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1 Introductory EM theory EMLAB Importance of electromagnetic theory • • 2 EM theory helps understand how electrical signals propagate along conductors as well as free space. Predicts voltages and currents using the concept of electric and magnetic field. EMLAB 3 Example : a simple DC circuit Charge distribution E/H-field distribution EMLAB 4 Generation of charges : battery An amount of positive charges are generated such that the terminal voltages are sustained. 2NH 4 2e 2 NH3 H 2 Electrons(-) are absorbed. (+) charges are generated Electrons(-) are generated. (+) charges are absorbed. Zn Zn 2 2e Electrons are generated via electro-chemical reaction. EMLAB 5 Electromagnetic theory EM-theory Material Electric field (E) Sources (q, J) Magnetic field (H) Electro-magnetic field (E,H ) Material (ε, μ) Mathematics Coordinate systems Vector calculus Green’s function Wave equation EMLAB Basic EM-theory 1. 6 Electric field ① Coulomb’s law ② Gauss’s law (divergence) ③ Electric potential (gradient) ④ Capacitance ⑤ Ohm’s law 2. Magnetic field ① Biot-Savart law ② Ampere’s law (curl) ③ Inductance EMLAB 7 3. Electro-magnetic field ① Faraday’s law ② Displacement current ③ Maxwell’s equations ④ Plane wave ⑤ Reflection/transmission 4. Transmission lines ① Impedance matching ② Smith chart ③ Waveguides 5. Radiation EMLAB Coulomb’s law 8 • This law is discovered by Coulomb experimentally. • In the free space, the force between two point charges is proportional to the charges of them, and is inversely proportional to the square of the distance between those charges. Fk q 1q 2 r 2 k 9 109 [ Nm2C2 ] rˆ 1 40 ε0 : permittivity of vacuum. If q1, q2 have the same polarity, the force is repulsive. +q1 +q2 R R 2 R1 R2 R1 Coulomb’s law only states that the force between two charge is related to the distance between them and their charges. It does not tells us how the interaction occurs. O EMLAB Electric field distribution near charged plates E 9 If a charge is brought into the plates, it will be accelerated along the direction of electric field. F ma qE EMLAB 10 Electric potential energy (V) • • • • As in the gravitational field, a potential energy for the electric field can be introduced. The potential energy in an electric field is defined as the energy levels of the test charge with +1C. The unit of potential energy is “voltage” named after the physicist Volta. The position far away from the source charge has zero potential energy. +qt Electric potential energy defined as the work to move a test charge with (+1C) +q rB rB rA rA W ( F) dr ( qt E) dr qtV rB V ( E) dr rA EMLAB 11 Potential energy in a gravitational field U (r2 ) r2 • The scalar field quantity of potential energy is introduced to represent energy levels inherent to positions in the space. • Differences between the energy levels can be obtained from the work that must applied for the object to move from the initial position to the final position. • The position that corresponds to zero energy level is one that is located far away from the earth. U (r2 ) r2 r1 U U (r2 ) U (r2 ) W O EMLAB 12 Work in a gravitational field r2 W ( F) dr GMm F 2 r̂12 r r2 r2 mass : m To move an object in the gravitational field, an external force must be applied that compensates the force due to gravity. r2 r1 mass : M r2 GMm rˆ rˆ dr 2 r2 r W F r O 2 1 1 1 GMm GMm r r2 r2 r2 GMm ( r2 r2 ) mgr. r2 GM g r 2 EMLAB 13 Magnetic field A charged particle in motion generates magnetic field nearby. In the same way, currents generate magnetic field nearby. EMLAB Motion of a charge in a magnetic field 14 F qv B Charged particles in motion are influenced by magnetic fields EMLAB 15 Biot-Savart law ˆ Ids R dH 4R 2 Current segment Id s r' R r r' r Direction of H-field The generated magnetic field can be predicted by Biot-Savart’s law EMLAB Electromagnetic law – Maxwell equations Maxwell equations B E t D H J t D B 0 16 1. Electromagnetic phenomena are explained by the four Maxwell equations. 2. Through the equations, electric field and magnetic field are coupled to each other. 3. Quantities on the right hand side are the source terms. 4. Quantities on the left side are the resulting phenomena. 5. The independent variables are current density vector J and charge density . EMLAB Ampere’s law 17 E H J t Current or increase of electric field strength E,J H EMLAB Faraday’s law 18 E H H t Increase of magnetic field E EMLAB Faraday’s law 19 The time-varying magnetic field generates electric field nearby. EMLAB Gauss’ law 20 E / E +Q -Q Electric field lines emanate from positive charges and sink into negative charges. EMLAB 21 H 0 Magnetic field lines always form closed loops EMLAB Example – Hertzian dipole antenna 22 spheres for storing electric charges Heinrich Hertz (1857-1894) arc monitoring EMLAB Schematic diagram of Hertz experiment 23 Transformer for high voltage generation EMLAB Propagation of electromagnetic wave 24 Electric field : red Magnetic field : blue EMLAB Radio communication 25 EMLAB Reception of EM wave 26 current E Transmitting antenna V Receiving antenna The charges on the receiving antenna move toward the antenna terminal, which causes voltage drop across them. EMLAB Example – Signal propagation over a line trace 27 V H-field due to moving charges t E V H E H J t ZL H E t EMLAB Electromagnetic problem 28 Source distributions are known over only a small area. EMLAB