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 = E e + E f (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. (E f = 0 outside the battery.) (2) Connecting wires are perfect conductors. (E = E e = 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 E e.
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. 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 its axis along the z-axis. The vector magnetic field over the surface of the loop is H = H o cos(tt)a z where H o = 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 = a z ds (then dl = a N 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 a z -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 = Ba 2 ). 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 a y. 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 = B o costt (!a z ). Choose dl counterclockwise Y ds out dl = dya y (on moving conductor) ds = dxdya z If we let x = 0 at t = 0 be our reference, then x = u o t and
Summary of Induction Formulas E t = transformer emf electric field E m = 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 Toroid sectional area = A = Ba2 mean length = l = 2BD o 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 = Ba 2 ). In a similar fashion, for the secondary,
Note that M 12 = M 21 = 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). i R (t)! conduction current i C (t)! displacement current From circuit theory Y In the resistor, the conduction current model is valid (J R = F R E R ). The ideal resistor electric field (E R ) and current density (J R ) 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 (F C = 0) so that no charge actually passes throught the dielectric [J C (t) = F C E C (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 conduction convection + current 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) = V o 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 90 o. 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), V s (T)] by instantaneous values [v(t), i(t)] ] phasor values [I s ( T), V s (T)] (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 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 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 [E x,e y,e z ] may be written in terms of the corresponding component phasors [E xs,e ys,e zs ] (scalar phasors). Note that E s (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 Maxwell s equations in instantaneous form ] time-harmonic form
(Time domain) (Frequency domain)
Example (Maxwell s Equations) The instantaneous magnetic field is H = 2cos(Tt!3y)a z 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). E s and H s 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 Ç 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 È, we may employ Gauss s law. If we take the curl of equation Ç, 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 µ and become Equations and 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