EE 207 Electromagnetics I � Static (time-invariant) fields

Electrostatic or magnetostatic fields are not coupled together. (one can exist without the other.)

Electrostatic fields � steady electric fields produced by stationaryelectric charge.

Magnetostatic fields � steady magnetic fields produced by steady(DC) currents or stationary magnetic materials.

EE 307 Electromagnetics II � Dynamic (time-varying) fields

Dynamic electric and magnetic fields are coupled together.(one field cannot exist without the other for time-varying fields.)

Electromagnetic fields � produced by time-varying currents orcharges, or static sources in motion.

Maxwell’s equations � four laws which govern the behavior of allelectromagnetic fields [Gauss’s law, Faraday’s law, Gauss’s law formagnetic fields, Ampere’s law].

The static versions of Faraday’s law and Ampere’s law must be modifiedto account for dynamic fields.

Complete Form of Faraday’s Law (Dynamic Fields)

The complete form of Faraday’s law, valid for both static anddynamic fields, is defined in terms of a quantity known as theelectromotive force (emf). In an electric circuit, the emf is the force whichsets the charge in motion (forcing function for the current).

Example (emf in a battery/resistor circuit)

E = Ee + Ef (total electric field)

In general, the integral of the total electric field around a closedcircuit yields the total emf in the integration path.

Assumptions for the battery circuit:(1) The emf electric field is confined to the battery.

(Ef = 0 outside the battery.)(2) Connecting wires are perfect conductors.

(E = Ee = 0 inside the wires.)

The emf voltage in terms of the electric field components is

0

The resistor voltage in terms of the electric field components may bedetermined using the conservative property of Ee.

Potential Difference Definition

General Equations for EMF and Potential Difference

Given the definition of electromotive force, we may now write thedynamic form of Faraday’s law.

Faraday’s Law � a time-changing magnetic flux through a closedcircuit induces an emf in the circuit (closed circuit � induced current, opencircuit � induced voltage).

The emf is an equal and opposite reaction tothe flux change (Lenz’s law)

The unit normal associated withthe differential surface ds isrelated to the unit vector of thedifferential length dl by the righthand rule.

Note that when static fields are assumed, the time derivative on the righthand side of the dynamic (complete) version of Faraday’s law goes to zeroand the equation reduces to the electrostatic form.

Example (Faraday’s law induction, wire loop in a time changing B)

For the closed loop, the flux produced by the induced current opposesthe change in B.

For the open-circuited loop, the polarity of the induced emf is definedby the emf line integral.

Induction Types

1. Stationary circuit / time-varying B (transformer induction).2. Moving circuit / static B (motional induction).3. Moving circuit / time-varying B

(general case, transformer and motional induction).

Example (transformer induction � AM antenna)

A circular wire loop of radius a = 0.4m lies in the x-y plane with itsaxis along the z-axis. The vector magnetic field over the surface of theloop is H = Hocos(�t)az where Ho = 200 �A/m and f = 1 MHz. Determinethe emf induced in the loop.

Since the loop is stationary, ds isnot time-dependent so that thederivative with respect to timecan be brought inside theintegral.

The time derivative is written as a partial derivative since the magnetic fluxdensity is, in general, a function of both time and space. The polarity ofthe induced emf is assigned when the direction of ds is chosen. If wechoose ds = az ds (then dl = a� dl for the line integral of E), the polarity ofthe induced emf is that shown above. For this problem, both B [B = �oH]and ds are az-directed so that the dot product in the transformer inductionintegral is one.

Since the partial derivative of H with respect to time is independent ofposition, it can be brought outside the integral. The resulting integral ofds over the surface S yields the area of the loop so that

where A is the area of the loop (A = �a2).

A typical AM antenna achieves a larger induced emf by employingmultiple turns of wire around a ferrite core.

Example (motional induction � moving conductor / static B)

A particle of charge Q moving with velocity u in a uniform Bexperiences a force given by

From Faraday’s law,

Choosing dl counterclockwise assigns the induced emf polarity as shownabove. On the moving conductor, dl = dy ay.

Note that a uniform velocity yields a DC voltage. An oscillatorymotion (back and forth) could be used to produce a sinusoidal voltage.

Example (General induction � moving conductor / time-varying B)

Using the same geometry as the last example, assume that themagnetic flux density is B = Bocos�t (�az).

Choose dl counterclockwise � ds out dl = dyay (on moving conductor) ds = dxdyaz

If we let x = 0 at t = 0 be our reference, then x = uot and

Summary of Induction Formulas

Et = transformer emf electric field

Em = motional emf electric field

E = total emf electric field

Faraday’s Law (Differential Form)

The differential form of Faraday’s law can be found by applyingStoke’s theorem to the integral form.

By applying Stoke’s theorem, the line integrals in the various forms ofFaraday’s law can be transformed into surface integrals. The integrandsof the surface integrals can then be equated to find the correspondingdifferential form of the equation.

Transformer(Toroidal core)

Toroid cross Toroidsectional area

= A = �a2

mean length = l = 2��o

Faraday’s law applied to the primary winding yields

where the surface integration is over the cross-section of the toroid. Thepolarity assumed for the primary voltage yields ds = ds a� (dl is the pathalong the primary winding from the “�” terminal to the “+” terminal). Thepartial derivative of B and ds are in opposite directions so that

The partial derivatives of the magnetic flux density components are

Note that these partial derivatives are independent of position so that theycan be brought outside the surface integral. The resulting surface integralof ds over S yields the cross-sectional surface area of the toroid (A = �a2).

In a similar fashion, for the secondary,

Note that M12 = M21 = M.

Displacement Current(Maxwell’s contribution to Maxwell’s equations)

The concept of displacement current can be illustrated by consideringthe currents in a simple parallel RC network (assume ideal circuitelements, for simplicity).

iR(t) � conductioncurrent

iC(t) � displacementcurrent

From circuit theory �

In the resistor, the conduction current model is valid (JR = �RER). Theideal resistor electric field (ER) and current density (JR) are assumed to beuniform throughout the volume of the resistor.

The conduction current model does not characterize the capacitorcurrent. The ideal capacitor is characterized by large, closely-spaced platesseparated by a perfect insulator (�C = 0) so that no charge actually passesthrought the dielectric [JC(t) = �CEC(t)]. The capacitor current measuredin the connecting wires of the capacitor is caused by the charging anddischarging the capacitor plates. Let Q(t) be the total capacitor charge onthe positive plate.

Based on these results, the static version of Ampere’s law must bemodified for dynamic fields to include conduction current ANDdisplacement current. Note that displacement current does not exist understatic conditions. The general form for current density in the dynamic fieldproblem is

displacement conduction convection current current

+ current

Complete Form of Ampere’s Law (Dynamic Fields)

Given the definition of displacement current, the complete form ofAmpere’s law for dynamic fields can be written.

The corresponding differential form of Ampere’s law is found usingStoke’s theorem.

Since the two surface integrals above are valid for any surface S, we mayequate the integrands.

Example (Ampere’s law, non-ideal capacitor)

The previously considered parallel RC network represents theequivalent circuit of a parallel plate capacitor with an imperfect insulatingmaterial between the capacitor plates (finite conductivity).

Capacitor with imperfectinsulating material

(assume E, J are uniform)

Equivalent circuit

C � models charge storage (displacement current)

R � models leakage current (conduction current)

Let the applied voltage be a sinusoid. � V(t) = Vosin�t

The resulting electric field in the capacitor is given by

Note that:1. The peak conduction current density is independent of

frequency.2. The peak displacement current density is directly proportional

to frequency.3. The displacement current density leads the conduction current

density by 90o.

Since typical material permittivities are in the 1-100 pF/m range, thedisplacement current density is typically negligible at low frequencies incomparison to the conduction current density (especially in goodconductors). At high frequencies, the displacement current densitybecomes more significant and can even dominate the conduction currentdensity in good insulators.

Maxwell’s Equations(Dynamic fields)

In addition to his contribution of displacement current, Maxwellbrought together the four basic laws governing electric and magnetic fieldsinto one set of four equations which, as a set, completely describe thebehavior of any electromagnetic field. All of the vector field, flux, currentand charge terms in Maxwell’s equations are, in general, functions of bothtime and space [e.g., E(x,y,z,t)]. The form of these quantities is referredto as the instantaneous form (we can describe the fields at any point intime and space). The instantaneous form of Maxwell’s equations may beused to analyze electromagnetic fields with any arbitrary time-variation.

Maxwell’s Equations [instantaneous, differential form]

Maxwell’s Equations [instantaneous, integral form]

Constitutive Relations (linear, homogeneous, isotropic media)

Boundary Conditions

Note that the unit normal n points into region 2.

Time-Harmonic Fields

Given a linear circuit with a sinusoidal source, all resulting circuitcurrents and voltages have the same harmonic time dependence so thatphasors may be used to simplify the mathematics of the circuit analysis.In the same way, given electromagnetic fields produced by sinusoidalsources (currents and charges), the resulting electric and magnetic fieldshave the same harmonic time dependence so that phasors may be used tosimplify the analysis of the fields.

For the circuit analysis example, based on Euler’s identity(ejx=cosx+jsinx), the instantaneous voltage and current [v(t), i(t)] arerelated to the phasor voltage and current [I s(�), Vs(�)] by

instantaneous values [v(t), i(t)] � phasor va lues [ I s (� ) ,Vs(�)]

(Time domain) (Frequency domain)

The voltage equations for a resistor, inductor and capacitor are

Note that the time-domain derivative and integral yield terms of j� and(j�)�1 respectively, in the frequency domain according to

The time-harmonic electromagnetic field problem is somewhat morecomplicated than the circuit problem since we must deal with vectorelectric and magnetic fields rather than scalar voltages and currents. Also,these electric and magnetic fields are, in general, functions of position andspace. However, the basic principles of phasor analysis still hold true. Thegeneral instantaneous vector electric field [E(x,y,z,t)] may be defined by

Each of the component scalars of the instantaneous vector electric field[Ex,Ey,Ez] may be written in terms of the corresponding componentphasors [Exs,Eys,Ezs] (scalar phasors).

Note that Es (x,y,z) is a vector phasor defined by three complex vector

components which are each defined by a magnitude and a phase.To transform the instantaneous (time-domain) Maxwell’s equations intothe time harmonic (frequency-domain) Maxwell’s equations, we use thesame techniques used to transform the time-domain circuit equations intotheir frequency-domain phasor form. We replace all sources and fieldquantities by their phasor equivalents and replace all time-derivatives ofquantities with j� times the phasor equivalent.

Maxwell’s Equations [time-harmonic, differential form]

Maxwell’s Equations [time-harmonic, integral form]

Maxwell’s equations in Maxwell’s equations in instantaneous form � time-harmonic form

(Time domain) (Frequency domain)

Example (Maxwell’s Equations)

The instantaneous magnetic field is H = 2cos(�t�3y)az A/m in amedium characterized by � = 0, � = 2�o, � = 5�o. Calculate � and E(assume a source-free region).

The phasor electric and magnetic fields are related by the time-harmonicMaxwell’s equations in a source-free region (J=0, �=0). Es and Hs mustsatisfy all four equations.

0

Time Varying Potentials

Maxwell’s equations define electromagnetic fields in terms of fieldquantities (E,H,D,B) and sources (J,�). Solving Maxwell’s equationsdirectly for the electric and magnetic fields is difficult for mostapplications due to the complicated integration which must be performed.The integration required to determine the fields can be simplified throughthe use of potentials (magnetic vector potential �A, electric scalarpotential �V).

We have shown previously that electrostatic fields can be determined usingthe electric scalar potential while magnetostatic fields can be determinedusing the magnetic vector potential.

Electrostatic Fields (electric scalar potential�V )

Magnetostatic Fields (magnetic vector potential�A )

Electromagnetic Fields (both A and V are required)

Begin with Gauss’s law for magnetic fields (same for static anddynamic fields).

Insert � into Faraday’s law.

The potentials at this point have been defined using only two of the fourMaxwell’s equations (the potentials are not completely described yet). Ifwe take the divergence of equation �, we may employ Gauss’s law.

If we take the curl of equation �, we may incorporate Ampere’s law.

To completely describe any vector, both the divergence (lamellarcomponents) and the curl (solenoidal components) must be defined. So farwe have defined the curl of A, but not the divergence of A. We maychoose the divergence of A in such a way as to simplify the mathematics.

Equations � and � become

Equations � and � are the governing partial differential equations whichrelate the potentials to the sources and have the basic form of waveequations (fundamental equations defining wave behavior).

For the special case of time-harmonic fields (e j�t) , each partialderivative yields a j� factor and the wave equations defining the potentialsreduce to