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ECE 3336 Introduction to Circuits & Electronics

Lecture Set #7Inductors and Capacitors

Fall 2012,TUE&TH 5:30-7:00pmDr. Wanda Wosik

Notes developed by Dr. Dave Shattuck

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Overview of this Part Inductors and Capacitors

In this part, we will cover the following topics:• Defining equations for inductors and capacito

rs• Power and energy storage in inductors and c

apacitors• Parallel and series combinations• Basic Rules for inductors and capacitors

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Basic Elements, Review

We are now going to pick up the remaining basic circuit elements that we will be covering in these modules.

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Circuit Elements• In circuits, we think about basic circuit

elements that are the basic “building blocks” of our circuits. This is similar to what we do in Chemistry with chemical elements like oxygen or nitrogen.

• A circuit element cannot be broken down or subdivided into other circuit elements.

• A circuit element can be defined in terms of the behavior of the voltage and current at its terminals.

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The 5 Basic Circuit ElementsThere are 5 basic circuit elements:1. Voltage sources2. Current sources3. Resistors4. Inductors5. CapacitorsWe defined the first three elements in a

previous module. We will now introduce inductors or capacitors.

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Inductors• An inductor is a two terminal circuit

element that has a voltage across its terminals which is proportional to the derivative of the current through its terminals.

• The coefficient of this proportionality (INDUCTANCE, L) is the defining characteristic of an inductor.

• An inductor is the device that we use to model the effect of magnetic fields on circuit variables.

• The energy stored In magnetic fields has effects on voltage and current. We use the inductor component to model these effects.

In many cases a coil of wire can be modeled as an inductor.

A Current Can Generate a Magnetic Field

Hans Christian Oersted(1777-1851)

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• http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html#c3

• http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html#c1

• http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/amplaw.html#c1

• http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html#c1

Faraday’s Law of Induction

emfEdtd

Faraday’s law:

sH ˆd

where

H

Magnetic flux (unit: Weber)

• It has to be varying with time.• The faster the change of flux is, the

larger is the current.

sd

Electromagnetic induction gives us current in a conductor moving in the magnetic field

Electromotive force

A decade after Oersted’s discovery:

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Inductors

There is an inductance whenever we have magnetic fields produced, and there are magnetic fields whenever current flows.

ac current produces a voltage which counteracts the changes of this current

Direction of B field and I current – right hand rule

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• An inductor obeys the expression

where vL is the voltage across the inductor, and iL is the current through the inductor, and LX is called the inductance.

• In addition, it works both ways. If something obeys this expression, we can think of it, and model it, as an inductor.

• The unit ([Henry] or [H]) is named for Joseph Henry, and is equal to a [Volt-second/Ampere].[H]=[V]•[s]/[A]

Inductors – Definition and Units

There is an inductance whenever we have magnetic fields produced, and there are magnetic fields whenever current flows. However, this inductance is often negligible except in coils.

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Schematic Symbol for InductorsThe schematic symbol that we use for

inductors is shown here.The schematic symbol can be labeled either with a variable, like LX, or a value, with some number, and units. An example might be 390[mH].

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Inductor Polarities

• Previously, we have emphasized the important of reference polarities of current sources and voltages sources. There is no corresponding polarity to an inductor.

• As in Resistors.

• And as for resistor, direction matters with respect to the voltage and current in the passive sign convention.

The direction of currents induced by magnetic field

Heinrich Friedrich Emil Lenz1804 –1865

The induced current creates a magnetic field opposing the change of the inducing magnetic field

The rule of resistance to change

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Passive and Active Sign Convention for Inductors

The sign of the equation that we use for inductors depends on whether we have used the passive sign convention or the active sign convention.

Passive Sign Convention Active Sign Convention

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Defining Equation, Integral Form, Derivation

The defining equation for the inductor,

can be rewritten in another way. If we want to express the current in terms of the voltage, we can integrate both sides. We get

We pick t0 and t for limits of the integral, where t is time, and t0 is an arbitrary time value, often zero. The inductance, LX, is constant, and can be taken out of the integral. To avoid confusion, we introduce the dummy variable s in the integral. We get

We finish the derivation in the next slide.

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We can take this equation and perform the integral on the right hand side. When we do this we get

Thus, we can solve for iL(t), and we have two defining equations for the inductor,

Defining Equations for Inductors

and

Remember that both of these are defined for the passive sign convention for iL and vL. If not, then we need a negative sign in these equations.

Initial conditions

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Voltage vL≠0 only if iL=f(t)

The implications of these equations are significant. For example, if the current is not changing, then the voltage will be zero. This current could be a constant value, and large, and an inductor will have no voltage across it. This is counter-intuitive for many students. That is because they are thinking of actual coils, which have some finite resistance in their wires. For us, an ideal inductor has no resistance; it simply obeys the laws below.

We might model a coil with both inductors and resistors, but for now, all we need to note is what happens with these ideal elements.

and

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Current Change is Limited so

Another implication is that we cannot change the current through an inductor instantaneously. If we were to make such a change, the derivative of current with respect to time would be infinity, and the voltage would have to be infinite. Since it is not possible to have an infinite voltage, it must be impossible to change the current through an inductor instantaneously.

and

But large voltages can be produced

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Dummy Variable s Replaces Variable t in the Integral Limits It is not really necessary to introduce a dummy variable. It really does

n’t matter what variable is integrated over, because when the limits are

inserted, that variable goes away.

and

The independent variable t is in the limits of the integral. This is indicated by the iL(t) on the left-hand side of the equation.

Remember, the integral here is not a function of s. It is a function of t.

This is a constant.

Inductance for a Coil The role of core

lANL

2

AlNiHA

dtdiA

lN

dtdvemf turn1

dtdi

lANNvv emfNemf

2

turn1turns

dtdiLv

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Energy in Inductors, DerivationWe can find the energy stored in the magnetic field associated with the

inductor. First, we note that the power is voltage times current, as it has always been.

Now, we can multiply each side by dt, and integrate both sides to get

We need limits. We know that when the current is zero, there is no magnetic field, and therefore there can be no energy in the magnetic field. That allowed us to use 0 for the lower limits. The upper limits came since we will have the energy stored, wL, for a given value of current, iL.

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Energy in Inductors, FormulaWe had the integral for the energy,

Now, we perform the integration. Note that LX is a constant, independent of the current through the inductor, so we can take it out of the integral. We have

We simplify this, and get the formula for energy stored in the inductor,

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Series Inductors Equivalent CircuitsTwo series

inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

iL1(t)=iL2(t)=iLEQ(t)

(∑Li is as for resistors)

Obtained from KVLvLEQ(t)=VL1(t)+VL2(t)

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More than 2 Series InductorsThis rule can be

extended to more than two series inductors. In this case, for N series inductors, we have

(∑Li reminds of ∑Ri for resistors)

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Series Inductors Equivalent Circuits: A ReminderTwo series inductors,

L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

Remember that these two equivalent circuits are equivalent only with respect to the circuit connected to them.

(In yellow here.)

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Series Inductors Equivalent Circuits: Initial Conditions

Two series inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be

the same as well. That is,

iL1(t0) must equal

iLEQ(t0).

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Parallel Inductors Equivalent Circuits

Two parallel inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

Obtained from KCL

∑(1/Li) as for resistors ∑(1/Ri)

iLEQ(t)=iL1(t)+iL2(t)

vL1(t)=vL2(t)=vLEQ(t)

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More than 2 Parallel Inductors

This rule can be extended to more than two parallel inductors. In this case, for N parallel inductors, we have

The product over sum rule only works for two inductors.

Analogy to RESISTORS

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Parallel Inductors Equivalent Circuits: A Reminder

Remember that these two equivalent circuits are equivalent only with respect to the circuit connected to them.

Two parallel inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

(In yellow here.)

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Parallel Inductors Equivalent Circuits: Initial Conditions

• To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is,

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Notes1.We took some mathematical liberties in this derivation. For example, we do

not really multiply both sides by dt, but the results that we obtain are correct here. 2.Note that the energy is a function of the current squared, which will be

positive. We will assume that our inductance is also positive, and clearly ½ is positive. So, the energy stored in the magnetic field of an inductor will be positive.

3.These three equations are useful, and should be learned or written down.

Go back to Overview

slide.