ECE 3313 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 stationary
electric charge.
Magnetostatic fields ! steady magnetic fields produced by steady
(DC) currents or stationary magnetic materials.
ECE 3324 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 or
charges, or static sources in motion.
Maxwell’s equations ! four laws which govern the behavior of all
electromagnetic fields [Gauss’s law, Faraday’s law, Gauss’s law for
magnetic fields, Ampere’s law].
The static versions of Faraday’s law and Ampere’s law must be modified
to account for dynamic fields.
Complete Form of Faraday’s Law (Dynamic Fields)
The complete form of Faraday’s law, valid for both static and
dynamic fields, is defined in terms of a quantity known as the electromotive
force (emf). In an electric circuit, the emf is the force which sets 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 closed
circuit 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 be
determined 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 the
dynamic form of Faraday’s law.
Faraday’s Law ! a time-changing magnetic flux through a closed
circuit induces an emf in the circuit (closed circuit ! induced current, open
circuit ! induced voltage).
The emf is an equal and opposite reaction to
the flux change (Lenz’s law)
The unit normal associated with
the differential surface ds is
related to the unit vector of the
differential length dl by the right
hand rule.
Note that when static fields are assumed, the time derivative on the right
hand side of the dynamic (complete) version of Faraday’s law goes to zero
and 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 opposes
the change in B.
For the open-circuited loop, the polarity of the induced emf is defined
by the emf line integral.
Induction Types
1.
2.
3.
Stationary circuit / time-varying B (transformer induction).
Moving circuit / static B (motional induction).
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 its
axis along the z-axis. The vector magnetic field over the surface of the loop
is H = Ho cos(Tt)az where Ho = 200 :A/m and f = 1 MHz. Determine the
emf induced in the loop.
Since the loop is stationary, ds is
not time-dependent so that the
derivative with respect to time
can be brought inside the
integral.
The time derivative is written as a partial derivative since the magnetic flux
density is, in general, a function of both time and space. The polarity of the
induced emf is assigned when the direction of ds is chosen. If we choose
ds = az ds (then dl = aN dl for the line integral of E), the polarity of the
induced emf is that shown above. For this problem, both B [B = :o H] and
ds are az-directed so that the dot product in the transformer induction
integral is one.
Since the partial derivative of H with respect to time is independent of
position, it can be brought outside the integral. The resulting integral of ds
over the surface S yields the area of the loop so that
where A is the area of the loop (A = Ba2 ).
A typical AM antenna achieves a larger induced emf by employing
multiple 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 B
experiences a force given by
From Faraday’s law,
Choosing dl counterclockwise assigns the induced emf polarity as shown
above. On the moving conductor, dl = dy ay.
Note that a uniform velocity yields a DC voltage. An oscillatory
motion (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 the
magnetic flux density is B = Bo cos Tt (!az).
Choose dl counterclockwise
dl = dyay (on moving conductor)
Y
ds out
ds = dxdyaz
If we let x = 0 at t = 0 be our reference, then x = uo t 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 applying
Stoke’s theorem to the integral form.
By applying Stoke’s theorem, the line integrals in the various forms of
Faraday’s law can be transformed into surface integrals. The integrands of
the surface integrals can then be equated to find the corresponding
differential form of the equation.
Transformer
(Toroidal core)
Toroid cross
= A = B a2
sectional area
Toroid
= l = 2BDo
mean length
Faraday’s law applied to the primary winding yields
where the surface integration is over the cross-section of the toroid. The
polarity assumed for the primary voltage yields ds = ds a N (dl is the path
along the primary winding from the “!” terminal to the “+” terminal). The
partial 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 they
can be brought outside the surface integral. The resulting surface integral
of ds over S yields the cross-sectional surface area of the toroid (A = Ba2).
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 considering
the currents in a simple parallel RC network (assume ideal circuit elements,
for simplicity).
iR(t) ! conduction
current
iC(t) ! displacement
current
From circuit theory
Y
In the resistor, the conduction current model is valid (JR = FR ER ). The ideal
resistor electric field (ER) and current density (JR) are assumed to be
uniform throughout the volume of the resistor.
The conduction current model does not characterize the capacitor
current. The ideal capacitor is characterized by large, closely-spaced plates
separated by a perfect insulator (FC = 0) so that no charge actually passes
throught the dielectric [JC (t) = FC EC (t)]. The capacitor current measured
in the connecting wires of the capacitor is caused by the charging and
discharging the capacitor plates. Let Q(t) be the total capacitor charge on
the positive plate.
Based on these results, the static version of Ampere’s law must be modified
for dynamic fields to include conduction current AND displacement
current. Note that displacement current does not exist under static
conditions. The general form for current density in the dynamic field
problem is
displacement
current
conduction
convection
+
current
current
Complete Form of Ampere’s Law (Dynamic Fields)
Given the definition of displacement current, the complete form of
Ampere’s law for dynamic fields can be written.
The corresponding differential form of Ampere’s law is found using
Stoke’s theorem.
Since the two surface integrals above are valid for any surface S, we may
equate the integrands.
Example (Ampere’s law, non-ideal capacitor)
The previously considered parallel RC network represents the
equivalent circuit of a parallel plate capacitor with an imperfect insulating
material between the capacitor plates (finite conductivity).
Capacitor with imperfect
insulating 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.
Y
V(t) = Vo sin Tt
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, the
displacement current density is typically negligible at low frequencies in
comparison to the conduction current density (especially in good
conductors). At high frequencies, the displacement current density
becomes more significant and can even dominate the conduction current
density in good insulators.
Maxwell’s Equations
(Dynamic fields)
In addition to his contribution of displacement current, Maxwell
brought together the four basic laws governing electric and magnetic fields
into one set of four equations which, as a set, completely describe the
behavior of any electromagnetic field. All of the vector field, flux, current
and charge terms in Maxwell’s equations are, in general, functions of both
time and space [e.g., E(x,y,z,t)]. The form of these quantities is referred to
as the instantaneous form (we can describe the fields at any point in time
and space). The instantaneous form of Maxwell’s equations may be used
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 circuit
currents and voltages have the same harmonic time dependence so that
phasors may be used to simplify the mathematics of the circuit analysis. In
the same way, given electromagnetic fields produced by sinusoidal sources
(currents and charges), the resulting electric and magnetic fields have the
same harmonic time dependence so that phasors may be used to simplify
the analysis of the fields.
For the circuit analysis example, based on Euler’s identity
jx
(e =cosx+jsinx), the instantaneous voltage and current [v(t), i(t)] are
related to the phasor voltage and current [I s(T), Vs (T)] by
instantaneous values [v(t), i(t)]
(Time domain)
]
phasor values [I s(T), Vs (T)]
(Frequency domain)
The voltage equations for a resistor, inductor and capacitor are
Note that the time-domain derivative and integral yield terms of jT and
(jT)!1 respectively, in the frequency domain according to
The time-harmonic electromagnetic field problem is somewhat more
complicated than the circuit problem since we must deal with vector
electric and magnetic fields rather than scalar voltages and currents. Also,
these electric and magnetic fields are, in general, functions of time and
space. However, the basic principles of phasor analysis still hold true. The
general 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 component phasors
[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 into the
time harmonic (frequency-domain) Maxwell’s equations, we use the same
techniques used to transform the time-domain circuit equations into their
frequency-domain phasor form. We replace all sources and field quantities
by their phasor equivalents and replace all time-derivatives of quantities
with jT times the phasor equivalent.
Maxwell’s Equations [time-harmonic, differential form]
Maxwell’s Equations [time-harmonic, integral form]
Maxwell’s equations in
instantaneous form
(Time domain)
]
Maxwell’s equations in
time-harmonic form
(Frequency domain)
Example (Maxwell’s Equations)
The instantaneous magnetic field is H = 2cos( Tt ! 3y)az A/m in a
medium characterized by F = 0, : = 2:o, , = 5,o. Calculate T and E
(assume a source-free region).
The phasor electric and magnetic fields are related by the time-harmonic
Maxwell’s equations in a source-free region (J=0, D =0). Es and Hs must
satisfy all four equations.
0
Time Varying Potentials
Maxwell’s equations define electromagnetic fields in terms of field
quantities (E,H,D,B) and sources (J, D). Solving Maxwell’s equations
directly for the electric and magnetic fields is difficult for most applications
due to the complicated integration which must be performed. The
integration required to determine the fields can be simplified through the
use of potentials (magnetic vector potential !A, electric scalar potential !).V
We have shown previously that electrostatic fields can be determined using
the electric scalar potential while magnetostatic fields can be determined
using 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 and
dynamic fields).
Insert â into Faraday’s law.
The potentials at this point have been defined using only two of the four
Maxwell’s equations (the potentials are not completely described yet). If
we take the divergence of equation ã, we may employ Gauss’s law.
If we take the curl of equation â, we may incorporate Ampere’s law.
To completely describe any vector, both the divergence (lamellar
components) and the curl (solenoidal components) must be defined. So far
we have defined the curl of A, but not the divergence of A. We may choose
the divergence of A in such a way as to simplify the mathematics.
Equations Ð and Ñ become
Equations Ò and Ó are the governing partial differential equations which
relate the potentials to the sources and have the basic form of wave
equations (fundamental equations defining wave behavior).
For the special case of time-harmonic fields (e jTt) , each partial
derivative yields a jT factor and the wave equations defining the potentials
reduce to