EE 207 Electromagnetics I ! Static (time-invariant) fields
Electrostatic or magnetostatic fields are not coupled together.
(one can exist without the other.)
Electrostatic fields
electric charge.
! steady electric fields produced by stationary
Magnetostatic fields ! steady magnetic fields produced by steady
(DC) currents or stationary magnetic materials.
Continuity Equation
The continuity equation defines the basic conservation of charge
relationship between current and charge. That is, a net current in or out of
a given volume must equal the net increase of decrease in the total charge
in the volume. If we define a surface S enclosing a volume V, the net
current out of the volume (Iout) is defined by
where ds = dsan and an is the
outward pointing normal. If the
current I is a DC current, then the
net current out of the volume is
zero (as much current flows out as flows in). For a time-varying current,
the net current out of the volume may be non-zero and can be expressed in
terms of the change in the total charge within the volume (Q).
The previous equation is the integral form of the continuity equation. The
differential form of the continuity equation can be found by applying the
divergence theorem to the surface integral and expressing the total charge
in terms of the charge density.
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 or
charges, or static sources in motion.
The second and last terms in the equation above yield integrals that are
valid for any volume V that we may choose.
Since the previous equation is valid for any volume V, we may equate the
integrands of the integrals (the only way for the integrals to yield the same
value for any volume V is for the integrands to be equal). This yields the
continuity equation.
The continuity equation is given in differential form and relates the current
density at a given point to the charge density at that point. For steady
currents (DC currents), the charge density does not change with time so
that
the divergence of the current density is always zero.
Relaxation Time
If some amount of charge is placed inside a volume of conducting
material, the Coulomb forces on the individual charges cause them to
migrate away from each other (assuming the charge is all positive or all
negative). The end result is a surface charge on the outer surface of the
conductor while the inside of the conductor remains charge-neutral. The
time required for the conductor to reach this charge-neutral state is related
to a time constant designated as the relaxation time. The relaxation time
can be determined by inserting the relationship for the current density in
terms of electric field
into the continuity equation
The continuity equation is the basis for Kirchhoff’s current law.
Given a circuit node connecting a system of N wires (assuming DC
currents) enclosed by a spherical surface S, the integral form of the
continuity equation gives
which yields
The divergence of the electric field is related to the charge density by
Inserting this result into previous equation yields
The integral form of the continuity equation (and thus Kirchhoff’s current
law) also holds true for time-varying (AC) currents if we let the surface S
shrink to zero around the node.
or
The solution to this homogeneous, first order PDE is
where Tr is the relaxation time given by.
The relaxation time is a time constant that describes the rate of decay of the
charge inside the conductor. After a time period of Tr, the charge has
decayed to 36.8 percent (1/e) of its original value.
Example (Relaxation time)
Determine the relaxation time for copper (,r = 1, F = 5.8×107 ®/m)
and fused quartz (,r = 5, F = 10!17 ®/m).
Copper
Fused Quartz
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.
Faraday’s Law
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)
Potential Difference Definition
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
General Equations for EMF and Potential Difference
The resistor voltage in terms of the electric field components may be
determined using the conservative property of Ee.
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).
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.
The emf is an equal and opposite reaction to
the flux change (Lenz’s law)
For the open-circuited loop, the polarity of the induced emf is defined
by the emf line integral.
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.
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)
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
A particle of charge Q moving with velocity u in a uniform B
experiences a force given by
From Faraday’s law,
If we let x = 0 at t = 0 be our reference, then x = uo t and
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.
Summary of Induction Formulas
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.
Et = transformer emf electric field
Em = motional emf electric field
E = total emf electric field
The partial derivatives of the magnetic flux density components are
Transformer
(Toroidal core)
Toroid cross
= A = B a2
sectional area
Toroid
= l = 2 BD o
mean length
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).
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
In a similar fashion, for the secondary,
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
Note that M12 = M21 = M.
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.
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.
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
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
(e jx =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)
]
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 position 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).
p h a s or va lu es [ 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.
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.
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).
Maxwell’s Equations [time-harmonic, differential form]
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.
Maxwell’s Equations [time-harmonic, integral form]
0
Maxwell’s equations in
instantaneous form
]
Maxwell’s equations in
time-harmonic form