Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Transcription

1 Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage as functin f time. Mst cmmn wavefrms are sinusidal that are naturally generated in pwer plants by cnverting the water, wind, r nuclear energy t electrical energy carried by sinusidal wavefrm. The cnversin is dne thrugh rtating pwer turbine and generatrs. Sinusidal wavefrms change their plarity (in case f vltage) r their directin (in case f current) with time s the term Alternating Current r AC was tssed fr them in the early develpment. Imprtance f sinusidal wavefrm is tw-flded, the wavefrms generated by rtating generatrs are sinusidal and als the behavir f the real systems t sinusidal excitatin r input remains sinusidal (mre abut this later). Prbably the mst cmmn term in electrnic engineering is signal ; it is gd t knw right nw that a signal is a wavefrm, mst likely a sinusidal, that carries sme useful r desired infrmatin. That ne is the stuff that makes yur phne calls, Internet surfing, MP3 players, and many mre pssible.. In algebra, sinusidal functins (als knwn as trignmetric functins) are usually pltted versus angles in degrees r radians as shwn in Figure and starting at 4 8 degrees in Figure. (Recall that each radian is 57.3.) π Figure. y = sin( α ) degrees Lecture Ntes, Page

2 Figure. y = sin ( α + 4 ) degrees Nte: In electrnic labs the wavefrms are generated by Electrnic Functin Generatrs and are viewed and measured by Oscillscpes. 3. A sinusidal wavefrm in electrnic has a few mre parameters that is the reasn it becmes mre useful because mre parameters means mre flexibility. Let us start with main parameters: Amplitude, Phase, and Frequency. Cnsider the sinusidal wavefrm specified by equatin v( t) = Asin ( π f t + φ). In this equatin A is the amplitude, f is the frequency, andφ is the phase r phase angle. Time is being represented by t, which is cnsidered t start frm zer time fr cnvenience. Study Sectins 5. t 5.6 f the textbk and write the definitin and/r explanatin f the fllwings. Time scale: Instantaneus Value: Lecture Ntes, Page

3 Frequency: Perid: Angular velcity r Angular frequency: Instantaneus Amplitude and Peak value: Peak-t-peak value: Relatinship betweenω, f, and t : Given frequency f 6 Hz, and 8 khz, find the angular frequency. ω = π f Given angular frequencies in the previus, find the crrespnding wavefrm π perids. T = ω Lecture Ntes, Page 3

4 In rder t see the different phase angles, let us graph wavefrms.5sin( π t ) and.5sin( t + 7 ) (in red) π (in green) fr time frmt = t t =.5sec., what is the frequency f these wavefrms? Figure time, in secnds In rder t see the different amplitudes, let us graph wavefrms sin ( π t ) (in red), sin( π t) (in green), and 3 sin( π t) (in blue) fr time frmt = t t =.5sec. is the angular velcity r angular frequency f these wavefrms? Figure 4., what time, in secnds Lecture Ntes, Page 4

5 In rder t see the different frequencies, let us graph wavefrms ( t) sin( π t red), v () t = sin( 5π t) (in green), and v ( t) sin( π t) v = ) (in 3 = (in blue) fr time frm t = t t =.5 sec., what is the frequency and perid f each wavefrms? Figure time, in secnds Recall that the difference between sine and csine waves is just a phase angle. Fpr examples check the fllwings. Asin Acs 3sin cs ( π f t) = Acs( π f t + 9 ) ( π f t) = Asin( π f t 9 ) = Asin( ( 8π t) = sin( 8π t) + sin( 8π t) ( 4π t + ) = sin( 4π t 8 ) π f t + 7 Ntice that in all the abve wavefrms the frequency is the same it is nt usually easy t mix the wavefrms f different frequencies. The wavefrms cntaining different frequencies are called cmpsite wavefrm and ne has t study their spectrum using a spectrum analyzer. T refresh yur algebra and trignmetry try t represent the expressin, sin( 4π t).5cs( 4π t) in ne term, a ( 4π t + φ) sin. ) Lecture Ntes, Page 5

6 4. An imprtant technique t represent sinusidal wavefrms is by using phasrs. In essence phasrs are similar t vectrs with ne significant difference that phasrs rtate in cunter-clck-wise directin with the angular velcityω therefre their lcatins r their directins as ne might say, depends n time. A snap-sht taken frm a phasr P in -space plane (sn we will refer t this as cmplex plane) at t = is shwn in Figure 6. Figure 6. V The length r amplitude f this phasr is specified by V, the phase angle (with respect t cs axis) is Φ. There are 3 different ways t represent this phasr. r a) Cmplex cmpnents: P = ( V csφ) + j ( V sinφ) in this representatin V csφ is the real part and V sin φ is the imaginary part, where j r i =. This representatin des nt include the time dependence and is very useful when cmparisn f several phasrs (see Figure 7). b) Prjectin f the phasr n cs axis: v( t) = V cs ( ω t + φ) = V cs( π t + φ). This includes the time dependence and is useful fr cmputing the instantaneus value f the wavefrms. c) Prjectin f the phasr n sin axis: v( t) = V ( ω t + 9 φ) = V sin( π t + 9 φ) sin. This is similar t the cs representatin. Ntice the phase angle with respect t sin axis is 9 φ. Lecture Ntes, Page 6

7 Phasr diagrams are very useful fr the analysis f the circuits cntaining R, L, and C such as the ne in Figure 7. Figure 7. Figure 8 shws a typical phasr representatin fr the RLC circuit f Figure 7 (actual phasr diagram depends n the value f the cmpnents and the input vltage surce). This phasr diagram illustrates a snap-sht f the rtatin phasrs say at time zer. Ntice that all the phasrs are rtating tgether by angular frequency fω that is exactly the angular frequency f the input surce vltage. We mentined smething abut this prperty in part abve. Figure 8. Lecture Ntes, Page 7

8 Fur phasr are shwn in Figure 8 (there culd be mre), E represents the input sinusidal vltage surce phasr, I is the phasr fr the circuit current, V C and V L are vltage phasr acrss the capacitr and the inductr respectively. When the length r the value f these phasrs and their relative phase angles are knwn, then their cmplete time dependent sinusidal expressins can be written. The frequency fr all f the expressins will be the ne frm the input surce. 5. Althugh sinusidal wavefrms have very nice prperties, nt all the wavefrms are sinusidal. We require that the wavefrms be at least peridic, i.e., they repeat the same wave shapes peridically (every T secnds). The practically imprtant peridic wavefrms are square wave, triangular, saw-tth, etc. Figure 9 shws sme examples. Figure V 5 7 time, ms -.5 V Lecture Ntes, Page 8

9 The same parameters we mentined fr sinusidal wavefrms are als defined fr ther peridic wavefrm such as amplitude, frequency, perid, etc. We will intrduce a cuple f new parameters. Harmnic frequencies: Fr the wavefrms that are nt pure sinusidal harmnic frequencies exist are defined as multiples f the fundamental frequency. Suppse that the perid f the wavefrm is T. Then the fundamental frequency r the first harmnic is given by f =. Other harmnics are: nd harmnic frequency is: f =, 3 rd harmnic T T 3 frequency is: 3 f =, and generally n th n harmnic frequency is: nf =. T T Average: average, DC cmpnent, r ffset f the wavefrm with the perid f T is mathematically defined as t+t v () t dt fr any arbitrary t. We are interested in T t cmputing average f the wavefrms graphically. (See Sectin 5.8 f the textbk.) 6. Effective r RMS value f a sinusidal wavefrm is simply an equivalent DC vltage that energy-wise has the same effect as the sinusidal wavefrm. Ntice that RMS value is defined nly fr sinusidal wavefrms. RMS stands fr Rt-Mean-Square that is mathematically defined as v () t dt. We are nt cncerned with the math T T frmula because the result fr sinusidal is surprisingly simple. Cnsider a sinusidal wavefrm with the peak amplitude f A. The crrespnding RMS value is A simply.77a. (See Sectin 5.9 f the textbk.) Fe example, cnsider a vltage wavefrm f 4 V peak amplitude, and then its RMS value is apprximately.83 V. Figure V time, in secnds Lecture Ntes, Page 9

10 7. Cmplex numbers: Cmplex numbers have tw real cmpnents and written as C = a + j b, where a and b are real numbers and j =. The cmpnent a is called real cmpnent, and b is the imaginary cmpnent f the cmplex number C. There are tw methds f representatin fr cmplex numbers in the cmplex plane. Imprtance f cmplex numbers in the cntext f AC circuit analysis is due t the fact that phasrs can be easily represented in the cmplex plane by their real and imaginary cmpnents. Figure. Imag (j) C b r a φ Real The basic representatins and transfrmatin: a) Rectangular frm: C = a + j b. b) Plar frm: C = r φ. b c) Rectangular t plar transfrmatin: r = a + b φ = tan. a d) Plar t rectangular transfrmatin: a r cs( φ) b = r sin( φ) =. Lecture Ntes, Page

11 All algebraic peratins are pssible n cmplex numbers. We are mre cncerned with the additin and subtractin f phasrs represented by cmplex numbers. These peratins (additin and subtractin) are usually perfrmed in rectangular frm. Cnsider tw cmplex numbersc = a + j b and C = a + j b. Then we can cmpute C ± C = ( a ± a ) + j ( b ± b ). Similarly the additin and subtractin can be dne n mre than cmplex numbers. We will illustrate these peratins graphically in cmplex plane fr tw phasrs in Figure. Figure. Lecture Ntes, Page