 # Unit 14: Inductors in DC Circuits

For most of this course you've studied circuits with resistors and DC voltage sources. In Unit 13 we added a new component, the capacitor. This unit will add another new component, the inductor. As you'll see, the techniques and equations for analyzing circuits with inductors are very similar to the techniques and equations that you've learned for capacitors.

Recall that resistors oppose the flow of current. And capacitors store charge. What about inductors? They oppose changes in current. What does that mean? Suppose that a circuit with an inductor has a certain current flowing through it. If you try to increase or decrease that current, then the inductor will fight against you, and will try to keep the current at its initial value. Eventually the inductor will lose this fight, and the current will change, but this will take some time to happen. This is different from what happens in a circuit with no inductor: if there's no inductor, then the amount of current can increase or decrease immediately. If there is an inductor, changes in current take a while to happen.

##### Preview of Electromagnetism
• In a future course you'll study two related phenomena called electromagnetism and electromagnetic induction. These two principles are key to understanding how an inductor operates, so let's take a quick look at them.
• In 1820, Hans Oersted discovered that electrical current creates a magnetic field.  This phenomenon is called electromagnetism. Oersted also realized that the you can increase the strength of the magnetic field surrounding a current-carrying wire by winding the wire into a series of closely spaced loops. A wire that is looped in this way is called a coil.
• A few years later, Michael Faraday discovered that a voltage is induced in a wire whenever there's a change in the size of the magnetic field surrounding the wire. This phenomenon is called electromagnetic induction. Also, the induced voltage will be greater if you use a coil of wire rather than a straight piece of wire.
##### Self-Inductance
• Now here comes the part that we're really interested in. Suppose you've got some current running through a coil of wire. According to the principle of electromagnetism, this current creates a magnetic field around the coil. What will happen if you change the size of the current? Well, that will change the strength of the magnetic field. But according to the principle of electromagnetic induction, when the magnetic field surrounding a wire changes, a voltage will be induced across that wire.
• And it turns out that this voltage will always oppose the change you're making to the current. In other words, if you increase the current, then a voltage will be induced that will try to decrease the current. On the other hand, if you decrease the current, then a voltage will be induced that will try to increase the current.
• The bottom line is: Whenever the current in a coil increases or decreases, a voltage is induced in the coil, and this induced voltage opposes the change in current.
• This is called self-inductance; as we've seen, it's the result of electromagnetism and electromagnetic induction working at the same time.
##### Inductors
• An inductor is a device designed to have a certain amount of inductance.
• Here's the schematic symbol for an inductor: • Most of the inductors in our labs look similar to this: • Typical inductors found in electronic equipment are in the microhenry (μH) or millihenry (mH) range. Recall that micro- means 10-6 and milli- means 10-3.
• ##### Winding Resistance
• Usually we treat wire as having zero resistance, but in reality wire does have some resistance. And the longer and thinner a piece of wire is, the greater its resistance.
• An inductor is simply a coiled piece of very long, very thin wire. Therefore an inductor will have some resistance, which we call the inductor's winding resistance.
• The symbol for winding resistance is RW, and it is measured in ohms.
• You can measure an inductor's winding resistance simply by connecting an ohmmeter to its two leads, just as you would measure a resistor's resistance.
• This winding resistance can be fairly large--for example, it's not unusual to have an inductor whose resistance is 50 Ω or more. But 50 Ω is still not huge, and we might be able to ignore it if the inductor is in a circuit whose resistors are much larger.
• For example, suppose you've got an inductor whose winding resistance is 50 Ω, and suppose this inductor is in a series circuit whose total resistance is 2 kΩ. Then the inductor's winding resistance is only 2.5% of the circuit's total resistance, small enough that you can probably ignore it when you're calculating the circuit's current and voltage drops.
• As a general rule of thumb, if an inductor's winding resistance is less than about 5% of the resistance that it's in series with, then you can ignore it.
• An ideal inductor has no winding resistance. In other words, RW = 0 Ω for an ideal inductor. In most of the following discussion we'll assume that inductors are ideal, but in a few places we'll mention the effect of a real inductor's winding resistance.
• ##### Inductor Types
• Inductors are classified by the materials used for their cores.
• Common core materials are air, iron, and ferrites.
• Variable inductors are also available. The schematic symbol has an arrow to show that the component's value can be adjusted: ##### Chokes and Coils
• Inductors used in high-frequency AC circuits are often called chokes, or simply coils.

##### Energy Stored in an Inductor
• Recall that resistors dissipate energy as heat, but that capacitors store energy.
• Like a capacitor, an inductor stores energy, which can later be returned to the circuit.
• An ideal inductor (with zero winding resistance) doesn't dissipate any energy as heat.
• Since RW ≠ 0 Ω for a real inductor, a real inductor does dissipate some energy as heat, but generally it's small enough to ignore. We'll return to this point later when we discuss power in an inductor.
• A capacitor stores energy in the electric field that exists between the positive and negative charges stored on its opposite plates. But an inductor stores energy in the magnetic field that is created by the current flowing through the inductor.
• The energy W stored by an inductance L is given by

W = ½ LI 2

where I is the current through the inductor.
• ##### Shortcut Rules for Inductors in Parallel
• In Unit 8 you learned two shortcut rules that you can use for special cases of parallel resistors. The same basic rules apply to inductors in parallel.
• The first shortcut rule said that if you have just two resistors in parallel, you can find their total resistance using the product-over-sum rule: RT = (R1 × R2) ÷ (R1 + R2). Similarly, if you have just two inductors in parallel, you can find their total inductance using the use the product-over-sum rule:

LT = (L1 × L2) ÷ (L1 + L2)

• The second shortcut rule applied to several parallel resistors that all have the resistance. This rule said that if you have n parallel resistors, each with resistance R, the total resistance is given by RT = R ÷ n. Similarly, if you have n parallel inductors, each with inductance L, the total inductance is given by:

LT = L÷ n

• Of course, the reciprocal formula given earlier applies to all cases of inductors in parallel, so it will give the same answer that the shortcut rules give for these special cases.
• ##### Series RL Network
• A resistor and inductor connected in series are called a series RL network. ##### DC RL Circuit
• An RL circuit is any circuit containing, in addition to a power supply, just resistors and inductors.
• In this course we'll restrict our attention to RL circuits containing DC voltage sources. We'll refer to such circuits as DC RL circuits.
• Examples: A very simple DC RL circuit just has a resistor, an inductor, and a voltage source in series: • Here's a more complicated DC RL circuit: ##### Behavior of Inductors in DC Circuits
• Just as we have rules of thumb that let us analyze the behavior of capacitors when they're fully charged or fully discharged, we have similar rules for inductors.
• Here's an important rule of thumb that you must memorize:

When an inductor with no current flowing through it is first switched into a circuit, it behaves like an open circuit.

• So to find currents and voltages in a DC RL circuit whose inductors have just been switched into the circuit, replace all inductors with open circuits. Then you'll be left with a circuit containing just a power supply and resistors, which you can analyze using the skills you learned earlier in this course.
• Here's another important rule of thumb:

When a constant, unchanging current is flowing through an ideal inductor, the inductor behaves like a short circuit.

• So to find currents and voltages in a DC RL circuit whose inductors are carrying a constant, unchanging current, replace all inductors with short circuits (in other words, with wires). Then you'll be left with a circuit containing just a power supply and resistors, which you can analyze using the skills you learned earlier in this course.
• Notice that this second rule of thumb applies to ideal inductors (with zero winding resistance). On the other hand, when a constant, unchanging current is flowing through a real inductor (with RW ≠ 0 Ω), the inductor behaves like a resistor whose resistance is equal to RW. But as we've seen, this RW is often small enough that we can ignore it and treat the inductor as a short.
##### Initial, Transient, Steady-State
• In most practical DC RL circuits, the values of current and voltage change with time as the current through each inductor changes. Typically such circuits contain a switch that is initially open, and you're interested in finding the values of voltage and current after the switch has been closed.
• To remind ourselves of this fact, we often include an open switch in schematic drawings of DC RL circuits, as in the following picture: • Just as with DC RC circuits, we distinguish three time periods in the behavior of any DC RL circuit:
1. the initial period
2. the transient period
3. the steady-state period
• During the transient period, the circuit's currents and voltages are changing from their initial values to their final (steady-state) values.
##### Initial Currents and Voltages
• The currents and voltages in a circuit at the instant when a switch is first closed are called the initial currents and initial voltages.
• In most cases, at this initial instant the circuit's inductors have no current flowing through them. Therefore, using the first rule of thumb you learned above, you'll find the circuit's initial values of voltage and current by replacing the inductors with opens.
• This is the opposite of capacitors, which initially behave like short circuits (assuming that they start out being fully discharged, which is normally the case).
• ##### Steady-State Currents & Voltages
• When the switch in a DC RL circuit has been closed for a long time, currents and voltages have reached their steady-state values.
• According to the second rule of thumb you learned above, an ideal inductor (with zero winding resistance) behaves like a short circuit in the steady state. So you find steady-state currents and voltages in an RL circuit by replacing all ideal inductors with short circuits.
• Usually we treat inductors as being ideal. But if you want to take a real inductor's winding resistance into account, then to find steady-state values you should replace the inductor with a resistor whose resistance is equal to RW, instead of replacing the inductor with a short.
• Again, this is the opposite of capacitors, which behave like open circuits in the steady state.
• ##### Transient Currents & Voltages
• When a switch is first closed (or opened) in a DC RL circuit, currents and voltages change for a short time from their initial values to their steady-state values.  This is very similar to what happens in a DC RC circuit, and the equations are also similar.
• ##### Transients: Closing Switch
• In the circuit shown below, if we close the switch at time t = 0, the current  will gradually increase from its initial value (zero) to its steady-state value (which is equal to VS÷R). • We wish to be able to calculate i, vR, and vL. In other words, we want to be able to calculate the current, the resistor's voltage drop, and the inductor's voltage drop at any particular time after the switch has been closed.
##### Time Constant
• The quantity L÷R in the equation above is called the time constant of the series RL network.  It is represented by the Greek letter τ, and it is measured in seconds (s):

τ = L ÷ R

• So we can rewrite our equation for i in a slightly simpler form:

i = (VS÷R) (1 − e−t÷τ)

• Be careful not to confuse t with τ. Remember, for any particular circuit, τ is a constant that depends on the size of the inductors and resistors, but t is a variable that represents time.
• ##### How Long to Reach Steady State?
• The time constant τ is an indicator of how long i takes to increase from zero to its steady-state value.
• Here is a useful rule of thumb:

For most practical purposes, we may assume that all quantities in a DC RL circuit have reached their steady-state values after five time constants.

• So if a circuit has a time constant of 1 millisecond, then it will take about 5 milliseconds for the circuit's currents and voltages to reach their steady-state values.
• Since one time constant is equal to L÷R, we can write this rule of thumb as an equation:

Time to reach steady state ≈ 5×L÷R

• Notice that in this equation I used a "squiggly equals sign" ≈ to indicate that this is an approximation. Actually, after five time constants the current will have risen to about 99.3% of its steady-state value. For most practical purposes, that's close enough.
• ##### Calculating vR and vL
• Let's continue our analysis of a simple series DC RL circuit. • We've seen that the current in this circuit, after the switch is closed at time t = 0, is given by the equation:

i = (VS÷R) (1 − e−t÷τ)

• In the same circuit, the voltage drop across the resistor is given by the equation:

vR = VS (1 − e −t ÷ τ)

and the voltage drop across the inductor is given by:

vL = VS e−t ÷ τ

• Don't think of these as three separate equations that you have to remember. Once you've got the equation for i, you can easily use Ohm's Law to derive the expression for vR.
• Remember, Ohm's Law says that a resistor's voltage is equal to its current times its resistance. Do you see that if you take the expression above for i and multiply it by R, you'll get the expression above for vR?
• And once you've got the equation for vR, you can easily use Kirchhoff's Voltage Law (KVL) to derive the expression for vL.
• Remember, KVL says that the sum of the voltage drops around a loop must equal the sum of the voltage rises around that loop. Applying KVL to our simple series DC RL circuit gives us VS = vR + vL. Do you see how this lets you derive the expression above for vL from the expression above for vR?
• ##### More Exponential Curves
• If we plot these equations for i, vR, and vL, we get exponential curves similar to the curves we saw in the previous unit for DC RC circuits.
• In the plots below, the values on the horizontal and vertical axes will change depending on the values of resistance, inductance, and source voltage in a particular circuit, but the shape of the curves will be the same for all series DC RL circuits.
• For example, the current in a DC RL circuit starts at 0 and rises to its final value: • Ohm's law tells us that a resistor's voltage is directly proportional at all times to its current. So we know that a graph of the resistor's voltage has the same shape as the graph of the current: • On the other hand, the inductor's voltage starts at its maximum value and then decreases to 0: • Notice again that, in each of these graphs, the values change very quickly at first, and then gradually approach a final value.
• ##### De-Energizing an Inductor
• Up to now we've been talking about energizing an inductor. Similar comments, but in reverse, apply to the case of discharging a capacitor. In this case we get what's called an inductive kick, which has some interesting practical applications , as you'll read about in this animation:
• ##### Series-Parallel Transients
• When an inductor is connected to a series-parallel resistor network, the equations given above for transient current and voltages still work, as long as you first replace the series-parallel network with its Thevenin equivalent.
• For example, in the series-parallel circuit shown below, which you will analyze in the Self-Test questions, we have an inductor connected to a network of three resistors and a voltage source. Thevenin's theorem lets you "collapse" those three resistors and the voltage source down to a single resistor and voltage source connected in series with the inductor. You can then use the equations you've learned above to analyze that "collapsed" circuit, and the results you get from this analysis will be correct for the original circuit as well. • ##### Unit 14 Review
• This e-Lesson has covered several important topics, including:
• inductance
• types of inductors
• energy stored in an inductor
• inductors in series, in parallel, and in series-parallel
• calculating initial values, steady-state values, and transient values in a DC RL circuit
• time constant of a DC RL circuit.
• To finish the e-Lesson, take this self-test to check your understanding of these topics.
• Congratulations! You've completed the e-Lesson for this unit. This concludes your study of DC circuits. Congratulations on making it through to the end! For a good review, I suggest that you go back and re-take each of the Unit Review self-tests (located at the end of the e-Lessons).

Of course, there's plenty more to learn. In this course we've concentrated on analyzing circuits that contain DC voltage sources and DC current sources. In later courses, you'll learn about circuits with AC sources instead of (or in addition to) DC sources. To get a head start on these topics, take a look at the material that you'll study in EET 1155.