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Introduction SI units: m, kg, s, A, V, , K, … Conversion factors: 1” = 2.54 cm 1 lb. = 0.454 kg 1 gallon = 3.785 liter Prefixes T G M k base m n p f 1012 109 106 103 1 10-3 10-6 10-9 10-12 10-15 Notations: Scalars: a, A, … Vectors: a, A, … Unit vector: x̂ , ŷ , … ~ ~ Phasors: E , H , … Fundamental forces: Nuclear force (strongest) EM force (strong)*** Weak-interaction force (weak) Gravitational force (weakest) Electric field Fe = Electrical force: The source of electrical force is electric charge. Elementary charge e = 1.6 10-19 (C) Coulomb’s law: The magnitude of the force (Fe21) on q2 due to q1 is given by: q1q2 F . 40 R122 The direction of the force points from q1 to q2. is called the permittivity and 0 = 8.854 10-12 F/m is for free space. If q1 and q2 are like charges, the resultant force will try to push q2 away from q1. Otherwise, the resultant force will try to pull q2 to q1. If a system of electric charges is placed in space, it will exert a force to any surrounding charges. Since this force depends on the magnitude and polarity of the surrounding charges, the concept of E-field, which equals to the force applied on a unit charge, is used to describe the electrical properties of the system of charges. The E-field for a point charge in free space is given by: q E R̂ (V/m) 40 R 2 The direction of the E-field points away from the point charge (i.e. toward the point charge if q is negative). Two important properties of electric charges: Conservation and superposition. The E-field in a material composed of atoms is smaller because a fraction of the force is needed to align (polarize) the atoms. Permittivity is used to describe the material effect on E-field. The relative permittivity or dielectric constant r =/0 is often used: A material with r = 10 reduces the E-field by 10 times. r for free space (vacuum) = 1. Electric flux density: D =E (C/m2) D is material independent. Magnetic field Fm = magnetic force: The sources of magnetic force are electric current or magnetic poles. Magnetic poles cannot be separated (not yet). Biot-Savart law:The magnetic flux density induced by a current I flowing in the z-direction is given by: I B ˆ 0 (T) 2r is called the permeability and 0 = 4 10-7 H/m is for free space. Permittivity is used to describe the material effect. The relative permeability: r =/0 r for free space (vacuum) = 1. Magnetic field intensity: B =H Magnetic field is intensified in materials with high relative permeability. Static fields Q E and I H Since I = dQ/dt, E and H are independent of each other. Electrostatics: q/t = 0 Magnetostatics: I/t = 0 Dynamic fields Time-varying Both E- and H- fields are present and related to each other. Traveling waves Carries energy Speed c = 3 108 m/s for EM waves 330 m/s for sound waves Linear wave: EM and sound waves; nonlinear wave: fluid Transient and continuous harmonic waves (sinusoidal) 1-D (transmission lines), 2-D, 3-D waves Plane waves, cylindrical waves, spherical waves Waves in medium 2t 2x 0 Lossless medium: y ( x, t ) A cos T A: amplitude of the wave T: time period of the wave : wavelength of the wave 0: reference of the wave Phase velocity: The speed of the wave measured at a fixed phase. up = / T = f Phase constant: The amount of phase shift (in radian) per meter. Hence: = 2/ = 2f/ up) = / up x y ( x, t ) A cost x 0 A cos t 0 u p Direction of propagation: The coefficients of t and x have opposite signs indicate that the wave is traveling in the + x direction. The coefficients of t and x have the same signs indicate that the wave is traveling in the - x direction. 2t 2x 0 Lossy medium: y ( x, t ) Ae x cos T The coefficient is called the attenuation factor with a unit of Np/m (Np is dimensionless). A more practical unit is dB/m = 8.686 dB (power ratio) 3 dB loss = 50% left, 10 dB loss = 10 % left, … – 1x 0.5 x 0.25 x 0.125 x 0.1 x 0.01 x 0.001 x dBm (power unit) + 1 mW 0.5 mW 0.25 mW 0.125 mW 0.1 mW 0.01 mW 0.001 mW dB 0 3 6 9 10 20 30 + 1x 2x 4x 8x 10 x 100 x 1000 x dBm 0 3 6 9 10 20 30 1 mW 2 mW 4 mW 8 mW 10 mW 100 mW 1000 mW Other dB units include dBW, dB, … The EM spectrum Opacity: Atmosphere opaque and ionosphere opaque Windows: optical, IR, and RF. -ray, X-ray, UV, visible, IR, and RF. Radio bands -wave: 300 MHz to 300 GHz; mm-wave: 30 to 300 GHz Complex mathematics j = -1 z = x + jy = |z| ej = |z| Euler’s identity: ej = cos + j sin z = |z| ej = |z| cos + j |z| sin Re{z} = x = |z| cos , Im{z} = y = |z| sin ; |z| = x2 + y2 , = tan(y/x) Complex conjugate: z* = x – jy = |z| e-j = |z| - ; |z| = z* z Operations Useful relations: 1 j 0 e j 0 e j 2 10 1360 j 1 e j e j j 4 2 190 (1 j ) 2 j 2 1 e j 1180 j e j 4 (1 j ) 2 Equality: z1 z2 Re{z1} Re{ z2 } and Im{z1} Im{z2 } or z1 z2 and 1 2 Add/subtract: z1 z2 (Re{ z1} Re{z2 }) j (Im{z1} Im{z2 }) Multiply/divide: z1 z 2 z1 z 2 e j ( 1 2) z1 z 2 cos( 1 2 ) j sin( 1 2 ) z z z1 1 e j ( 1 2) 1 cos( 1 2 ) j sin( 1 2 ) z2 z2 z2 Powers and roots: z n z1 e jn z1 cos(n ) j sin( n ) n z cos j sin 2 2 Phasors: A shortcut for solving linear differential equations (DE) with sinusoidal excitations. Because of the unique property of the exponential function, deax/dt= aeax, DEs can be transformed into ordinary algebraic equations in the phasor-domain. Exponential (phasor) representation of a sinusoidal signal z ze n j 2 A cos t Ae A A sin t Ae j0 A cost 0 Ae j0 2 jA A sin t 0 Ae A cos t x 0 Ae j ( x 0 ) dz Zˆ jZˆ z dt dt j d cos t 0 je j0 dt j j 0 2 Ae x cos t x 0 Ae x e j ( x 0 ) sin t dt 0 e j 0 2 j A more practical example: Use phasor method to find the total current of the following circuit: f 100KHz v 10e ZL 2f L e v IR R j 90 deg IR 10 mA IT IR IL IC ZT R IR 1 ZL L 1mH V ZL 0.628i k v IL ZL ZC C 1000pF 1 2f C IL 15.915i mA IT 10 9.632i mA 1 1 j 0deg 1 e j 90deg IC R 1K ZC 1.592i k v ZC IC 6.283i mA arg IT 43.927 deg ZT 0.519 0.5i k IT v ZT IT 10 9.632i mA ZC + IL + IC = IT IR IC IL Itotal(t) = 13.885 cos(t + 0) mA = 2f = 6.28310+5 rad/sec, 0 = -43.927 IT 13.885 mA IT Another practical example: Use phasor method to find the output voltage of the following circuit: f 100KHz ZL 2f L e v 10e j 90 deg 1 Zout 1 ZL Vout v Zout ZT 1 j 0deg L 1mH V ZL 0.628i k ZC Zout 1.038i k C 1000pF 1 2f C e R 1K j 90deg ZC 1.592i k ZT R Zout ZT 1 1.038i K ZC Vout 5.187 4.996i V arg Vout 43.927 deg Vout 7.202 V Vout Vout(t) = 7.202 cos(t + 0) V V1 = 2f = 6.28310+5 rad, 0 = 43.927 6 1.2 10 s 1 360deg 43.2 deg vp v p 14.1383V v rms Vout_p 10.2760V Vout_rms v rms 9.997 V 2 f Vout_p 2 Vout_rms 7.266 V