1、 ECTE170 Lecture 8/111Chapter 14 BoylestadSinusoidal response of resistor,inductor and capacitorLow and high frequency response of inductors and capacitorsAverage power and Power FactorBasic Elements and Phasors2Introduction_ The response of the basic R,L,and C elements to a sinusoidal voltage and c
2、urrent will be examined with a special note of how frequency will affect the“opposing”characteristic of each element._ Phasor notation will then be introduced to establish a method of analysis.3The Derivative_ To understand the response of the basic R,L,and C elements to a sinusoidal signal,you need
3、 to examine the concept of the derivative._ The derivative dx/dt is defined as the rate of change of x with respect to time.If x fails to change at a particular instant,dx=0,and the derivative is zero._ For the sinusoidal waveform,dx/dt is zero only at the positive and negative peaks(wt=p/2 and 3p/2
4、)since x fails to change at these instants of time.45The Derivative_ The derivative dx/dt is actually the slope of the graph at any instant of time._ The greatest change in x will occur at the instants wt=0,p,and 2p._ For various values of wt between these maxima and minima,the derivative will exist
5、 and will have values from the minimum to the maximum inclusive._ The derivative of a sine wave is a cosine wave;it has the same period and frequency as the original sinusoidal waveform.67Sinusoidal Response:ResistorRIVorRVIwheretsinIRtsinVRvimmmmmmww For a resistor the voltage and current are in ph
6、ase and are related by Ohms law8The voltage and current of a resistive element are in phase.9Sinusoidal Response:InductorsLIVX)90tsin(V)90tsin(LIv)90tsin(ItcosIdtditsinIiwheredtdiLvmmLomomommmwwwwwwwww For an inductor the current lags the voltage by 90 degrees XL=wL is called the inductive reactance
7、-unit 10Sinusoidal Response:WaveformsInductor11Sinusoidal Response:CapacitorscmmomomommmXC1IV)90tsin(I)90tsin(CVi)90tsin(VtcosVdtdvtsinVvwheredtdvCiwwwwwwwww For a capacitor the current leads the voltage by 90 degrees Xc=1/wC is called the capacitive reactance unit 12Sinusoidal Response:WaveformsCap
8、acitor13Sinusoidal ResponseThe current through a 5 ohm resistor is i=40sin(377t+30)A.Find the expression for voltage across it.14Sinusoidal ResponseThe current through a 0.1H coil is i=7sin(377t 70)A.Find the voltage across it.1516Capacitor Example1718More questions192021Low and High Frequency Respo
9、nse for InductorsInductors:XL=L At low frequencies,and especially DC,the reactance of a inductor is very low(zero for DC)Hence at very low frequencies,an inductor may be considered as a short circuit As input frequencies become very high,the reactance of an inductor approaches infinity Hence at very
10、 high frequencies,an inductor may be considered as an open circuitBoylestad,Prentice Hall 201922Low and High Frequency Response for CapacitorsCapacitors:Xc=1/C At low frequencies,and especially DC,the reactance of a capacitor is very high(infinite for DC)Hence at very low frequencies,a capacitor may
11、 be considered as an open circuit As input frequencies become very high,the reactance of a capacitor approaches 0 Hence at very high frequencies,a capacitor may be considered as a short circuit23Average Power and Power Factor Second term has a zero average value over a cycle and causes no average po
12、wer First term is independent of(a)time and is constant(b)whether v leads or lags i,and will be the Average Power or the Real PowerIn general v=Vm sinwt and I=Im sin(wt-)t2cos(2IVcos2IVp2B)cos(A-B)-cos(AsinAsinBidentity tric trigoneme theUsing)tsin(ItsinVpin timeinstant any at deliveredPower mmmmmmw
13、ww24Average Power and Power Factor)t2cos(2IVcos2IVpppower ousinstantanefor expression theofdisplay GraphicalmmmmwBoylestad,Prentice Hall 201925Average Power and Power FactorcosVIcos2I2Vcos2IVPPowerAveragemmmm where V and I are rms values of the sinusoidal voltage and current respectively The factor(
14、cos)which controls the average power flow is called the Power Factor.For a resistor the Power Factor is unity For an inductor or capacitor Power Factor is zero Another way of finding the Power Factor is to use the expressionVIPFactorPower26Average Power and Power Factor When the power factor is stat
15、ed it is important to state whether it is leading or lagging in addition to its value(note that it lies between 0 and 1.0)27Average Power and Power Factor28Average Power and Power Factor2930Power Factor Fp Power Factor=Fp=cos The term leading or lagging is often written in conjunction with the power
16、 factor.They are defined by the current through the load.If the current lead the voltage then its a leading power factor If the current lags the voltage then its a lagging power factor Capacitive circuits have leading power factors,while Inductive circuits have lagging power factors31Example Power f
17、actor3233Complex NumbersAs an essential tool complex numbers will be used in solving ac circuits Rectangular form Z=a+j b where j is an operator which turns the real number b by 90 in the anti-clockwise direction on the complex plane Polar form Z=Z/Addition is convenient in rectangular form Division
18、/multiplication is convenient in polar formajb34Complex NumbersDefining the rectangular form.35Polar formDefining the polar form.Demonstrating the effect of a negative sign on the polar form.3637Complex Conjugate38Conversion Between Forms39Complex Numbers40Complex Numbers41Maths Operations4243Polar
19、form operations44Phasors and Phasor DiagramsOften addition/subtraction of sinusoidal voltages and currents is required in ac circuit analysis.For example consider the addition of two sinusoidal signalsv1=Vm1 sin(wt+)and v2=Vm2 sin wt We can add the two waveforms on a point-by-point basis as shown(a
20、tedious process!)to obtain vT.Boylestad45Phasors and Phasor Diagrams However it is very convenient to represent the two sinusoidal waveforms v1=Vm1 sin(wt+)and v2=Vm2 sin wt as vectors and then add the vectors.These vectors are called phasors.V2=Vm2/0VmTPhasor diagram of V1,V2 and VTNote the phasors
21、 with underscore in the book shown in boldV1=Vm1/46Phasors and Phasor Diagrams Phasors are characterised by a magnitude and a phase angle It is common to represent the magnitude of a phasor as an rms quantity rather than a peak value They represent a snap shots of the rotating vectors at t=0 They do not carry information about the frequency V2=V2/0VTV1=V1/47Phasor Addition484950
