly marked terminal of the generator, but also the same ground value that the Thornton scope signal input banana plugs hold on the tabbed terminal. The circuit of Fig. 5 shows the generator output being displayed on channel A of the scope, while the voltage across the inductor is monitored on channel B. From your familiarity with the method of obtaining the display gained in the RC study, you should be able to compare the actual and theoretical values of the RL time constant. Recall that exp(-­‐1) is approximately 0.368. RL Circuits Driven by a Sinusoidal Voltage. When the voltage impressed on an R-­‐L circuit is sinusoidal, the current through the circuit will also be sinusoidal. Again, as in the R-­‐C case we ignore the transient effects of the type studied above, which will die out in a time comparable to the ratio L/R, and consider only the so-­‐called steady state behavior. Using the notation of the notes in MGM44, we write the driving sinusoidal voltage as v(t) = V expj ωt Eq. (4) by which we denote that the arbitrary phase constant for this driving voltage is set to zero. This means that the current in the R-­‐L circuit must, in general, be given an explicit phase constant. The driving voltage, as indicated, has an amplitude of V, a real number. 140625 6 Brown University
Physics Department
PHYS 0060
LAB B - 180
The steady state current then has the form i(t) = I expj (ωt + θ ) Eq. (5) showing that it is a sinusoidal function with an amplitude I, and a phase difference θ relative to the driving voltage. We now consider the circuit of Fig. 6 , which is identical to that of Fig. 5, expect that the generator voltage had been made sinusoidal, and the resistor R and inductor L have changes places. This enables us to observe the current on the B-­‐channel of the scope, since the voltage dropped across R is just R times the circuit current. The A-­‐channel will monitor the applied voltage, so we can look for phase differences between v(t) and i(t) by comparing the two displayed signals. V The voltage dropped across the inductor has the value L (di/dt) = jωLI expj (ωt + θ ) Eq. (6) while, as we noted above, the voltage dropped across the resistance is IR = IR expj (ωt + θ ) Eq. (7) so that the equality of voltage generated and voltages dropped across the impedances gives V expj (ωt) = (R + jωL) ⋅ I ⋅ expj (ωt + θ ) The factor expj (ωt) can be factored from both sides, then the real and imaginary parts of the complex equation separately equated. This gives the pair of equations in which all numbers are real: V = IR cos θ -­‐ ωLI sin θ Eq. (8) 0 = ωL cos θ + R sin θ The last equation shows that the phase angle between the current and the impressed voltage: 140625 7 Brown University
Physics Department
PHYS 0060
LAB B - 180
tan θ = -­‐ωL/R Eq. (9) When this is used in the first equation, we find that the magnitude of the current (the maximum amplitude) is I =VR/(R2 + ω2L2 ) Eq. (10) The relations for the phase angle and for the maximum current can be examined as a function of the variables R and L with the circuit of Fig, 6. In particular, observe the frequency dependence of the current amplitude. Will the current increase or decrease with frequency if R and L are kept at some particular pair of values? Note: Order of magnitude estimate to start: for f=104, ω=2 π f Phase shift of ≈ 45° obtained if R ≈ 104 k Ω , L ≈ 0.16 H 140625 8