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| JAPAN - MALAYSIA TECHNICAL INSTITUTE MANPOWER DEPARTMENT MINISTRY OF HUMAN RESOURCE
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| INFORMATION SHEET
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| Date: 19/07/2007 | ||||
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| DEPARTMENT | ELECTRONICS ENGINEERING TECHNOLOGY | Ref. No | EDL-04 | ||
| SECTION | ELECTRONICS INDUSTRY | Code | TKE 5504 | ||
| COURSE | ADV. DIP. IN ELECTRONICS ENG. TECHNOLOGY | NOSS Ref | NOT AVAILABLE | ||
| BLOCK | OPTICAL ELECTRONICS |
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| UNIT | INTRODUCTION TO ELECTROMAGNETIC WAVES | SEM | 3 | ||
| SEGMENT | WAVES AND PHASORS |
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| DUTY | - | TASK | - | ||
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TITLE: WAVES AND PHASORS INSTRUCTION AIMS:
Upon completion of this course, the students will be able to understand:
1. Basics Electric Fields. 2. Basics Magnetic Fields. 3. Basics Static and Dynamic Fields. 4. Defined Traveling Waves. 5. Electromagnetic Spectrum. 6. Complex Numbers. 7. Calculate Phasors.
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Basic Electric Fields In electromagnetics, we work with scalar and vector quantities. For scalar quantities medium-weight italic fonts denote scalar quantities such as R for resistance. For vector we use a boldface roman font such as E for electric field.
Dimension 1 E = E is the magnitude of E, electric field vector and
Letters denoting phasor are printed with a tilde ~ over the letter. Thus
Facts 1 The electromagnetic force consists of an electrical force Fe and a magnetic force Fm.
Facts 2 The source of the gravitational field is mass but the source of the electrical field is electric charge. Dimension 2 e = 1.6 x 10-19 ( coulomb)
Rule 1 1. Two like charges repel one another, whereas two charges of opposite polarity attract.
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Dimension 3 Electrical permittivity of free space, ε0 = 8.854 x 10-12 farad per meter (F/m)
Definition 1 The force acting on charge q1 by q2 is defined as Fe12 or Fe21 if the acting on charge q2 by q1, thus we can say that Fe12 = - Fe21.
If there is an electric field exist between these two point charges, the electric field intensity can be shown to be:
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Rule 2 The electric charge can neither be created nor destroyed. If a volume contains np protons and ne electrons then the total charge is q = npe - nee ( coulomb).
Rule 3 Principle of linear superposition states that: The total vector electric field at a point in space due to a system of point charges is equal to the vector sum of the electric fields at that point due to the individual charges.
Table 1 Important prefixes
Phenomena 1 Placing a point charge surrounded by atoms cause them to become distorted. The center of symmetry of the electron cloud i9s altered with respect to the nucleus, with one pole of the atom becoming more positively charged while the other side negatively charged.
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Rule 4 The degree of polarization depends on the distance between the atom and the isolated point charge and the orientation of the dipole is such that the dipole axis connecting its two poles directed toward the point charge.
Derived Formula 1 E = Replace the permittivity of free space ε0 with ε, where ε is now the permittivity of the material in which the electric field is measured. Thus, E = Often, ε is expressed in the form ε = εr ε0 (Farad/meter) where εr is called the relative permittivity or dielectric constant.
Dimension 4 εr = 1 for vacuum εr = 1.0006 for air near earth surface
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| Basic magnetic fields
Formula 3 D = εE (C/m2) D is called the electric flux density.
Phenomena 2 Like poles of different magnets repel each other while unlike poles attract each other.
Definition 2 The magnetic lines encircling a magnet are called magnetic field lines and represent the existence of a magnetic field called the magnetic flux density B.
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Formula 4 B = This formula is also known as the Bio-Savart law leads to the result of a magnetic flux density B by a constant current flowing in wire in z –direction. R is the radial distance from the current and Dimension 5 μ0 = 4π x 10-7 henry per meter (H/m)
Dimension 6 c = Dimension 7 For nonmagnetic material μ = μ0 .
Facts 3 Magnetization properties of a material is significantly shown by μ.
Dimension 8 μ = μr μ0 (H/m)
Formula 5 Magnetic flux density, B and magnetic field intensity, H are related by the formula: B = μH
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| Rule 5 Electric and magnetic fields are independent of each other as long as the current that flows in the wire remains a constant. Rule 6 The electric field is governed by the charge q and the magnetic field by current, I.
Facts 4 Since magnetic field depends on current or rate of charge since I = dq/dt then magnetic field remains the same but electric field always changing.
Definition 5 Electrostatics and magnetostatics deals with stationary charges and steady currents respectively. Definition 6 Dynamics are dealing with time varying fields induced by time varying sources that is current and charge densities. Phenomena 5 If the current associated with the beam of moving charged particles varies with time then the amount of charges present in a given section of a beam also varies with time vice versa. These phenomena will make the current and magnetic field to be coupled to each other. Facts 5 A time varying electric field will generate a time varying magnetic field.
Definition 7 The conductivity, σ of a material is measured in siemens per meter (S/.m). If σ = 0, the charges do not move and the material is said to be a a perfect dielectrics. If σ = ∞, the charges move freely throughout the materials and the material is said to be a perfect conductor.
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| Definition 8 A medium is said to be homogenous if all the three parameters ε, μ and σ in the medium are constant.
Questions 1. What are the four fundamental forces of nature and what are their relative strengths?
2. What is Coulomb’s law and state its properties?
3. What are the two important properties of electric charge?
4. What do the electrical permittivity and magnetic permeability of a material account for?
5. What are the three branches and associated conditions of electromagnetics?
Table 2 3 branches of electromagnetics
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Traveling Waves Definition 9 All various types of waves exhibit the following properties: 1. Moving waves carry energy from one point to another. 2. Waves have velocity. 3. Some waves exhibit linear property. Waves that do not affect the passage of other waves are called linear because they pass right through each other and the total of two linear waves is the sum of two waves. Definition 10 Waves are of two types: transient waves caused by short-duration disturbance and continuous harmonic waves generated by an oscillating source.
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| Figure 8 – A two dimensional waves propagates like ripples as shown in (a) A three dimensional waves travel like in Figure 8 (b) and (c). It may take different shapes plane waves, cylindrical waves and spherical waves.
Definition 11 A medium is said to be lossless if it does not attenuate the amplitude of the wave traveling within or on its surface. Mathematical Form 1 y denotes the height of the water surface relative to the mean height and x denotes the distance the wave travel. A is the amplitude of the wave, T is its time period, λ is its spatial wavelength, and
Mathematical Form 2 y(x,t) = Acos(x,t) where The angle
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The direction of wave propagation is determined by inspecting the signs of the t and x terms in the expression for the phase
Rule 7 If one of the signs t or x is positive while the other is negative then the waves is traveling in the positive x-direction and if both signs are positive or negative then the wave is traveling in the negative x-direction. Facts 6 The constant phase reference
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| Formula 7 Frequency of a sinusoidal wave is given by: f =
Derived Formula 2 Combined formula 6 and 7 yields: up = fλ (m/s)
Derived Formula 3 Insert formula 6 into mathematical form 1 yields: = A cos(ωt – βx) Where ω is the angular velocity and β is the phase constant or wavenumber.
Formula 8 ω = 2πf
Formula 9 β =
Derived Formula 4 up = fλ = =
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| Figure 10 Putting back the reference phase into its place the derived formula 3 can be written as:
Derived Formula 5 y(x,t) = Acos(ωt – βx +
Figure 10 illustrates the use of derivede formula 5. When
Definition 12 Attenuation factor is defined as the amplitude loss of the wave when it travels in a lossy medium. And is denoted by e-αx where α is the attenuation constant. The unit is neper per meter.
Mathematical Form 3 The wave amplitude now is Ae-αx.
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Review Questions :
1. How can you tell if a wave is traveling in the positive x-direction or negative x-direction?
2. How does the envelope of the wave pattern vary with distance in a) Loseless medium b) Lossy medium
3. Why does a negative value of
Exercise :
1. The electric field of a traveling electromagnetic wave is given by Determine a) Direction of wave propagation b) Wave frequency f c) Wavelength λ d) Phase velocity up
2. An electromagnetic wave propagating in the z-direction in a lossy medium with attenuation constant α = 0.5 Np/m. If the wave electric field amplitude is 100 V/m at z = 0, how far can the wave travel before its amplitude will have been reduced to (a) 10 V/m (b) 1 V/m (c) 1 μV/m ?
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| Definition 13 An electromagnetic wave has the following fundamental properties: 1. An EM wave consists of electric and magnetic field intensities that oscillate at the same frequency f. 2. The phase velocity of an EM wave propagating in vacuum is a universal constant given by the velocity of light c. 3. In vacuum, the wavelength λ of an EM wave is related to its oscillation frequency f by Figure 11 A very narrow wavelength range extending between λ = 0.4 μm (violet) and λ = 0.7 μm (red) for visible part. As we move progressively toward shorter wavelengths, we encounter the ultraviolet, x-ray, and gamma-ray bands. On the other side of the visible spectrum lie the infrared band and then the radio region. Because of the link between λ and f, each of these spectral ranges may be specified in terms of its wavelength range or alternatively in terms of its frequency range. In practice, however, a wave is specified in terms of its wavelength λ if λ <> 1mm (i.e., in the radio region).
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| A wavelength of 1 mm corresponds to a frequency of 3 x 1011 Hz = 300 GHz in free space. The radio spectrum consists of several individual bands, as shown in the chart of Fig. 12. Figure 12
Each band covers one decade of the radio spectrum and has a letter designation based on a nomenclature defined by the International Telecommunication Union. Different frequencies have different applications because they are excited by different mechanisms, and the properties of an EM wave propagating, in a material may vary considerably from one band to another. The extremely low frequency (ELF) band from 3 to 30 Hz is used primarily for the detection of buried metal objects. Lower frequencies down to 0.1 Hz are used in magnetotelluric sensing of the structure of the earth, and frequencies in the range from I Hz to 1 kHz sometimes are used for communications with submerged submarines and for certain kinds of sensing of Earth's ionosphere.
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| The very low frequency (VLF) region from 3 to 30 kHz is used both for submarine communications and for position location by the Omega navigation system. The low-frequency (LF) band, from 30 to 300 kHz, is used for some forms of communication and for the Loran C position-location system. Some radio beacons and weather broadcast stations used in air navigation operate at frequencies in the higher end of the LF band. The medium-frequency (MF) region from 300 kHz to 3 MHz contains the standard AM broadcast band from 0.5 to 1.5 MHz. Long-distance communications and short-wave broadcasting over long distances use frequencies in the high-frequency (HF) band from 3 to 30 MHz because waves in this band are strongly affected by reflections by the ionosphere and least affected by absorption in the ionosphere. The next frequency region, the very high frequency (VHF) band from 30 to 300 MHz, is used primarily for television and FM broadcasting over line-of-sight distances and also for communicating with aircraft and other vehicles. Some early radio-astronomy research was also conducted in this range. The ultrahigh frequency (UHF) region from 300 MHz to 3 GHz is extensively populated with radars, although part of this band also is used for television broadcasting and mobile communications with aircraft and surface vehicles. The radars in this region of the spectrum are normally used for aircraft detection and tracking. Some parts of this region have been reserved for radio astronomical observation. Many point-to-point radio communication systems and various kinds of ground-based radars and ship radars operate at frequencies in the super-high frequency (SHF range from 3 to 30 GHz. Some aircraft navigation systems operate in this range as well. Most of the extremely high frequency (EHF) band from 30 to 300 GHz is used less extensively, primarily because the technology is not as well developed and because of excessive absorption by the atmosphere in some parts of this band. Some advanced communication systems are being developed for operation at frequencies in the "atmospheric windows," where atmospheric absorption is not a serious problem, as are automobile collision-avoidance radars and some military imaging radar systems.
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| Mathematics Definition 1 A complex number is written in the form z = x + jy, where x and y are the real (Re) and imaginary (Im) parts of z and j =
Mathematics Definition 2 Z can also be written in a polar form, z = |z|ejө = |z|
Mathematics Formula 1 ejө = cosө + jsinө
Mathematics Derivation 1 Combine mathematics definition 2 and mathematics formula 1 we get; z = |z|cosө + j|z|sinө
Mathematics Formula 2 x = |z|cosө y = |z|sinө |z| = Ө = tan-1 (y/x)
Mathematics Diagram 1
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Mathematics Derivation 2 Take mathematics definition 1 and replace j with – j and we get; z* = x – jy = |z|e-jө = z and |z| =
Mathematics Rule 1 Equality – If two complex numbers z1 and z2 are given by z1 = x1 + jy1 = |z|ejө1 z2= x2 + jy2= |z|ejө2 then z1 = z2 if and only if x1 = x2 and y1 = y2 or equivalently |z1| = |z2| and Ө1 = Ө2
Mathematics Rule 2 Addition z1 + z2 = (x1 +x2) + j(y1 + y2)
Mathematics Rule 3 Multiplication
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| Mathematics Rule 4 Mathematics Rule 5 Mathematics Rule 6
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| Phasor analysis is a useful mathematical tool for solving problems involving linear systems in which the excitation is a periodic time function. Many engineering problems are cast in the form of linear integro-differential equations. If the excitation, more commonly known as the forcing function, varies sinusoidally with time, the use of phasor notation to represent time-dependent variables allows us to convert the integro-differential equation into a linear equation with no sinusoidal functions, thereby simplifying the method of solution. After solving for the desired variable, such as the voltage or current in a circuit, conversion from-the phasor domain back to the time domain provides the desired result. The phasor technique can also be used for analyzing linear systems when the forcing function is any arbitrary (nonsinusoidal) periodic time function, such as a square wave or a sequence of pulses. By expanding the forcing function into a Fourier series of sinusoidal components, we can solve for the desired variable using phasor analysis for each Fourier component of the forcing function separately. According to the principle of superposition, the sum of the solutions due to all of the Fourier components gives the same result as one would obtain had the problem been solved entirely in the time domain without the aid of Fourier representation. The obvious advantage of the phasor-Fourier approach is simplicity. Moreover, in the case of nonperiodic source functions, such as a single pulse, the functions can be expressed as Fourier integrals, and a similar application of the principle of superposition can be used as well. Figure 13 The simple RC circuit shown in Fig. 13 contains a sinusoidally time-varying voltage Mathematical Form 4 vs(t) = V0sin(ωt + | |
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Method 1 You are given a sinusoidally time varying circuit and you must find expression for the current i(t) the steps that you should follow is therefore; 1. Adopt a cosine reference. 2. Express time-dependant variables as phasors. 3. Recast the differential/integral equation in phasor form. 4. Solve the phasor domain equation. 5. Find the instantaneous value.
Example 1 Using the same circuit in figure 13 you must find the expression for current i(t).
Applying Kirchoff’s voltage law yields; Ri(t) + 1. Adopt a cosine reference. So the system sinusoidal equation should be converted into cosine form. Thus, We used the properties sinx = cos (π/2 – x) and cos(-x) = cos x.
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2. Express time-dependant variables as phasors. Voltage given by step 1 can be expressed in the form, where Next we define the unknown variable (t) in terms of phasor For equations that contain derivative and integrals we use
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3. Recast the differential/integral equation in phasor form. Fill up the equation Ri(t) + With those from step 5 yields, This is simplifies to
4. Solve the phasor domain equation. The phasor current Convert the right hand side of the above equation into the form I0ejө. Thus
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(phasor domain)
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5. Find the instantaneous value. Multiply the phasor current by ejωt, then take the real part we get, Table 3
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