The inductor is one of the ideal circuit elements. We learn more about how an inductor behaves by looking closely at its $i$-$v$ equation.

Written by Willy McAllister.

### Contents

### Where we’re headed

The derivative form and integral form of the inductor $i$-$v$ equation are,

$v = \text L\,\dfrac{di}{dt}\qquad$ and $\quad \displaystyle i = \dfrac1{\text L}\, \int_{\,0}^{\,T} v\,\text dt + i_0$

- We learn why current in an inductor does not change instantaneously.
- We learn when an inductor acts like a short circuit (if its current is constant).

### Inductor $i$-$v$ equations

$v = \text L\,\dfrac{di}{dt}\quad$ and $\quad\displaystyle i = \dfrac1{\text L}\, \int_{\,0}^{\,T} v\,\text dt + i_0$

These are the derivative form and integral form of the inductor equations.

$\text L$ is the *inductance*, a physical property of the inductor.

$\text L$ is the scale factor between $v$ and $di/dt$.

$\text L$ tells you how much $v$ gets generated for a given amount of $di/dt$.

$i_0$ is the initial current flowing in the inductor, at $t=0$.

### Inductor voltage is proportional to the *rate of change* of current

When we learned about resistors, Ohm’s Law told us the voltage across a resistor is proportional to the current through the resistor: $v = \text R\,i$.

Now we have an inductor with its $i$-$v$ equation: $v = \text L\,\dfrac{di}{dt}$.

This tells us the voltage across the inductor is proportional not to current but rather the *rate of change* of current through the inductor. That’s what $di/dt$ means.

For real-world resistors, we have to take care that voltage and current don’t get too big for the resistor to handle. For real-world inductors, we have to be careful the *rate of change of current* does’t get too big for the inductor to handle. This can be tricky. It is very easy to create a big change of current when you open or close a switch. In the article on inductor kickback we design for this situation.

### Inductor and current source

Now we look at a few different simple circuits to get a feel for what the inductor $i$-$v$ equations mean. Along the way we will see how the equations teach us some simple rules of thumb about the inductor.

The first example is an inductor connected to an ideal current source.

The current source provides a constant current to the inductor, $i = \text I$.

For example, let $i = 2 \,\text{mA}$.

**What is the voltage across the inductor?**

The inductor equation tells us:

$v = \text L\,\dfrac{di}{dt}$

The current source provides a constant current of $2\,\text{mA}$. That’s interesting, but what we really need to know is, what is the rate of change of the current?

$\dfrac{di}{dt} = \dfrac{d2}{dt} = 0\qquad$ (everybody knows $2$ doesn’t change with time)

Therefore, the voltage across the inductor is:

$v = \text L\cdot 0 = 0$

If current is constant in an inductor, then $v = \text L\,di/dt = 0$. Zero volts appear across the inductor. This is true for any value of current and any value of inductor.

### Simulation model

Here’s a simulation model of a $5\,\mu\text H$ inductor with constant $2\,\text{mA}$ current source. Open the link and click on TRANS in the top menu bar to run a transient simulation. The result is pretty boring. The voltage across the inductor is $0$. Change the inductor or current to anything you want, the answer is always $0$.

### Sometimes an inductor “looks like” a short

It’s very popular to paint mental pictures of an inductor by saying it “looks like” something. Here’s our first example.

When the current is constant, the voltage difference between the ends of an inductor is $0\,\text V$. In this condition, an inductor acts just like an ideal wire. (An ideal wire has $0\,\text V$ between its ends no matter what.)

An inductor “looks like” a short circuit when its current is constant.

### Inductor and voltage source

Now let’s the current source to an ideal constant voltage source and see what the $i$-$v$ equation predicts. Let’s get specific and say $\text V = 3\,\text V$ and $\text L = 10\,\text{mH}$.

If we put these values into the inductor equation we get,

$v = \text L\,\dfrac{di}{dt}$

$3 = 10 \,\text{mH} \, \dfrac{di}{dt}$

or, solving for $di/dt$,

$\dfrac{di}{dt}= \dfrac{3}{10 \times 10^{-3}} = \,\text{amperes}/\text{sec}$

That means the current through the inductor will have a rising slope of $\,\text{amperes}/\text{second}$.

That is kind of amazing, but that’s what the equation predicts. Needless to say, this is not a practical circuit. We just want see what happens with a constant voltage. If we build this circuit the current would ramp up until our real-world voltage source couldn’t keep up with the demand for more current. But over a short time span, this is how real inductors work.

### Simulation model

Here’s a simulation model of the inductor with voltage source circuit. Open the link and click on TRANS in the top menu to run a transient simulation.

### Example

We can actually come up with something more useful than an infinitely-rising current ramp. If we change the voltage source so it reverses direction every once in a while, we get a more interesting and potentially useful circuit.

Here’s a challenge for you: Assume the square wave amplitude is $\pm1\,\text V$ and the frequency is $1\,\text{MHz}$ $($has a period of $1\,\mu\text{sec})$.

**Sketch the shape of the current waveform and find the peak values.**

See if you can do this before looking at the answer or simulation model.

Hint: Work out $di/dt$ for the two different states of the input voltage.

### Simulation model

Simulation model of the pulsed voltage source circuit. Open the link and click on TRANS to run a transient simulation.

The inductor is “integrating” the voltage over time, as indicated by the value of the current at any moment. This is an exact “dual” of a capacitor integrating current.

### Explore

What happens to the current if you …?

- Change the value of the inductor by a little bit $(2\times)$ and by a lot $(\times)$.
- Change the amplitude or frequency of the voltage source.
- Change the voltage levels so they are not symmetric.
- Change the voltage duty cycle from $50\%$ to another value, like $40\%$.

### Analogy to mass

This is the most useful mental image to have when looking at an inductor.

Inductance, $\text L$, is analogous to mass or inertia in a mechanical system. The energy in the magnetic field of an inductor doesn’t allow the current to change instantaneously, just like the heavy mass of the car tends to resist changes in velocity. A car cannot start or stop instantaneously. It takes time to accelerate or brake. An inductor is basically the electrical version of Newton’s First Law of Motion (also called the Law of Inertia): A body in motion tends to stay in motion. For an inductor it goes like this: A current in motion tends to stay in motion.

The current in an inductor does not (will not) change instantaneously.

### Summary

The current in an inductor does not change instantaneously.

When its current is constant, an inductor looks like a short circuit.

Be careful making circuits with an inductor. A sudden changed in current, like a switch thrown open, breaking a current path, that means the derivative of current, $di/dt$, can become very large. The inductor equation tells us there can be a large voltage generated across the inductor.

One way to deal with potentially destructive inductor voltage is to design a path for the current, so you don’t get a large $di/dt$. We showed how to add a diode to provide a current path and clamp the inductor voltage to an acceptable value when a switch was thrown open.

### Inductance

Induction is the process in which an emf is induced by changing magnetic flux, such as a change in the current of a conductor.

### Learning Objectives

Describe properties of an inductor

### Key Takeaways

#### Key Points

- In the case of electronics, inductance is the property of a conductor by which a change in current in the conductor creates a voltage in both the conductor itself, called self-inductance, and any nearby conductors, called mutual inductance.
- From Lenz&#;s law, a changing electric current through a circuit that has inductance induces a proportional voltage which opposes the change in current.
- Mutual inductance is illustrated by. A change in the current
*I*in one device, coil 1 in the figure, induces an_{1}*emf*in the other. We express this in equation form as [latex]\text{emf}_2=-\text{M} \frac{\Delta \text{I}_1}{\Delta \text{t}}[/latex]. M is the same for the reverse process._{2} - Self-inductance is the effect of Faraday&#;s law of induction of a device on itself. The induced emf is related to the physical geometry of the device and the rate of change of current given by [latex]\text{emf}=-\text{L} \frac{\Delta \text{I}}{\Delta \text{t}}[/latex].
- A device that exhibits significant self-inductance is called an inductor, and given the symbol in.

#### Key Terms

**mutual inductance**: The ratio of the voltage in a circuit to the change in current in a neighboring circuit.**self-inductance**: The ratio of the voltage to the change in current in the same circuit.**inductor**: A passive device that introduces inductance into an electrical circuit.

### Inductance

### OVERVIEW

Induction is the process in which an emf is induced by changing magnetic flux. Specifically in the case of electronics, inductance is the property of a conductor by which a change in current in the conductor creates a voltage in both the conductor itself (self-inductance) and any nearby conductors (mutual inductance). This effect derives from two fundamental observations of physics: First, that a steady current creates a steady magnetic field and second, that a time-varying magnetic field induces a voltage in a nearby conductor (Faraday&#;s law of induction). From Lenz&#;s law, a changing electric current through a circuit that has inductance induces a proportional voltage which opposes the change in current (if this wasn&#;t true one can easily see how energy could not be conserved, with a changing current reinforcing the change in a positive feedback loop).

### MUTUAL INDUCTANCE

Mutual inductance is the effect of Faraday&#;s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See, where simple coils induce emfs in one another.

**Mutual Inductance in Coils**: These coils can induce emfs in one another like an inefficient transformer. Their mutual inductance M indicates the effectiveness of the coupling between them. Here a change in current in coil 1 is seen to induce an emf in coil 2. (Note that &#;E2 induced&#; represents the induced emf in coil 2. )

In the many cases where the geometry of the devices is fixed, flux is changed by varying current. We therefore concentrate on the rate of change of current, Δ*I*/Δ*t*, as the cause of induction. A change in the current* I _{1}* in one device, coil 1 in the figure, induces an

*emf*in the other. We express this in equation form as

_{2}[latex]\text{emf}_2=-\text{M} \frac{\Delta \text{I}_1}{\Delta \text{t}}[/latex]

where* M* is defined to be the mutual inductance between the two devices. The minus sign is an expression of Lenz&#;s law. The larger the mutual inductance *M*, the more effective the coupling. Units for *M* are (V⋅s)/A=Ωs, which is named a henry (H), after Joseph Henry (discovered of self-inductance). That is, 1 H=1Ωs.

Nature is symmetric here. If we change the current* I _{2}* in coil 2, we induce an

*emf*in coil 1, which is given by

_{1}[latex]\text{emf}_1=-\text{M} \frac{\Delta \text{I}_2}{\Delta \text{t}}[/latex]

where *M* is the same as for the reverse process. Transformers run backward with the same effectiveness, or mutual inductance *M.*

A large mutual inductance *M* may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance* M* is to counterwind coils to cancel the magnetic field produced. (See ).

**Counterwinding**: The heating coils of an electric clothes dryer can be counter-wound so that their magnetic fields cancel one another, greatly reducing the mutual inductance with the case of the dryer.

### SELF-INDUCTANCE

Self-inductance, the effect of Faraday&#;s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also increase, inducing a counter emf, as required by Lenz&#;s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a fixed geometry, and so the change in flux is due entirely to the change in current *ΔI* through the device. The induced emf is related to the physical geometry of the device and the rate of change of current. It is given by

[latex]\text{emf}=-\text{L} \frac{\Delta \text{I}}{\Delta \text{t}}[/latex]

where *L* is the self-inductance of the device. A device that exhibits significant self-inductance is called an inductor, and given the symbol in.

**Inductor Symbol**

The minus sign is an expression of Lenz&#;s law, indicating that emf opposes the change in current. Units of self-inductance are henries (H) just as for mutual inductance. The larger the self-inductance *L* of a device, the greater its opposition to any change in current through it. For example, a large coil with many turns and an iron core has a large *L* and will not allow current to change quickly. To avoid this effect, a small L must be achieved, such as by counterwinding coils as in.

### SOLENOIDS

It is possible to calculate* L* for an inductor given its geometry (size and shape) and knowing the magnetic field that it produces. This is difficult in most cases, because of the complexity of the field created. The inductance *L* is usually a given quantity. One exception is the solenoid, because it has a very uniform field inside, a nearly zero field outside, and a simple shape. The self-inductance of a solenoid of cross-sectional area A and length ℓ is

[latex]\text{L}=\frac{\mu _0 \text{N}^{2} \text{A} }{\mathscr{\text{l}}}[/latex](solenoid).

It is instructive to derive this equation, but this is left as an exercise to the reader. (Hint: start by noting that the induced emf is given by Faraday&#;s law of induction as emf=−N(Δ/Δt) and, by the definition of self-inductance is given as as emf=−L(ΔI//Δt) and equate these two expressions). Note that the inductance depends only on the physical characteristics of the solenoid, consistent with its definition.

### RL Circuits

An RL circuit consists of an inductor and a resistor, in series or parallel with each other, with current driven by a voltage source.

### Learning Objectives

Describe current-voltage relationship in the RL circuit and calculate energy that can be stored in an inductor

### Key Takeaways

#### Key Points

- The energy stored in an inductor is [latex]\text{E}=\frac{1}{2}\text{LI}^{2}[/latex]. It takes time to build up stored energy in a conductor and time to deplete it.
- When a resistor and an inductor in series are connected to a voltage source, the time-dependent current is given by [latex]\text{I}=\text{I}_{0}(1-\text{e}^{\frac{-\text{t}}{\tau}})[/latex]. The final current after a long time is [latex]\text{I}_0[/latex].
- The characteristic time constant is given by [latex]\tau=\frac{\text{L}}{\text{R}}[/latex], where R is resistance and L is inductance. This represents the time necessary for the current in a circuit just closed to go from zero to [latex]\cdot \text{I}_0[/latex].
- When the voltage source is disconnected from the inductor, the current will decay according to [latex]\text{I}=\text{I}_{0}\text{e}^{\frac{-\text{t}}{\tau}}[/latex]. In the first time interval τ the current falls by a factor of [latex]\frac{1}{\text{e}}[/latex] to [latex]\cdot \text{I}_0[/latex].

#### Key Terms

**characteristic time constant**: Denoted by $\tau$, in RL circuits it is given by $ \tau=\frac{L}{R}$ where R is resistance and L is inductance. When a switch is closed, it is the time it takes for the current to decay by a factor of 1/e.**inductor**: A device or circuit component that exhibits significant self-inductance; a device which stores energy in a magnetic field.

### RL Circuits

A resistor-inductor circuit (RL circuit) consists of a resistor and an inductor (either in series or in parallel ) driven by a voltage source.

### Review

Recall that induction is the process in which an emf is induced by changing magnetic flux. Mutual inductance is the effect of Faraday&#;s law of induction for one device upon another, while self-inductance is the the effect of Faraday&#;s law of induction of a device on itself. An inductor is a device or circuit component that exhibits self-inductance.

### Energy of an Inductor

We know from Lenz&#;s law that inductors oppose changes in current. We can think of this situation in terms of energy. Energy is stored in a magnetic field. It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic field is directly proportional to current and to the inductance of the device. It can be shown that the energy stored in an inductor E_{ind} is given by:

[latex]\text{E}_{\text{ind}}=\frac{1}{2}\text{LI}^{2}[/latex].

### Inductors in Circuits

We know that the current through an inductor L cannot be turned on or off instantaneously. The change in current changes the magnetic flux, inducing an emf opposing the change (Lenz&#;s law). How long does the opposition last? Current *will* flow and *can* be turned off, but how long does it take? The following figure shows a switching circuit that can be used to examine current through an inductor as a function of time.

**Current in an RL Circuit**: (a) An RL circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.

When the switch is first moved to position 1 (at *t=0*), the current is zero and it eventually rises to I_{0}=*V*/*R*, where *R* is the total resistance of the circuitand V is the battery &#;s voltage. The opposition of the inductor *L* is greatest at the beginning, because the change in current is greatest at that time. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. This is the hallmark of an exponential behavior, and it can be shown (with calculus) that

[latex]\text{I}=\text{I}_{0}(1-\text{e}^{\frac{-\text{t}}{\tau }})[/latex]

is the current in an *RL* circuit when switched on. (Note the similarity to the exponential behavior of the voltage on a charging capacitor.) The initial current is zero and approaches I_{0}=*V*/*R* with a characteristic time constant for an *RL* circuit, given by:

[latex]\tau=\frac{\text{L}}{\text{R}}[/latex],

where [latex]\tau[/latex] has units of seconds, since [latex]1\text{H}=1\Omega\cdot \text{s}[/latex]. In the first period of time [latex]\tau[/latex], the current rises from zero to I_{0}, since I=I_{0}(1−e^{−1})=I_{0}(1−)=I_{0}. The current will be of the remainder in the next time. A well-known property of the exponential function is that the final value is never exactly reached, but of the remainder to that value is achieved in every characteristic time [latex]\tau[/latex]. In just a few multiples of the time [latex]\tau[/latex], the final value is very nearly achieved (see part (b) of above figure).

The characteristic time [latex]\tau[/latex] depends on only two factors, the inductance* L* and the resistance *R*. The greater the inductance *L*, the greater it is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance R, the greater [latex]\tau[/latex] is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases (large *L* and small *R)* more energy is stored in the inductor and more time is required to get it in and out.

When the switch in (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. However, this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, (1/2)LI_{0}^{2}, stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is I^{2}R. Once again the behavior is exponential, and I is found to be

[latex]\text{I}=\text{I}_{0}\text{e}^{\frac{-\text{t}}{\tau}}[/latex]

In (c), in the first period of time [latex]\tau=\text{L}/\text{R}[/latex] after the switch is closed, the current falls to of its initial value, since I=I_{0}e^{−1}=I_{0}. In each successive time [latex]\tau[/latex], the current falls to of the preceding value, and in a few multiples of [latex]\tau[/latex], the current becomes very close to zero.

### RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram

Response of an RLC circuit depends on the driving frequency—at large enough frequencies, inductive (capacitive) term dominates.

### Learning Objectives

Distinguish behavior of RLC series circuits as large and small frequencies

### Key Takeaways

#### Key Points

- RLC circuits can be described by the (generalized) Ohm &#;s law. As for the phase, when a sinusoidal voltage is applied, the current lags the voltage by a 90º phase in a circuit with an inductor, while the current leads the voltage by 90∘ in a circuit with a capacitor.
- At large enough frequencies [latex](\nu \gg \frac{1}{\sqrt{2\pi \text{LC}}})[/latex], the circuit is almost equivalent to an AC circuit with just an inductor. Therefore, the rms current will be Vrms/XL, and the current lags the voltage by almost 90∘.
- At small enough frequencies [latex](\nu \ll \frac{1}{\sqrt{2\pi \text{LC}}})[/latex], the circuit is almost equivalent to an AC circuit with just a capacitor. Therefore, the rms current will be given as V
_{rms}/X_{C,}and the current leads the voltage by almost 90^{∘}.

#### Key Terms

**Lenz&#;s law**: A law of electromagnetic induction that states that an electromotive force, induced in a conductor, is always in such a direction that the current produced would oppose the change that caused it; this law is a form of the law of conservation of energy.**resonance**: The increase in the amplitude of an oscillation of a system under the influence of a periodic force whose frequency is close to that of the system&#;s natural frequency.**rms**: Root mean square: a statistical measure of the magnitude of a varying quantity.

In previous Atoms we learned how an RLC series circuit, as shown in, responds to an AC voltage source. By combining Ohm&#;s law (I_{rms}=V_{rms}/Z; I_{rms} and V_{rms} are rms current and voltage) and the expression for impedance Z, from:

**Series RLC Circuit**: A series RLC circuit: a resistor, inductor and capacitor (from left).

[latex]\text{Z} = \sqrt{\text{R}^2 + (\text{X}_\text{L} - \text{X}_\text{C})^2}[/latex][latex](\text{X}_\text{L} = 2\pi \nu \text{L}, \text{X}_\text{C} = \frac{1} {2\pi \nu \text{C}})[/latex],

we arrived at: [latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\sqrt{\text{R}^2 + (\text{X}_\text{L} - \text{X}_\text{C})^2}}[/latex].

From the equation, we studied resonance conditions for the circuit. We also learned the phase relationships among the voltages across resistor, capacitor and inductor: when a sinusoidal voltage is applied, the current lags the voltage by a 90º phase in a circuit with an inductor, while the current leads the voltage by 90^{∘} in a circuit with a capacitor. Now, we will examine the system&#;s response at limits of large and small frequencies.

### At Large Frequencies

At large enough frequencies [latex](\nu \gg \frac{1}{\sqrt{2\pi \text{LC}}})[/latex], X_{L} is much greater than X_{C}. If the frequency is high enough that X_{L} is much larger than R as well, the impedance Z is dominated by the inductive term. When [latex]\text{Z} \approx \text{X}_\text{L}[/latex], the circuit is almost equivalent to an AC circuit with just an inductor. Therefore, the rms current will be V_{rms}/X_{L}, and the current lags the voltage by almost 90^{∘}. This response makes sense because, at high frequencies, Lenz&#;s law suggests that the impedance due to the inductor will be large.

### At Small Frequencies

The impedance Z at small frequencies [latex](\nu \ll \frac{1}{\sqrt{2\pi \text{LC}}})[/latex] is dominated by the capacitive term, assuming that the frequency is high enough so that X_{C} is much larger than R. When [latex]\text{Z} \approx \text{X}_\text{C}[/latex], the circuit is almost equivalent to an AC circuit with just a capacitor. Therefore, the rms current will be given as V_{rms}/X_{C,} and the current leads the voltage by almost 90^{∘}.

### Resistors in AC Circuits

In a circuit with a resistor and an AC power source, Ohm&#;s law still applies (*V *= *IR*).

### Learning Objectives

Apply Ohm&#;s law to determine current and voltage in an AC circuit

### Key Takeaways

#### Key Points

- With an AC voltage given by: [latex]\text{V} = \text{V}_0 \sin(2\pi \nu \text{t})[/latex] the current in the circuit is given as: [latex]\text{I} = \frac{\text{V}_0}{\text{R}} \sin(2\pi \nu \text{t})[/latex] This expression comes from Ohm &#;s law: [latex]\text{V}=\text{IR}[/latex].
- Most common applications use a time-varying voltage source instead of a DC source. Examples include the commercial and residential power that serves so many of our needs.
- Power dissipated by the AC circuit with a resistor in the example is: [latex]\text{P} = \frac{\text{V}_0^2}{\text{R}} \cdot \sin(2\pi \nu \text{t})[/latex] Therefore, average AC power is: [latex]\frac{\text{V}_0^2}{2\text{R}}[/latex].

#### Key Terms

**Ohm&#;s law**: Ohm&#;s observation is that the direct current flowing in an electrical circuit consisting only of resistances is directly proportional to the voltage applied.

Direct current (DC) is the flow of electric charge in only one direction. It is the steady state of a constant-voltage circuit. Most well known applications, however, use a time-varying voltage source. Alternating current (AC) is the flow of electric charge that periodically reverses direction. If the source varies periodically, particularly sinusoidally, the circuit is known as an alternating-current circuit. Examples include the commercial and residential power that serves so many of our needs. shows graphs of voltage and current versus time for typical DC and AC power. The AC voltages and frequencies commonly used in homes and businesses vary around the world.

**Sinusoidal Voltage and Current**: (a) DC voltage and current are constant in time, once the current is established. (b) A graph of voltage and current versus time for Hz AC power. The voltage and current are sinusoidal and are in phase for a simple resistance circuit. The frequencies and peak voltages of AC sources differ greatly.

We have studied Ohm&#;s law:

[latex]\text{I} = \frac{\text{V}}{\text{R}}[/latex]

where *I* is the current, *V* is the voltage, and *R* is the resistance of the circuit. Ohm&#;s law applies to AC circuits as well as to DC circuits. Therefore, with an AC voltage given by:

[latex]\text{V} = \text{V}_0 \sin(2\pi \nu \text{t})[/latex]

where *V*_{0} is the peak voltage and [latex]\nu[/latex] is the frequency in hertz, the current in the circuit is given as:

[latex]\text{I} = \frac{\text{V}_0}{\text{R}} \sin(2\pi \nu \text{t})[/latex]

In this example, in which we have a resistor and the voltage source in the circuit, the voltage and current are said to be in phase, as seen in (b). Current in the resistor alternates back and forth without any phase difference, just like the driving voltage.

Consider a perfect resistor that brightens and dims times per second as the current repeatedly goes through zero. (A Hz flicker is too rapid for your eyes to detect. ) The fact that the light output fluctuates means that the power is fluctuating. Since the power supplied is *P* = *IV*, if we use the above expressions for *I* and *V*, we see that the time dependence of power is:

[latex]\text{P} = \frac{\text{V}_0^2}{\text{R}} \cdot \sin(2\pi \nu \text{t})[/latex]

To find the average power consumed by this circuit, we need to take the time average of the function. Since:

[latex]\frac{1}{\pi}\int_{0}^{\pi}\sin^2(\text{x})\text{dx} = \frac{1}{2}[/latex]

we see that:

[latex]\text{P}_{\text{avg}} = \frac{\text{V}_0^2}{2 \text{R}}[/latex]

### Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams

The voltage across a capacitor lags the current. Due to the phase difference, it is useful to introduce phasors to describe these circuits.

### Learning Objectives

Explain the benefits of using a phasor representation

### Key Takeaways

#### Key Points

- When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current.
- If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will &#;peak&#; at the same time.
- The rms current in the circuit containing only a capacitor C is given by another version of Ohm &#;s law to be [latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\text{X}_\text{C}}[/latex], where [latex]\text{X}_\text{c}[/latex] is the capacitive reactance.

#### Key Terms

**rms**: Root mean square: a statistical measure of the magnitude of a varying quantity.

In the previous Atom on &#;Resistors in AC Circuits&#;, we introduced an AC power source and studied how resistors behave in AC circuits. There, we used the Ohm&#;s law (V=IR) to derive the relationship between voltage and current in AC circuits. In this and following Atoms, we will generalize the Ohm&#;s law so that we can use it even when we have capacitors and inductors in the circuit. To get there, we will first introduce a very general, pictorial way of representing a sinusoidal wave, using phasor.

**Capacitors in AC Circuits with Phasors**

### Phasor

The key idea in the phasor representation is that a complex, time-varying signal may be represented as the product of a complex number (that is independent of time) and a complex signal (that is dependent on time). Phasors separate the dependencies on A (amplitude), [latex]\nu[/latex] ( frequency ), and θ ( phase ) into three independent factors. This can be particularly useful because the frequency factor (which includes the time-dependence of the sinusoid) is often common to all the components of a linear combination of sinusoids. In those situations, phasors allow this common feature to be factored out, leaving just the A and θ features. For example, we can represent [latex]\text{A}\cdot \cos(2\pi \nu \text{t} + \theta)[/latex] simply as a complex constant, [latex]\text{A} \text{e}^{\text{i}\theta}[/latex]. Since phasors are represented by a magnitude (or modulus) and an angle, it is pictorially represented by a rotating arrow (or a vector) in x-y plane.

**Fig 3**: A phasor can be seen as a vector rotating about the origin in a complex plane. The cosine function is the projection of the vector onto the real axis. Its amplitude is the modulus of the vector, and its argument is the total phase \omega t+\theta. The phase constant \theta represents the angle that the vector forms with the real axis at t = 0.

### Capacitors in AC circuits

Earlier in a previous Atom, we studied how the voltage and the current varied with time. If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will &#;peak&#; at the same time. We say that the current and voltage are in phase.

When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current. The current has its maximum (it peaks) one quarter of a cycle before the voltage peaks. Engineers say that the &#;current leads the voltage by 90^{∘}&#;. This is shown in.

**Fig 2**: The current peaks (has its maximum) one quarter of a wave before the voltage when a capacitor is connected to an alternating voltage.

For a circuit with a capacitor, the instantaneous value of V/I is not constant. However, the value of V_{max}/I_{max} is useful, and is called the capacitive reactance (X_{C}) of the component. Because it is still a voltage divided by a current (like resistance ), its unit is the ohm. The value of X_{C} (C standing for capacitor) depends on its capacitance (C) and the frequency (f) of the alternating current. [latex]\text{X}_\text{C} = \frac{1} {2\pi \nu \text{C}}[/latex].

The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current I_{rms} in the circuit containing only a capacitor C is given by another version of Ohm&#;s law to be [latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\text{X}_\text{C}}[/latex], where V_{rms} is the rms voltage. Note that X_{C} replaces R in the DC version of the Ohm&#;s law.

### Phase representation

Since the voltage across a capacitor lags the current, the phasor representing the current and voltage would be give as in. In the diagram, the arrows rotate in counter-clockwise direction at a frequency [latex]\nu[/latex]. (Therefore, current leads voltage. ) In the following Atoms, we will see how these phasors can be used to analyze RC, RL, LC, and RLC circuits.

**Fig 4**: Phasor diagram for an AC circuit with a capacitor

### Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams

In an AC circuit with an inductor, the voltage across an inductor &#;leads&#; the current because of the Lenz&#; law.

### Learning Objectives

Explain why the voltage across an inductor &#;leads&#; the current in an AC circuit with an inductor

### Key Takeaways

#### Key Points

- With an inductor in an AC circuit, the voltage leads the current by one-fourth of a cycle, or by a 90º phase angle.
- The rms current I
_{rms}through an inductor L is given by a version of Ohm &#;s law: [latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\text{X}_\text{L}}[/latex]. X_{L}is called the inductive reactance, given as [latex]\text{X}_\text{L} = 2\pi \nu \text{L}[/latex]. - Phasors are vectors rotating in counter-clockwise direction. A phasor for an inductor shows that the voltage lead the current by a 90º phase.

#### Key Terms

**Lenz&#;s law**: A law of electromagnetic induction that states that an electromotive force, induced in a conductor, is always in such a direction that the current produced would oppose the change that caused it; this law is a form of the law of conservation of energy.**rms**: Root mean square: a statistical measure of the magnitude of a varying quantity.**phasor**: A representation of a complex number in terms of a complex exponential.

Suppose an inductor is connected directly to an AC voltage source, as shown in. It is reasonable to assume negligible resistance because in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. The graph shows voltage and current as functions of time. (b) starts with voltage at a maximum. Note that the current starts at zero, then rises to its peak after the voltage driving it (as seen in the preceding section when DC voltage was switched on).

**AC Voltage Source in Series with an Inductor**: (a) An AC voltage source in series with an inductor having negligible resistance. (b) Graph of current and voltage across the inductor as functions of time.

When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c where it begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. Hence, when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a 90º phase angle.

Current lags behind voltage, since inductors oppose change in current. Changing current induces an emf. This is considered an effective resistance of the inductor to AC. The rms current I_{rms} through an inductor L is given by a version of Ohm&#;s law: [latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\text{X}_\text{L}}[/latex] where V_{rms} is the rms voltage across the inductor and [latex]\text{X}_\text{L} = 2\pi \nu \text{L}[/latex] with [latex]\nu[/latex] the frequency of the AC voltage source in hertz. X_{L} is called the inductive reactance. Because the inductor reacts to impede the current, X_{L} has units of ohms (1 H=1 Ωs, so that frequency times inductance has units of (cycles/s)(Ωs)=Ω), consistent with its role as an effective resistance.

### Phasor Representation

The voltage across an inductor &#;leads&#; the current because of the Lenz&#;s law. Therefore, the phasor representing the current and voltage would be given as in. Again, the phasors are vectors rotating in counter-clockwise direction at a frequency [latex]\nu[/latex] (you can see that the voltage leads the current). Subsequent Atoms will discuss how these phasors can be used to analyze RC, RL, LC, and RLC circuits.

**Phasor Diagram**: Phasor diagram for an AC circuit with an inductor.

**Phasors for Inductors in AC Circuits**

### Resonance in RLC Circuits

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies—in an RLC series circuit, it occurs at [latex]\nu_0 = \frac{1}{2\pi\sqrt{\text{LC}}}[/latex].

### Learning Objectives

Compare resonance characteristics of higher- and lower-resistance circuits

### Key Takeaways

#### Key Points

- Resonance condition of an RLC series circuit can be obtained by equating X
_{L}and X_{C}, so that the two opposing phasors cancel each other. - At resonance, the effects of the inductor and capacitor cancel, so that Z=R, and I
_{rms}is a maximum. - Higher- resistance circuits do not resonate as strongly compared to lower-resistance circuits, nor would they be as selective in, for example, a radio receiver.

#### Key Terms

**reactance**: The opposition to the change in flow of current in an alternating current circuit, due to inductance and capacitance; the imaginary part of the impedance.**rms**: Root mean square: a statistical measure of the magnitude of a varying quantity.**impedance**: A measure of the opposition to the flow of an alternating current in a circuit; the aggregation of its resistance, inductive and capacitive reactance. Represented by the symbol Z.

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system&#;s resonance frequencies. To study the resonance in an RLC circuit, as illustrated below, we can see how the circuit behaves as a function of the frequency of the driving voltage source.

**RLC Series Circuit**: An RLC series circuit with an AC voltage source. f is the frequency of the source.

Combining Ohm &#;s law, I_{rms}=V_{rms}/Z, and the expression for impedance Z from

[latex]\text{Z} = \sqrt{\text{R}^2 + (\text{X}_\text{L} - \text{X}_\text{C})^2}[/latex] gives

[latex]\text{I}_{\text{rms}} = \frac{\text{V}_{\text{rms}}}{\sqrt{\text{R}^2 + (\text{X}_\text{L} - \text{X}_\text{C})^2}}[/latex],

where I_{rms} and V_{rms} are rms current and voltage, respectively. The reactances vary with frequency [latex]\nu[/latex], with X_{L} large at high frequencies and X_{C} large at low frequencies given as:

[latex]\text{X}_\text{L} = 2\pi \nu \text{L}, \text{X}_\text{C} = \frac{1} {2\pi \nu \text{C}}[/latex].

At some intermediate frequency [latex]\nu_0[/latex], the reactances will be equal and cancel, giving Z=R —this is a minimum value for impedance, and a maximum value for I_{rms} results. We can get an expression for [latex]\nu_0[/latex] by taking X_{L}=X_{C}. Substituting the definitions of X_{L} and X_{C} yields:

[latex]\nu_0 = \frac{1}{2\pi \sqrt{\text{LC}}}[/latex].

[latex]\nu_0[/latex] is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At [latex]\nu_0[/latex], the effects of the inductor and capacitor cancel, so that Z=R, and I_{rms} is a maximum. Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined as a forced oscillation (in this case, forced by the voltage source) at the natural frequency of the system.

The receiver in a radio is an RLC circuit that oscillates best at its [latex]\nu_0[/latex]. A variable capacitor is often used to adjust the resonance frequency to receive a desired frequency and to reject others. is a graph of current as a function of frequency, illustrating a resonant peak in I_{rms} at [latex]\nu_0 = \text{f}_0[/latex]. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus higher-resistance circuits do not resonate as strongly, nor would they be as selective in, for example, a radio receiver.

**Current vs. Frequency**: A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at f0, but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude V0.

### Power

Power delivered to an RLC series AC circuit is dissipated by the resistance in the circuit, and is given as [latex]\text{P}_{\text{avg}} = \text{I}_{\text{rms}} \text{V}_{\text{rms}} \cos{\phi}[/latex]. Here, [latex]\phi[/latex] is called the phase angle.

### Learning Objectives

Calculate the power delivered to an RLC-series AC circuit given the current and the voltage

### Key Takeaways

#### Key Points

- Phase angle ϕ is the phase difference between the source voltage V and the current I. See the phasor diagram in.
- At the resonant frequency or in a purely resistive circuit Z=R, so that cosϕ=1. This implies that ϕ=0º and that voltage and current are in phase.
- Average power dissipated in an RLC circuit can be calculated by taking a time average of power, P(t) = I(t)V(t), over a period.

#### Key Terms

**rms**: Root mean square: a statistical measure of the magnitude of a varying quantity.

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. However, the average power is not simply current times voltage, as is the case in purely resistive circuits. As seen in previous Atoms, voltage and current are out of phase in an RLC circuit. There is a phase angle ϕ between the source voltage V and the current I, given as

[latex]\cos{\phi} = \frac{\text{R}}{\text{Z}}[/latex], as diagramed in.

**Phasor Diagram for an RLC Series Circuit**: Phasor diagram for an RLC series circuit. \phi is the phase angle, equal to the phase difference between the voltage and current.

For example, at the resonant frequency [latex](\nu_0 = \frac{1}{2\pi \sqrt{\text{LC}}})[/latex] or in a purely resistive circuit, Z=R, so that cosϕ=1. This implies that ϕ=0º and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance, because voltage and current are out of phase and I_{rms} is lower.

The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is

[latex]\text{P}_{\text{avg}} = \text{I}_{\text{rms}} \text{V}_{\text{rms}} \cos{\phi}[/latex]

(an equation derived by taking a time average of power, P(t) = I(t)V(t), over a period. I(t) and V(t) are current and voltage at time t). Thus cosϕ is called the power factor, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, cosϕ=1.

Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output, but do not dissipate energy out of the circuit. Rather, they transfer energy back and forth to one another, with the resistor dissipating the exact amount that the voltage source gives the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor (such as radio waves).

The circuit is analogous to the wheel of a car driven over a corrugated road, as seen in. The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels&#; motion is a maximum if the bumps in the road are hit at the resonant frequency.

**Forced Damped Motion of a Wheel on a Car Spring**: The forced but damped motion of the wheel on the car spring is analogous to an RLC series AC circuit. The shock absorber damps the motion and dissipates energy, analogous to the resistance in an RLC circuit. The mass and spring determine the resonant frequency.

## Inductor

For inductors whose magnetic properties rather than electrical properties matter, see electromagnet.

Passive two-terminal electrical component that stores energy in its magnetic field

An **inductor**, also called a **coil**, **choke**, or **reactor**, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it.^{[1]} An inductor typically consists of an insulated wire wound into a coil.

When the current flowing through the coil changes, the time-varying magnetic field induces an electromotive force (*e.m.f.*) (voltage) in the conductor, described by Faraday's law of induction. According to Lenz's law, the induced voltage has a polarity (direction) which opposes the change in current that created it. As a result, inductors oppose any changes in current through them.

An inductor is characterized by its inductance, which is the ratio of the voltage to the rate of change of current. In the International System of Units (SI), the unit of inductance is the henry (H) named for 19th century American scientist Joseph Henry. In the measurement of magnetic circuits, it is equivalent to weber/ampere. Inductors have values that typically range from 1&#;µH (10^{−6}&#;H) to 20&#;H. Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves to increase the magnetic field and thus the inductance. Along with capacitors and resistors, inductors are one of the three passive linearcircuit elements that make up electronic circuits. Inductors are widely used in alternating current (AC) electronic equipment, particularly in radio equipment. They are used to block AC while allowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters to separate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.

### Description[edit]

An electric current flowing through a conductor generates a magnetic field surrounding it. The magnetic flux linkage generated by a given current depends on the geometric shape of the circuit. Their ratio defines the inductance .^{[2]}^{[3]}^{[4]}^{[5]} Thus

- .

The inductance of a circuit depends on the geometry of the current path as well as the magnetic permeability of nearby materials. An inductor is a component consisting of a wire or other conductor shaped to increase the magnetic flux through the circuit, usually in the shape of a coil or helix, with two terminals. Winding the wire into a coil increases the number of times the magnetic fluxlines link the circuit, increasing the field and thus the inductance. The more turns, the higher the inductance. The inductance also depends on the shape of the coil, separation of the turns, and many other factors. By adding a "magnetic core" made of a ferromagnetic material like iron inside the coil, the magnetizing field from the coil will induce magnetization in the material, increasing the magnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor of several thousand over what it would be without it.

### Constitutive equation[edit]

Any change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. By Faraday's law of induction, the voltage induced by any change in magnetic flux through the circuit is given by^{[5]}

Reformulating the definition of L above, we obtain^{[5]}

It follows, that

for L independent of time, current and magnetic flux linkage.

So inductance is also a measure of the amount of electromotive force (voltage) generated for a given rate of change of current. For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second. This is usually taken to be the constitutive relation (defining equation) of the inductor.

The dual of the inductor is the capacitor, which stores energy in an electric field rather than a magnetic field. Its current–voltage relation is obtained by exchanging current and voltage in the inductor equations and replacing *L* with the capacitance *C*.

### Circuit equivalence at short-time limit and long-time limit[edit]

In a circuit, an inductor can behave differently at different time instant. However, it's usually easy to think about the short-time limit and long-time limit:

- In the long-time limit, after the magnetic flux through the inductor has stabilized, no voltage would be induced between the two sides of the inductor; Therefore, the long-time equivalence of an inductor is a wire (i.e. short circuit, or 0 V battery).
- In the short-time limit, if the inductor starts with a certain current I, since the current through the inductor is known at this instant, we can replace it with an ideal current source of current I. Specifically, if I=0 (no current goes through the inductor at initial instant), the short-time equivalence of an inductor is an open circuit (i.e. 0 A current source).

### Lenz's law[edit]

Main article: Lenz's Law

The polarity (direction) of the induced voltage is given by Lenz's law, which states that the induced voltage will be such as to oppose the change in current.^{[6]} For example, if the current through an inductor is increasing, the induced voltage will be positive at the current's entrance point and negative at the exit point, tending to oppose the additional current.^{[7]}^{[8]}^{[9]} The energy from the external circuit necessary to overcome this potential "hill" is being stored in the magnetic field of the inductor. If the current is decreasing, the induced voltage will be negative at the current's entrance point and positive at the exit point, tending to maintain the current. In this case energy from the magnetic field is being returned to the circuit.

### Energy stored in an inductor[edit]

One intuitive explanation as to why a potential difference is induced on a change of current in an inductor goes as follows:

When there is a change in current through an inductor there is a change in the strength of the magnetic field. For example, if the current is increased, the magnetic field increases. This, however, does not come without a price. The magnetic field contains potential energy, and increasing the field strength requires more energy to be stored in the field. This energy comes from the electric current through the inductor. The increase in the magnetic potential energy of the field is provided by a corresponding drop in the electric potential energy of the charges flowing through the windings. This appears as a voltage drop across the windings as long as the current increases. Once the current is no longer increased and is held constant, the energy in the magnetic field is constant and no additional energy must be supplied, so the voltage drop across the windings disappears.

Similarly, if the current through the inductor decreases, the magnetic field strength decreases, and the energy in the magnetic field decreases. This energy is returned to the circuit in the form of an increase in the electrical potential energy of the moving charges, causing a voltage rise across the windings.

#### Derivation[edit]

The work done per unit charge on the charges passing the inductor is . The negative sign indicates that the work is done *against* the emf, and is not done *by* the emf. The current is the charge per unit time passing through the inductor. Therefore the rate of work done by the charges against the emf, that is the rate of change of energy of the current, is given by

From the constitutive equation for the inductor, so

In a ferromagnetic core inductor, when the magnetic field approaches the level at which the core saturates, the inductance will begin to change, it will be a function of the current . Neglecting losses, the energy stored by an inductor with a current passing through it is equal to the amount of work required to establish the current through the inductor.

This is given by: , where is the so-called "differential inductance" and is defined as: . In an air core inductor or a ferromagnetic core inductor below saturation, the inductance is constant (and equal to the differential inductance), so the stored energy is

For inductors with magnetic cores, the above equation is only valid for linear regions of the magnetic flux, at currents below the saturation level of the inductor, where the inductance is approximately constant. Where this is not the case, the integral form must be used with variable.

### Ideal and real inductors[edit]

The constitutive equation describes the behavior of an *ideal inductor* with inductance , and without resistance, capacitance, or energy dissipation. In practice, inductors do not follow this theoretical model; *real inductors* have a measurable resistance due to the resistance of the wire and energy losses in the core, and parasitic capacitance due to electric potentials between turns of the wire.^{[10]}^{[11]}

A real inductor's capacitive reactance rises with frequency, and at a certain frequency, the inductor will behave as a resonant circuit. Above this self-resonant frequency, the capacitive reactance is the dominant part of the inductor's impedance. At higher frequencies, resistive losses in the windings increase due to the skin effect and proximity effect.

Inductors with ferromagnetic cores experience additional energy losses due to hysteresis and eddy currents in the core, which increase with frequency. At high currents, magnetic core inductors also show sudden departure from ideal behavior due to nonlinearity caused by magnetic saturation of the core.

Inductors radiate electromagnetic energy into surrounding space and may absorb electromagnetic emissions from other circuits, resulting in potential electromagnetic interference.

An early solid-state electrical switching and amplifying device called a saturable reactor exploits saturation of the core as a means of stopping the inductive transfer of current via the core.

*Q* factor[edit]

The winding resistance appears as a resistance in series with the inductor; it is referred to as DCR (DC resistance). This resistance dissipates some of the reactive energy. The quality factor (or *Q*) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal inductor. High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonant circuit.

The Q factor of an inductor is defined as, where *L* is the inductance, *R* is the DCR, and the product *ωL* is the inductive reactance:

*Q* increases linearly with frequency if *L* and *R* are constant. Although they are constant at low frequencies, the parameters vary with frequency. For example, skin effect, proximity effect, and core losses increase *R* with frequency; winding capacitance and variations in permeability with frequency affect *L*.

At low frequencies and within limits, increasing the number of turns *N* improves *Q* because *L* varies as *N*^{2} while *R* varies linearly with *N*. Similarly increasing the radius *r* of an inductor improves (or increases) *Q* because *L* varies with *r*^{2} while *R* varies linearly with *r*. So high *Q* air core inductors often have large diameters and many turns. Both of those examples assume the diameter of the wire stays the same, so both examples use proportionally more wire. If the total mass of wire is held constant, then there would be no advantage to increasing the number of turns or the radius of the turns because the wire would have to be proportionally thinner.

Using a high permeability ferromagnetic core can greatly increase the inductance for the same amount of copper, so the core can also increase the Q. Cores however also introduce losses that increase with frequency. The core material is chosen for best results for the frequency band. High Q inductors must avoid saturation; one way is by using a (physically larger) air core inductor. At VHF or higher frequencies an air core is likely to be used. A well designed air core inductor may have a Q of several hundred.

### Applications[edit]

Inductors are used extensively in analog circuits and signal processing. Applications range from the use of large inductors in power supplies, which in conjunction with filter capacitors remove ripple which is a multiple of the mains frequency (or the switching frequency for switched-mode power supplies) from the direct current output, to the small inductance of the ferrite bead or torus installed around a cable to prevent radio frequency interference from being transmitted down the wire. Inductors are used as the energy storage device in many switched-mode power supplies to produce DC current. The inductor supplies energy to the circuit to keep current flowing during the "off" switching periods and enables topographies where the output voltage is higher than the input voltage.

A tuned circuit, consisting of an inductor connected to a capacitor, acts as a resonator for oscillating current. Tuned circuits are widely used in radio frequency equipment such as radio transmitters and receivers, as narrow bandpass filters to select a single frequency from a composite signal, and in electronic oscillators to generate sinusoidal signals.

Two (or more) inductors in proximity that have coupled magnetic flux (mutual inductance) form a transformer, which is a fundamental component of every electric utility power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect on the windings. The size of the core can be decreased at higher frequencies. For this reason, aircraft use hertz alternating current rather than the usual 50 or 60 hertz, allowing a great saving in weight from the use of smaller transformers.^{[12]} Transformers enable switched-mode power supplies that isolate the output from the input.

Inductors are also employed in electrical transmission systems, where they are used to limit switching currents and fault currents. In this field, they are more commonly referred to as reactors.

Inductors have parasitic effects which cause them to depart from ideal behavior. They create and suffer from electromagnetic interference (EMI). Their physical size prevents them from being integrated on semiconductor chips. So the use of inductors is declining in modern electronic devices, particularly compact portable devices. Real inductors are increasingly being replaced by active circuits such as the gyrator which can synthesize inductance using capacitors.

### Inductor construction[edit]

A ferrite core inductor with two 20&#;mH windings.

Large 50 Mvarthree-phase iron-core loading inductor at a utility substation

An inductor usually consists of a coil of conducting material, typically insulated copper wire, wrapped around a core either of plastic (to create an air-core inductor) or of a ferromagnetic (or ferrimagnetic) material; the latter is called an "iron core" inductor. The high permeability of the ferromagnetic core increases the magnetic field and confines it closely to the inductor, thereby increasing the inductance. Low frequency inductors are constructed like transformers, with cores of electrical steellaminated to prevent eddy currents. 'Soft' ferrites are widely used for cores above audio frequencies, since they do not cause the large energy losses at high frequencies that ordinary iron alloys do. Inductors come in many shapes. Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made by stringing a ferrite bead on a wire.

Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral pattern. Some such planar inductors use a planar core. Small value inductors can also be built on integrated circuits using the same processes that are used to make interconnects. Aluminium interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensions limit the inductance, and it is far more common to use a circuit called a *gyrator* that uses a capacitor and active components to behave similarly to an inductor. Regardless of the design, because of the low inductances and low power dissipation on-die inductors allow, they are currently only commercially used for high frequency RF circuits.

### Shielded inductors[edit]

Inductors used in power regulation systems, lighting, and other systems that require low-noise operating conditions, are often partially or fully shielded.^{[13]}^{[14]} In telecommunication circuits employing induction coils and repeating transformers shielding of inductors in close proximity reduces circuit cross-talk.

### Types[edit]

### Air-core inductor[edit]

The term *air core coil* describes an inductor that does not use a magnetic core made of a ferromagnetic material. The term refers to coils wound on plastic, ceramic, or other nonmagnetic forms, as well as those that have only air inside the windings. Air core coils have lower inductance than ferromagnetic core coils, but are often used at high frequencies because they are free from energy losses called core losses that occur in ferromagnetic cores, which increase with frequency. A side effect that can occur in air core coils in which the winding is not rigidly supported on a form is 'microphony': mechanical vibration of the windings can cause variations in the inductance.

#### Radio-frequency inductor[edit]

At high frequencies, particularly radio frequencies (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the Q factor of the circuit, broadening the bandwidth. In RF inductors, which are mostly air core types, specialized construction techniques are used to minimize these losses. The losses are due to these effects:

- Skin effect
- The resistance of a wire to high frequency current is higher than its resistance to direct current because of skin effect. Radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface. For example, at 6&#;MHz the skin depth of copper wire is about inches (25&#;µm); most of the current is within this depth of the surface. Therefore, in a solid wire, the interior portion of the wire may carry little current, effectively increasing its resistance.
- Proximity effect
- Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy currents in the wire of the coil, which causes the current in the conductor to be concentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance.
- Dielectric losses
- The high frequency electric field near the conductors in a tank coil can cause the motion of polar molecules in nearby insulating materials, dissipating energy as heat. So coils used for tuned circuits are often not wound on coil forms but are suspended in air, supported by narrow plastic or ceramic strips.
- Parasitic capacitance
- The capacitance between individual wire turns of the coil, called parasitic capacitance, does not cause energy losses but can change the behavior of the coil. Each turn of the coil is at a slightly different potential, so the electric field between neighboring turns stores charge on the wire, so the coil acts as if it has a capacitor in parallel with it. At a high enough frequency this capacitance can resonate with the inductance of the coil forming a tuned circuit, causing the coil to become self-resonant.

*(left)* Spiderweb coil *(right)* Adjustable ferrite slug-tuned RF coil with basketweave winding and litz wire

To reduce parasitic capacitance and proximity effect, high Q RF coils are constructed to avoid having many turns lying close together, parallel to one another. The windings of RF coils are often limited to a single layer, and the turns are spaced apart. To reduce resistance due to skin effect, in high-power inductors such as those used in transmitters the windings are sometimes made of a metal strip or tubing which has a larger surface area, and the surface is silver-plated.

- Basket-weave coils
- To reduce proximity effect and parasitic capacitance, multilayer RF coils are wound in patterns in which successive turns are not parallel but criss-crossed at an angle; these are often called
*honeycomb*or*basket-weave*coils. These are occasionally wound on a vertical insulating supports with dowels or slots, with the wire weaving in and out through the slots. - Spiderweb coils
- Another construction technique with similar advantages is flat spiral coils. These are often wound on a flat insulating support with radial spokes or slots, with the wire weaving in and out through the slots; these are called
*spiderweb*coils. The form has an odd number of slots, so successive turns of the spiral lie on opposite sides of the form, increasing separation. - Litz wire
- To reduce skin effect losses, some coils are wound with a special type of radio frequency wire called litz wire. Instead of a single solid conductor, litz wire consists of a number of smaller wire strands that carry the current. Unlike ordinary stranded wire, the strands are insulated from each other, to prevent skin effect from forcing the current to the surface, and are twisted or braided together. The twist pattern ensures that each wire strand spends the same amount of its length on the outside of the wire bundle, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

- Axial Inductor

Small inductors for low current and low power are made in molded cases resembling resistors. These may be either plain (phenolic) core or ferrite core. An ohmmeter readily distinguishes them from similar-sized resistors by showing the low resistance of the inductor.

### Ferromagnetic-core inductor[edit]

See also: Magnetic core

Ferromagnetic-core or iron-core inductors use a magnetic core made of a ferromagnetic or ferrimagnetic material such as iron or ferrite to increase the inductance. A magnetic core can increase the inductance of a coil by a factor of several thousand, by increasing the magnetic field due to its higher magnetic permeability. However the magnetic properties of the core material cause several side effects which alter the behavior of the inductor and require special construction:

- A time-varying current in a ferromagnetic inductor, which causes a time-varying magnetic field in its core, causes energy losses in the core material that are dissipated as heat, due to two processes:
- From Faraday's law of induction, the changing magnetic field can induce circulating loops of electric current in the conductive metal core. The energy in these currents is dissipated as heat in the resistance of the core material. The amount of energy lost increases with the area inside the loop of current.
- Changing or reversing the magnetic field in the core also causes losses due to the motion of the tiny magnetic domains it is composed of. The energy loss is proportional to the area of the hysteresis loop in the BH graph of the core material. Materials with low coercivity have narrow hysteresis loops and so low hysteresis losses.

- If the current through a magnetic core coil is high enough that the core saturates, the inductance will fall and current will rise dramatically. This is a nonlinear threshold phenomenon and results in distortion of the signal. For example, audio signals can suffer intermodulation distortion in saturated inductors. To prevent this, in linear circuits the current through iron core inductors must be limited below the saturation level. Some laminated cores have a narrow air gap in them for this purpose, and powdered iron cores have a distributed air gap. This allows higher levels of magnetic flux and thus higher currents through the inductor before it saturates.
^{[19]} - If the temperature of a ferromagnetic or ferrimagnetic core rises to a specified level, the magnetic domains dissociate, and the material becomes paramagnetic, no longer able to support magnetic flux. The inductance falls and current rises dramatically, similarly to what happens during saturation. The effect is reversible: When the temperature falls below the Curie point, magnetic flux resulting from current in the electric circuit will realign the magnetic domains of the core and its magnetic flux will be restored. The Curie point of ferromagnetic materials (iron alloys) is quite high; iron is highest at &#;°C. However, for some ferrimagnetic materials (ceramic iron compounds – ferrites) the Curie point can be close to ambient temperatures (below &#;°C).
^{[citation needed]}

#### Laminated-core inductor[edit]

Low-frequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to transformers. The core is made of stacks of thin steel sheets or laminations oriented parallel to the field, with an insulating coating on the surface. The insulation prevents eddy currents between the sheets, so any remaining currents must be within the cross sectional area of the individual laminations, reducing the area of the loop and thus reducing the energy losses greatly. The laminations are made of low-conductivity silicon steel to further reduce eddy current losses.

#### Ferrite-core inductor[edit]

Main article: Ferrite core

For higher frequencies, inductors are made with cores of ferrite. Ferrite is a ceramic ferrimagnetic material that is nonconductive, so eddy currents cannot flow within it. The formulation of ferrite is xxFe_{2}O_{4} where xx represents various metals. For inductor cores soft ferrites are used, which have low coercivity and thus low hysteresis losses.

#### Powdered-iron-core inductor[edit]

See also: Carbonyl iron

Another material is powdered iron cemented with a binder.

#### Toroidal-core inductor[edit]

Main article: Toroidal inductors and transformers

In an inductor wound on a straight rod-shaped core, the magnetic field lines emerging from one end of the core must pass through the air to re-enter the core at the other end. This reduces the field, because much of the magnetic field path is in air rather than the higher permeability core material and is a source of electromagnetic interference. A higher magnetic field and inductance can be achieved by forming the core in a closed magnetic circuit. The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a toroidal or doughnut-shaped ferrite core. Because of their symmetry, toroidal cores allow a minimum of the magnetic flux to escape outside the core (called *leakage flux*), so they radiate less electromagnetic interference than other shapes. Toroidal core coils are manufactured of various materials, primarily ferrite, powdered iron and laminated cores.^{[20]}

### Variable inductor[edit]

*(left)* Inductor with a threaded ferrite slug *(visible at top)* that can be turned to move it into or out of the coil, &#;cm high. *(right)* A variometer used in radio receivers in the s

Probably the most common type of variable inductor today is one with a moveable ferrite magnetic core, which can be slid or screwed in or out of the coil. Moving the core farther into the coil increases the permeability, increasing the magnetic field and the inductance. Many inductors used in radio applications (usually less than &#;MHz) use adjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certain tolerances (inaccuracy). Sometimes such cores for frequencies above &#;MHz are made from highly conductive non-magnetic material such as aluminum.^{[21]} They decrease the inductance because the magnetic field must bypass them.

Air core inductors can use sliding contacts or multiple taps to increase or decrease the number of turns included in the circuit, to change the inductance. A type much used in the past but mostly obsolete today has a spring contact that can slide along the bare surface of the windings. The disadvantage of this type is that the contact usually short-circuits one or more turns. These turns act like a single-turn short-circuited transformer secondary winding; the large currents induced in them cause power losses.

A type of continuously variable air core inductor is the *variometer*. This consists of two coils with the same number of turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned with respect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the same direction, the fields add and the inductance is maximum. When the inner coil is turned so its axis is at an angle with the outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned ° so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and the inductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used in antenna tuners and matching circuits to match low frequency transmitters to their antennas.

Another method to control the inductance without any moving parts requires an additional DC current bias winding which controls the permeability of an easily saturable core material. *See*Magnetic amplifier.

### Choke[edit]

A choke is an inductor designed specifically for blocking high-frequency alternating current (AC) in an electrical circuit, while allowing DC or low-frequency signals to pass. Because the inductor resistricts or "chokes" the changes in current, this type of inductor is called a choke. It usually consists of a coil of insulated wire wound on a magnetic core, although some consist of a donut-shaped "bead" of ferrite material strung on a wire. Like other inductors, chokes resist changes in current passing through them increasingly with frequency. The difference between chokes and other inductors is that chokes do not require the high Q factor construction techniques that are used to reduce the resistance in inductors used in tuned circuits.

### Circuit analysis[edit]

The effect of an inductor in a circuit is to oppose changes in current through it by developing a voltage across it proportional to the rate of change of the current. An ideal inductor would offer no resistance to a constant direct current; however, only superconducting inductors have truly zero electrical resistance.

The relationship between the time-varying voltage *v*(*t*) across an inductor with inductance *L* and the time-varying current *i*(*t*) passing through it is described by the differential equation:

When there is a sinusoidalalternating current (AC) through an inductor, a sinusoidal voltage is induced. The amplitude of the voltage is proportional to the product of the amplitude (*I*_{P}) of the current and the frequency (*f*) of the current.

In this situation, the phase of the current lags that of the voltage by π/2 (90°). For sinusoids, as the voltage across the inductor goes to its maximum value, the current goes to zero, and as the voltage across the inductor goes to zero, the current through it goes to its maximum value.

If an inductor is connected to a direct current source with value *I* via a resistance *R* (at least the DCR of the inductor), and then the current source is short-circuited, the differential relationship above shows that the current through the inductor will discharge with an exponential decay:

### Reactance[edit]

The ratio of the peak voltage to the peak current in an inductor energised from an AC source is called the reactance and is denoted *X*_{L}.

Thus,

where *ω* is the angular frequency.

Reactance is measured in ohms but referred to as *impedance* rather than resistance; energy is stored in the magnetic field as current rises and discharged as current falls. Inductive reactance is proportional to frequency. At low frequency the reactance falls; at DC, the inductor behaves as a short circuit. As frequency increases the reactance increases and at a sufficiently high frequency the reactance approaches that of an open circuit.

### Corner frequency[edit]

In filtering applications, with respect to a particular load impedance, an inductor has a corner frequency defined as:

### Laplace circuit analysis (s-domain)[edit]

When using the Laplace transform in circuit analysis, the impedance of an ideal inductor with no initial current is represented in the *s* domain by:

where

- is the inductance, and
- is the complex frequency.

If the inductor does have initial current, it can be represented by:

### Inductor networks[edit]

Main article: Series and parallel circuits

Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (*L*_{eq}):

The current through inductors in series stays the same, but the voltage across each inductor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total inductance:

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.

#### Mutual inductance[edit]

Mutual inductance occurs when the magnetic field of an inductor induces a magnetic field in an adjacent inductor. Mutual induction is the basis of transformer construction.

where M is the maximum mutual inductance possible between 2 inductors and L_{1} and L_{2} are the two inductors. In general

as only a fraction of self flux is linked with the other. This fraction is called "Coefficient of flux linkage (K)" or "Coefficient of coupling".

### Inductance formulas[edit]

See also: Inductance §&#;Self-inductance of thin wire shapes

The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.

### See also[edit]

### Notes[edit]

**^**Nagaoka’s coefficient (*K*) is approximately 1 for a coil which is much longer than its diameter and is tightly wound using small gauge wire (so that it approximates a current sheet).

### References[edit]

**^**Alexander, Charles; Sadiku, Matthew.*Fundamentals of Electric Circuits*(3&#;ed.). McGraw-Hill. p.&#;**^**Singh, Yaduvir ().*Electro Magnetic Field Theory*. Pearson Education India. p.&#; ISBN&#;.**^**Wadhwa, C. L. ().*Electrical Power Systems*. New Age International. p.&#; ISBN&#;.**^**Pelcovits, Robert A.; Josh Farkas ().*Barron's AP Physics C*. Barron's Educational Series. p.&#; ISBN&#;.- ^
^{a}^{b}^{c}Purcell, Edward M.; David J. Morin ().*Electricity and Magnetism*. Cambridge Univ. Press. p.&#; ISBN&#;. **^**Shamos, Morris H. ().*Great Experiments in Physics: Firsthand Accounts from Galileo to Einstein*. Courier Corporation. ISBN&#;.**^**Schmitt, Ron ().*Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics*. Elsevier. pp.&#;75– ISBN&#;.**^**Jaffe, Robert L.; Taylor, Washington ().*The Physics of Energy*. Cambridge Univ. Press. p.&#; ISBN&#;.**^**Lerner, Lawrence S. ().*Physics for Scientists and Engineers, Vol. 2*. Jones and Bartlet Learning. p.&#; ISBN&#;.**^**Bowick, Christopher ().*RF Circuit Design, 2nd Ed*. Newnes. pp.&#;7–8. ISBN&#;.**^**Kaiser, Kenneth L. ().*Electromagnetic Compatibility Handbook*. CRC Press. pp.&#;– ISBN&#;.**^**"Aircraft electrical systems". Wonderquest.com. Retrieved**^**Ott, Henry W. ().*Electromagnetic Compatibility Engineering*. John Wiley and Sons. p.&#; ISBN&#;.**^**Violette, Norman ().*Electromagnetic Compatibility Handbook*. Springer. pp.&#;– ISBN&#;

No, I don't like it. I shook my head. She took hold of the pole, threw her leg over it and leaned back beautifully. Then I ran my free hand over my chest, squeezed it lightly and slowly rose. No, I saw that, I criticized her.

## Across inductor voltage

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Lesson 2 - Voltage Across An Inductor, Part 2 (Engineering Circuits)" My first woman was a 20-year-old second-year student named Masha. We met her in the summer when we worked together on attractions. I was 18 then, and I really wanted sex. She constantly instigated me with this.

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