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
  • TGMkbasemnpf

1012109106103110-310-610-910-1210-15

  • Notations:
  • Scalars: a, A, …
  • Vectors: a, A, …
  • Unit vector: , , …
  • Phasors: , , …
  • 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:

.

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:
  • 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:
  •  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
  • Lossless medium:

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/ = 2f/ up) =  / up

  • 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.
  • Lossy medium:
  • 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, …

–dB+

1 x01 x

0.5 x32 x

0.25 x64 x

0.125 x98 x

0.1 x1010 x

0.01 x20100 x

0.001 x301000 x

  • dBm (power unit)

+dBm-

1 mW01 mW

0.5 mW32 mW

0.25 mW64 mW

0.125 mW98 mW

0.1 mW1010 mW

0.01 mW20100 mW

0.001 mW301000 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:


  • Equality:

  • Add/subtract:
  • Multiply/divide:

  • Powers and roots:
  • 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
  • A more practical example: Use phasor method to find the total current of the following circuit:

IR + IL+ IC= IT IR

IC IT

IL

Itotal(t) = 13.885 cos(t + 0) mA

= 2f = 6.28310+5 rad/sec, 0 = -43.927

  • Another practical example: Use phasor method to find the output voltage of the following circuit:

Vout(t) = 7.202 cos(t + 0) V

= 2f = 6.28310+5 rad, 0 = 43.927