Chapter 14 Waveforms PAGE 1

Transcription

1 Chapter 4 Waveform Chapter 4 Waveform Introduction Waveform Tranient Firt Order Tranient RL Circuit RC Circuit LaPlace LaPlace Operational Rule Partial Fraction Expanion Alternate Approach Fourier Serie Signal - Modulation Modulation Type Amplitude Modulation (AM) Angle Modulation Frequency Modulation (FM) Phae Modulation Sampled Meage Digital - Pule Modulation Signal tranmiion dbm Noie Propagation - Tranmit Reflection RLC Sytem Repone RLC Equation Sytem Repone Characteritic Tranfer Reonance Serie Parallel Duality Firt Order Exemplar Application PAGE

2 4. Introduction Signal and waveform are not normally part of the tudy for electric machine. However, with the growing amount of electronic control and with ditortion on the power line from witched mode power upplie, the waveform i often complex. Therefore, thi chapter i provided a a reference to ait with thoe challenge. Signal that are encountered can be a contant, direct current (DC), they can be repetitive, alternating current (AC), or they can be hort term, tranient. The circuit element repond differently to each type ignal. Thi chapter will addre waveform and tool to analyze their impact on ytem performance. The time domain ignal repone or olution contain all the component. yt () = F+ ( I Fe ) t τ co( ω t+ θ ) 4. Waveform By far, the inuoid i the mot common repetitive waveform in electrical ytem. It i the phyical reult due to the rotational motion of machine in a magnetic field. The waveform definition follow. voltage = v = V coωt frequency = f = ( hertz) T ω = π f ( radian /ec) o average value = VDC = dt = for inuoid T T o effective value = VRMS = = v dt T T V v(v) Vo = peak Vrm = effective Time T (ec) For multiple waveform, ue uperpoition. For effective or root mean quare (RMS), thi i quare root of the um of the quare. V V V RMS = RMS + RMS +... Generally, AC value of V & I are given in RMS. The frequency i aumed contant. For a Volt, 6Hz voltage waveform v= co(π 6 t) V = o VO = 4volt PAGE

3 4.3 Tranient Tranient are waveform that exit for a hort period of time. Waveform are determined by the circuit element. Since there are only three element, the mot complex circuit i a econd order. The characteritic olution for a ytem circuit i the time varying equation that decribe the exponential decay after a ignal i applied. The variable, y, can repreent either current or voltage. where yt () = F+ ( I Fe ) t τ co( ω t+ θ ) F = final value (t= ) I = Initial Value (t=) τ = time contant ω = LC 3.3. Firt Order Tranient Firt order ytem are very common, ince they are the model of a imple ytem. Firt order ytem have a reitor and either a capacitor or an inductor. Firt Order Circuit RC or RL Form: di v = L + Ri dt dv v i = C + dt R Characteritic Solution Repone to a tep input (DC) y() t = F + ( I F) e t τ For Capacitor: dv i = C dt Voltage doe not change intantaneouly Open circuit under DC condition Capacitor dicharge to VC ( ) = Initial voltage = ource voltage di For Inductor: v= L dt Current doe not change intantaneouly Short circuit under DC condition PAGE 3

4 Proce: Plot: Inductor diipate to I ( ) = I Initial current = ource current Find τ L τ = time contant= RC or L / R = time for exponent to be e - e ue equivalent circuit w/o ource to get RC or RL (Thevenin Impedance) deactivate all the ource and replace with internal Z reduce to ingle equivalent RC or RL Find y() ue circuit (KVL) w/ element a ource Find y(final) ue circuit (KVL) w/ element a limit Initial lope = F I τ Tranfer function = repone excitation = output input PAGE 4

5 4.3. RL Circuit Standard calculu form di Vo = L + Ri dt Inductor i hort circuit in final tate. il ( ) = Vo il ( ) f = R L τ = R General olution yt () = F+ ( I Fe ) t τ Current olution t Vo V o τ i = + ( ) e R R Rt V o L i = ( e ) R V o R i di L dt L τ = R Sec PAGE 5

6 4.3.3 RC Circuit Standard calculu form dv v i = C + (calculu form) dt R Capacitor i open circuit in final tate. i, v c cannot change intantaneouly vc ( ) = Vo vc ( ) f = τ = RC General olution Voltage olution yt () = F+ ( I Fe ) t τ v= V + ( V V ) e τ F I F t v c o t = V e τ Vo Vc RC ec PAGE 6

7 4.4 LaPlace A tandard waveform i defined in term of time and frequency. A mathematical tranform i often ued to provide a different mathematical tool. The phaor repreentation i one tranform that applie to teady tate alternating circuit. LaPlace tranform are ued for many manipulation of the inductor and capacitor element. The function can be tranformed from time to the domain, which repreent a tationary and rotational component. =σ+ jω The mot ued tranform pair are illutrated. f () t f () t F () I () f() t = m f () t = t t f () t = e αt.37 = e α e αt α + -α e α f () t = te αt α te αt ( α) + -α PAGE 7

8 inωt jω - ω inω t + ω - π ω jω inωt jω - - π ω coω t + ω jω αt e inωt π ω α e αt inωt ω ( + α) + ω α jω jω e αt coωt π ω α e αt coωt + α ( + α) + ω α jω jω PAGE 8

9 4.5 LaPlace Operational Rule The mathematical manipulation of the time function and the LaPlace tranform follow defined rule. f () t + f () t F() + F () ) ) af() t af() 3) 4) 5) df() t = f () t F() f() dt f t F f f t () ( ) () () f ( t ) dt F( ) PAGE 9

10 4.5. Partial Fraction Expanion F () N () N () K K D() ( p )( p ) K ( p ) ( p ) = = = + + N( p ) K = ( p ) F( ) = jk = KjK = K = p 3 ( p p)( p p3) K f() t = K e + K e +K pt pt K PAGE

11 4.5. Alternate Approach F(), F () are from the table, known tranform F () = KF () + KF() +K Right hand ide to common denominator, equate numerator, olve for K, K f() t = K f () t + K f () t +K Impedance in -domain Impedance Admittance V Z I I Y = Z = V Z = R Y R = R ZL = R = L L Z C Y L = YC = C = C PAGE

12 4.6 Fourier Serie Any alternating waveform can be repreented by the ummation of a fundamental ine wave and it multiple called harmonic. Thi ummation i called a Fourier erie. y = Y + Y in( ωt+ θ ) + Y in( ωt+ θ ) + K + Y in( nωt+ θ ) The term y i the intantaneou value at any time. It can be either current or voltage. The Y term i the contant offet, average, or DC component. The Y term are the maximum amplitude for each of the harmonic frequencie. The angular frequency ω i πf. The phae hift angle repreent the time delay between the reference voltage waveform and the current. The n ubcript and coefficient of frequency indicate the harmonic number. The time domain i a plot of the Y amplitude veru time for the curve. The frequency pectrum i a plot of harmonic amplitude veru harmonic frequency number. An odd function i created with the um of the odd harmonic. A ine wave i the baic example. If the waveform ha the pattern of a fundamental ine wave, then it i odd. y() t = y( t) An even function i created with the um of the even harmonic. A coine i the baic example. If the waveform ha the pattern of a fundamental coine wave, then it i even. y() t = y( t) A function that contain both even and odd harmonic will have pike. A pule and awtooth are example. n n PAGE

13 The very definition of Fourier erie indicate the erie can take everal form. A coine i an orthogonal hift to a ine wave. A a reult, a common repreentation i to ue coine term for the even harmonic and ine term for the odd. Then the even harmonic become odd coefficient for the coine term. Although thi i a common repreentation, it i not a eay to viualize or to obtain a pectrum a the imple inuoidal form. The Fourier erie can be decompoed into the um of even and odd part. f () t = f () t + f () t e o The even part can be repreented by the Fourier erie ao fe() t = + anco( nωt) n= The odd part can be repreented by the Fourier erie f () t = b in( nωt) o n= The coefficient are imilar. T ao = f() t dt T T n an = f()co( t nωt) dt T T bn = f()in( t nωt) dt T PAGE 3

14 4.7 Signal - Modulation Modulation i the proce of combining two ignal into one combined waveform. The combination can be through an adder or multiplier. The mathematical reult look very much like the Fourier erie. Modulation i imilar to wrapping a paper note around a rock and toing the combination. The carrier wave, or rock, provide a vehicle for paing the information. The information i on the paper note. In it baic form, the carrier i a ingle waveform. xt () = Ain( π ft+φ ) c c c A c = amplitude, f = frequency, and θ i the phae hift. Therefore only the amplitude, frequency, or phae can be changed or modulated. The meage, information, or baeband ha a imilar form. Uually the meage wave ha a fixed or zero phae hift. mt () = A in( π f t+φ ) m m m 4.7. Modulation Type There are numerou variation to the type of modulation. There are three analog modulation technique baed on the variable in the waveform.. Amplitude modulation (AM) Frequency modulation (FM) Phae modulation (PM) Special variation of thee technique have unique characteritic that affect bandwidth and power. Angle modulation include both frequency and phae modulation, ince they are operated on by the inuoid. Double ideband modulation (DSB) i AM with the carrier removed. Single-ideband modulation (SSB) i DSB with one of the ideband removed. There are three fundamental ampling or digital modulation technique. Pule amplitude modulation (PAM) Pule frequency modulation (PFM) Pule phae modulation (PPM) Variation of thee technique reult in a variety of keying procee. The proce of modulation and demodulation i called a modem. Pule code modulation include both frequency and phae. Amplitude hift key modulation (ASK) Frequency hift key modulation (FSK) Binary-phae hift key modulation (BPSK) Quadrature-phae hift key modulation (QPSK) Quadrature amplitude modulation (QAM) PAGE 4

15 4.7. Amplitude Modulation (AM) Amplitude modulation mixe the information or meage with the carrier amplitude. The general form of amplitude modulation i to add a function of the meage to the carrier amplitude. y() t = [ A + k m()]in( t π f t) c a c For a ingle waveform, k a i unity. k = a The amplitude varie with the carrier and the ignal. The expanded form illutrate the three component, carrier + lower ideband - upper ideband. y() t = [ A + m()]in( t πf t) c y( t) = [ A + A in( πf t]in( πf t) c m c c c yt ( ) = Acin( π ft c ) + Amco(πfc πfm) t Amco(π fc + πfm) t The modulation index i the depth of the variation around the original level of the carrier, A c. When multiplied by, it i the percent modulation. m am ΔA = = A c A A m c The power in an AM ignal i the um of the power in the carrier and the power in the ignal. P= P + P c m Ac = ( )(( km( t)) ave + ) AM ignal can be demodulated with an envelope detector or a ynchronou demodulator. A double ideband (DSB) ignal would contain the upper and lower ideband information but would not have the carrier. DSB ignal can be demodulated with a ynchronou demodulator. A Cota loop i a common technique. Single ideband (SSB) can be either the lower or upper ideband information only without the carrier or the other ideband. AM ignal can be demodulated with a ynchronou demodulator or by carrier reinertion and envelope detector. The bandwidth ha a lower frequency of f c -f m, center frequency f c, and an upper frequency of f c +f m. BW = f f h l PAGE 5

16 4.7.3 Angle Modulation Angle modulation mixe the ignal a a component of the carrier inuoid which include the frequency and phae term. In eence the ignal become the phae term. xt () = Ain( π ft+φ ) c c c y() t = A in[ π f t+ m()] t c c y( t) = A in( π f t)co( m( t)) + A co( πf t)in( m( t))] c c c c y( t) = A in( πf t) co( A in( π f t)) + A co( πf t) in( A in( πf t))] c c m m c c m m Thi i obviouly a very complex function with numerou frequency component. There are infinite ideband to the ignal. However, the amplitude of mot deteriorate quickly. Frequency modulation and phae modulation each ue elect component of thi waveform. The phae deviation or hift i a function of the meage or information. A dicued earlier, it i aumed that the meage phae hift i zero. The function, k p, i the phae modulation index. φ () t = k m() t p The intantaneou phae i the carrier angle added to the ignal. Thi i the angle within the carrier wave in term. φ () t = π f t+φ () t i c The intantaneou frequency i the change of intantaneou phae with time. The intantaneou frequency i the carrier frequency plu the frequency deviation. d fi = φi dt d = fc + φ() t dt = f +Δf The frequency deviation i the change in the phae, which i the change in the meage with time. d d Δω = φ () t = km() t dt dt The meage bandwidth i the frequency of modulation, f m. BW = f m c m The bandwidth of an FM & PM ignal i approximated uing Caron rule. BWy = ( Δ f + fm) = ( m + ) f fm m d ω= i φi dt d = π fc + φ( t) dt =ω +Δω c PAGE 6

17 4.7.4 Frequency Modulation (FM) Frequency modulation mixe the information or meage with the carrier frequency. The amplitude i contant. The reult i the carrier varie above and below it idle or normal frequency, f c. A the voltage amplitude of the modulating ignal increae in the poitive direction from A to B, the frequency of the carrier i increaed in proportion to the modulating voltage. Frequency modulation i adding the carrier frequency and a function of the meage. f () t = f + k m() t i c f The modulation index or factor i the maximum deviation in frequency, Δf, divided by the modulation frequency. When multiplied by, it i the percent modulation. Δf m fm = fm The frequency modulator contant i the frequency deviation divided by the amplitude of the modulating or meage ignal. Δf k f = A m PAGE 7

18 4.7.5 Phae Modulation Phae modulation i another component of angular modulation that i ometime referred to indirect FM. Note that the phae i part of the inuoid. Here, the amount of the carrier frequency hift i proportional to both the amplitude and frequency of the modulating ignal. The phae of the carrier i changed by the change in amplitude of the modulating ignal. The modulated carrier wave i lagging the carrier wave when the modulating frequency i poitive. Phae modulation i manipulation of the angle of the carrier and a function of the ignal. φ () t = π f t+ k m() t i c p The modulation index i the peak phae variation. m pm Δf =Δφ () t = f m The phae modulation contant depend on both the frequency and amplitude. It i the ratio of the phae deviation to the meage. φ() t k p = mt () PAGE 8

19 4.7.6 Sampled Meage A meage, m(t) can be recreated from uniformly paced ample. The ampling frequency, called the Nyquit frequency f N,, mut be at leat twice a fat a the highet frequency being recreated. f N = = f T Digital - Pule Modulation Pule or digital modulation i frequently ued to tranmit ampled meage. Analog to digital converion i a two tep proce. Firt, ampling change the analog ource to a erie of dicrete value, called ample. Second, quantization, convert each ample to a number. The number of quantization level, q, i the two power of the number of bit. q = n The bandwidth required i inverely proportional to the invere of twice the pule length or duration, T. Thi i called the Shannon bandwidth when the Dimenionality, D i included. For minimum bandwidth, D=. D BWS = T The meage bandwidth, W, and the number of bit determine the minimum modulated bandwidth, BW. BW nw = W log q PAGE 9

20 4.8 Signal tranmiion 4.8. dbm Signal power can be meaured in watt. However, comparion value and mall ignal are meaured in decibel. Pignal db = log ( ) P ref When the reference i on milliwatt, then the decibel are reference a dbm. P ignal dbm = log ( ) mw A a reult a milliwatt ignal i dbm. dbm = mw 4.8. Noie Noie i a random or background ignal that may interfere with the meage or information. Signal-to-noie ratio i an indication of the power ratio between the deired information and the background noie. The ymbol are SNR or S/N. P S/ N = P ignal noie Often the expreion i in term of decibel (db). Pignal Aignal S/ N( db) = log ( ) = log ( ) P A noie noie In a digital ignal, the number of bit in each value determine the SNR. Noie in a digital ignal i dependent on the converion proce. The dynamic range i an expreion of the SNR. n S / N( db) = DR( db) = log ( ) White noie create a thermal noie power, P, in watt that i dependent on the bandwidth, Δf in Hertz and temperature, T in degree Kelvin. Thi i alo the thermal noie that will be created by electron activity in a reitor and i called Johnon noie. P = K T Δf T B K J KB = = K O T = T K Boltzmann' contant C For current or voltage acro the reitor the power ha the tandard relationhip. V P= = R I R Thermal noie at room temperature i dependent on the bandwidth. The unit are decibel. PdB ( ) = 74 + log( Δ f) The total noie figure for a erie of tranfer function or amplifier i baed on the ratio of the noie figure for each tage, F, to the gain ratio of each tage, G. The noie figure and gain mut be converted to the power ratio from db. F F F3 FT = G GG PAGE

21 4.8.3 Propagation - Tranmit The velocity of propagation of a wave i the ditance the wave will travel in one time period. If the ditance i one wavelength, λ, then the velocity i the ratio of the wavelength to the frequency. u p = d t = λ f In free pace, the propagation velocity i the peed of light. 8 c = x The velocity of a wave on a tranmiion line i imply the ratio of the ditance to the time it take for the wave to propagate. For a reflected wave, the ditance i twice the length becaue of the trip length and back. u p d = t Tranmiion of wave involve the power denity in Watt per quare meter. It i the ratio of the power tranmitted to the orthogonal area that the waveform trike. A pherical hape i the normal pattern of an omni-directional wave. Ptranmitted P Pdenity = = x A 4 π R R = range from antenna, radiu of phere Antenna can direct power in pecific direction. The gain of the antenna i the radiation intenity in a particular direction divided by the power that would be radiated from an omni-directional or iotropic antenna. G = Effective radiated power Iotropic rated power Power i diipated a a waveform propagate. The attenuation or lo in free pace depend on the velocity of light. In other medium, the velocity of propagation hould be ued. The lo i db, ditance i m, and frequency i Hz. P f 4πd = log c/ f Characteritic impedance i the oppoition in a circuit that connected to the output terminal of a line will caue the line to appear infinitely long. It i the electric and magnetic property of the material that impact the velocity of propagation. Z μ = = = u pμ ε u ε p The electric property i permittivity in Farad per meter, Fd/m. It i a factor of the free air. ε=εε r ε = x Fd/m The magnetic property i permeability in Henrie per meter, Hy/m. μ=μμ μ = π r 7 4 x Hy/m From thee three preceding concept, the impedance of free pace air i calculated. Z = 377Ω PAGE

22 Becaue of the definition of inductance and capacitance in relation to permeability and permittivity, characteritic impedance can be found in term of circuit element. Z = L C Reflection Maximum power tranfer occur when the load i equal to the ource or characteritic impedance. When a dicontinuity occur on a line or a load i connected that doe not match the characteritic impedance, the waveform will be reflected and oppoe the meage ignal. The reflection coefficient decribe both the magnitude and phae hift of the reflection. The coefficient i the ratio of the complex forward voltage to the complex revere wave voltage. Z Z V f Z + Z V L Γ= = L r Standing wave ratio i the maximum power over the minimum power tranferred. SWR i dependent on the reflection coefficient. SWR = +Γ Γ Voltage SWR i the maximum voltage over the minimum voltage node. VSWR only contain the magnitude of reflection coefficient. V +Γ max VSWR V = = Γ min The reflection coefficient ha the following range of value. Γ = : maximum negative reflection, line i hort-circuited, Γ = : no reflection, when the line i perfectly matched, Γ = + : maximum poitive reflection, line i open-circuited. At the maximum node the wave interfere poitively and add. At the minimum node, the wave are colliding and ubtract. Vmax = Vf + Vr = Vf ( + Γ) V = V V = V ( Γ) min f r f Tranmiion line propertie are defined in term of propagation contant. Propagation contant i inverely proportional to the wavelength. The ditance i meaured from the load. π β= λ j d V( d) = V e + V e + β jβd j d Id ( ) = Ie + Ie = + β jβd Z + jz tan( βd) tan( ) L Zin Z Z + jz L β d PAGE

23 4.9 RLC Sytem Repone 4.9. RLC Equation The three element, RLC can be arranged in erie or it dual parallel. Thi i a econd order ytem. The analyi of the circuit can be made in many domain. Typically the time domain i the tarting point. However, the Calculu required make the mathematic interpretation difficult. For that reaon numerou tranform are ued. The math of the tranform will not be developed, but the correpondence i apparent from the table. The duality of the circuit i intriguing. Function Serie Parallel Reference Same current through all element Same voltage acro all element Diagram Error! Not a valid link. Fundamental dq dq vt () = L + R + q dt dt C Time di vt () = L Ri idt dt + + C LaPlace V() = ( L+ R+ )() I C Sinuoidal Steady State V( jω ) = ( jω L+ R+ ) I( jω) jωc d φ dφ = + + φ it () C dt R dt L dv it () = C Rv vdt dt + + C I() = ( C+ + ) V() R L I( j ω ) = ( j ω C + ) V( j ) R + jωl ω Several obervation can be made about the relationhip. d j dt = = ω dt = = jω dq q' i dt = = dφ dt = φ ' = v 4.9. Sytem Repone The ytem repone i the olution to the econd order equation. t yt () = F+ ( I Fe ) τ co( ω t+ θ ) Time contant i the time it take for a ignal to ettle o that the exponential decay. L τ= RC = = time contant R Characteritic Tranfer Tranfer function are often ued a a model for a ytem. PAGE 3

24 Function Serie Parallel Tranfer function I() Y() = V() Characteritic Y() = L + R + C Standard form Reonance Y() = Y() = / L R + + L LC / L +Δω +ω V() X() = I() Z () = C + + R L Z () = Z () = / C + + RC LC / C + Δω + ω Reonance Frequency i inverely related to time. Angular frequency i one complete revolution of cycle of the frequency. ω = π f Reonance i a very ignificant concept that may be a boon or ban to electrical ytem. Reonance i the frequency where the magnetic (or inductor) energy equal the electric (or capacitor) energy. ω = LC Since the energie are balanced, it flow from one to the other reulting in a inuoidal frequency. The natural frequency i the ocillation determined by the phyical propertie. Reonant frequency i a created ocillation that matche the natural frequency. Reonance i the frequency at which the input impedance i purely real or reitive. The frequency repone ha a roll-off on either ide. The tranition i called the cut-off frequency. ω =ωclω ch Bandwidth, Δω, i the range between the upper and lower cut-off frequencie. The bandwidth i alo called the pa band or bandpa. Δω = ω ω ch cl Δω ω cl =ω Δω ω ch =ω + Quality factor or electivity i the harpne of the peak at reonance. Q ω = Δω Damping i the effect of reitance on the rate that a ignal i tabilized to teady tate. Undamped implie that there i no reitance, R=. The damping coefficient i dependent on the natural frequency and i inverely proportional to twice the quality factor. Some author ue the ymbol alpha, α, rather than zeta, ς. Note thi i alo the real term of the LaPlace, σ. PAGE 4

25 ω R actual damping ζ= = = Q L/ C critical damping The range of value for the camping coefficient reflect how quickly the waveform will ettle and whether it will overhoot. Under-damping reult in ocillation or ringing, over-damping reult in a low exponential approach to tability, critical-damping i the tranition between ocillation and exponential. ζ < under-damped = ocillation ζ = critical-damped = tranition ζ > overdamped = exponential The relationhip between the variou factor can be decribed in term of the quality factor. Q = ω = ω Δω ζ Damped reonance, ω d, i a hift from the reonant frequency caued by the damping. ω =ω ζ d The root of the characteritic equation ha the real part a damping coefficient and the imaginary part a the damped reonance. For the econd order, there are two root. = ζ ± jω, d Serie Parallel Duality Comparion of the tandard form and the reonance equation reveal the duality of impedance and admittance. The ymmetry of the duality reolve to a reciprocal form at reonance. Function Serie Parallel Quality factor Q = X R Q = R X Quality factor L Q = R C C Q = R L Firt Order A firt order ytem ha a reitor and either a capacitor or inductor. Therefore, there i no ocillation. However, there i till a cut-off frequency that i the invere of the time contant. R( jω) + = ωc = Time Contant = RC C RC R L( jω) + R( jω) = ωc = Time Contant = L L R End of chapter PAGE 5

26 4. Exemplar An exemplar i typical or repreentative of a ytem. Thee example are repreentative of real world ituation. Problem Conider the circuit hown below. R and R are 5Ω reitor. R 3 i a Ω reitor and R 4 i a 5Ω reitor. Z i a μf capacitor, and V i a V ource. The time contant of the circuit i mot nearly (A) (B) (C) (D) 85 μs 38 μs 55 μs 4 μs SOLUTION: Redraw the circuit to make it eaier to ee The reitance can be combined to determine the equivalent reitance of the circuit. R = R + R + ( R // R ) eq 4 3 R = 5Ω+ Ω+ (5 Ω// 5 Ω ) eq R = 5Ω+.5Ω= 7.5Ω eq The time contant of a RC circuit i τ = R C = 7.5Ω μf = 55μS eq eq The anwer i (C) ( )( ) PAGE 6

27 Problem Conider the circuit hown in the problem above, and recreated below. R and R are 5Ω reitor. R3 i a Ω reitor and R4 i a 5Ω reitor. Z i a mh capacitor, and V i a V, 6Hz ource. The witch ha been cloed for a ignificant period of time. The voltage acro the inductor i mot nearly. (A) (B) (C) (D) SOLUTION Impedance of the Inductor Z Z = jπ 6( mh) = j7.54ω Redraw with all impedance RA = R3+ R4 + Z = 35 + j7.54ω R = 5 Ω // R =.6 + j.664ω B A V A ( RB ) V(.6 + j.664 Ω) = = = j.8 ( R + R ) (5.6 + j.664 Ω) B V ( Z ) ( j.8 V)( j7.54 Ω) A VZ = = =.83+ j.3v = V ( RA + Z) (35+ j7.54 Ω) The anwer i (B) PAGE 7

28 Problem 3 What i the time contant of the figure hown? V 3MΩ.μF SOLUTION: The time contant of an RC circuit i τ = RC 6 6 ( 3 )(. ) = =.6 econd PAGE 8

29 Problem 4 In the figure below, the witch ha been open for a ignificant period of time and i cloed at t=. What i the current in the capacitor at t= +?. SOLUTION: The capacitor, at t=+, act a a hort circuit. The current through the capacitor then i determined by the voltage and the reitance V V 6 i = c A t 6 Z = = + 6 Ω = PAGE 9

30 Problem 5 In the figure below, the witch ha been open for a ignificant period of time, and i then cloed at t=. What i the current through the two capacitor at t= +? 75Ω.F V.F 5Ω Ω SOLUTION: If the witch i opened for a ignificant period of time the capacitor on top of the circuit i charged to V, and the capacitor in the middle of the circuit i dicharged to V. At t= +, the capacitor are modeled a voltage ource with the charged voltage. The equivalent circuit i hown below The voltage acro the 5Ω reitor i V, o i B =A. KVL on the left loop i V i (75 Ω) V = V i = =.33A 75Ω KVL on the right loop i V + V + i ( Ω ) = V i = =.5A Ω KCL i i i i = A B.33A i.5a= i A A =.33.5 =.83A The current through the top capacitor i i =.5A The current through the middle capacitor i i A =.83A PAGE 3

31 Problem 6 In the figure below, the witch ha been open for a ignificant period of time. The witch i cloed at t=. Find the current through the reitor at t= +, and at t=.5. Find the energy in the inductor at t=.5. SOLUTION: The current in an inductor cannot change intantaneouly, o i ( ) L A + = The general olution for a firt order RL circuit i Rt V L it () = e R 5V i() = e Ω =.39A ( Ω)(.5) 8H The energy in the inductor i found uing WL = Li W = H = J L (8 )(.39).85 PAGE 3

32 Problem 7 A carrier wave of MHz i amplitude modulated by an audio ignal of.5 khz. What are the upper and lower limit of the reulting modulated ignal bandwidth? SOLUTION: fc 6 = Hz fm 3 =.5 Hz Lower ideband frequency - Upper ideband frequency fc fm =.5 Hz =,998,5Hz 6 3 fc fm = +.5 Hz =,,5Hz PAGE 3

33 Problem 8 A MHz carrier i frequency modulated by a 65kHz information ignal. The information ignal ha a V amplitude, and a frequency modulator contant of Hz/V. What i the bandwidth? SOLUTION: Caron Rule BW ( ak + f ) f m Hz 3 BW = ( V) + 65 Hz V = 3, Hz 3kHz 4. Application Application are an opportunity to demontrate familiarity, comfort, and comprehenion of the topic. PAGE 33

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