Chapter 33AC, the descriptionResistors in an AC Circuit, Ohm’s LawPhasor Diagram, a useful tool.The power for a resistive AC circuit and the rms current and voltageSlide 6Resistors in an AC Circuit, summaryInductors in an AC circuit, voltage and currentInductors in an AC circuit, voltage leads currentInductive Reactance, the “resistance” the inductor offers in the circuit.Capacitors in an AC circuit, voltage and currentCapacitors in an AC circuit, current leads the voltageCapacitive Reactance, the “resistance” the capacitor offers in the circuit.The RLC series circuit, current and voltageThe RLC series circuit, current and voltage, solved with Phasor DiagramsSlide 16Slide 17Determining the Nature of the CircuitChapter 33Alternating Current (AC)R, L, C in AC circuitsAC, the descriptionA DC power source, like the one from a battery, provides a potential difference (a voltage) that does not change its polarity with respect to a reference point (often the ground)An AC power source is sinusoidal voltage source which is described as Here ( )maxv V sin tD D w=maxvVDDwis the instantaneous voltage with respect to a reference (often not the ground).is the maximum voltage or amplitude. is the angular frequency, related to frequency f and period T as 22 fTpw p= =VDtVDSymbol in a circuit diagram:orvDvDThe US AC system is 110V/60Hz. Many European and Asian countries use 220V/50Hz.Resistors in an AC Circuit, Ohm’s LawThe voltage over the resistor:( )R maxv v V sin tD D D w= =Apply Ohm’s Law, the current through the resistor:( ) ( )maxRR maxVvi sin t I sin tR RDDw w= = �The current is also a sinusoidal function of time t. The current through and the voltage over the resistor are in phase: both reach their maximum and minimum values at the same time.PLAYACTIVE FIGUREThe power consumed by the resistor is ( )222 2maxRR R R RVvp v i i R sin tR RDDD w= � = = =We will come back to the power discussion later.Phasor Diagram, a useful tool.The projection of a circular motion with a constant angular velocity on the y-axis is a sinusoidal function. To simplify the analysis of AC circuits, a graphical constructor called a phasor diagram is used. A phasor is a vector whose length is proportional to the maximum value of the variable it representsThe phasor diagram of a resistor in AC is shown here. The vectors representing current and voltage overlap each other, because they are in phase. xyORtwThe projection on the y-axis is( )yR R sin tw=The power for a resistive AC circuit and the rms current and voltage( )2R maxp p sin tw=( )R maxv V sin tD D w=( )R maxi I sin tw=When the AC voltage source is applied on the resistor, the voltage over and current through the resistor are:Both average to zero.But the power over the resistor isAnd it does not average to zero. The averaged power is:PmaxPavPmaxP( )( )22220012122 4 2TTav maxTmax maxp p sin t dtp ptsin tTwwww=� �= - =� �� ��( )2122 4xsin xdx sin x= -�The power for a resistive AC circuit and the rms current and voltage22rmsav rms rms rmsVp V I I RRDD= � = =So the averaged power the resistor consumes isIf the power were averaged to zero, like the current and voltage, could we use AC power source here?The averaged power can also be written as:PmaxPavPmaxP12 2maxav max maxpp V ID= = �221 12 2maxav maxVp I RRD= =Define a root mean square for the voltage and current:2 2 2 21 1 and 2 2rms max rms maxV V , I ID D= =or2 and 2max rms max rmsV V , I ID D= =One get back to the DC formula equivalent:Resistors in an AC Circuit, summaryOhm’s Law applies. Te current through and voltage over the resistor are in phase.The average power consumed by the resistor isFrom this we define the root mean square current and voltage. AC meters (V or I) read these values.The US AC system of 110V/60Hz, here the 110 V is the rms voltage, and the 60 Hz is the frequency f, so ( )R maxv V sin tD D w=( )R maxi I sin tw=22rmsav rms rms rmsVp V I I RRDD= � = = and 2 2max maxrms rmsV IV , IDD = =2 156 Vmax rmsV VD D= =12 377 secfw p-= =Inductors in an AC circuit, voltage and currentThe voltage over the inductor is( )L maxv v V sin tD D D w= =To find the current i through the inductor, we start with Kirchhoff’s loop rule:0Lv vD D+ =( )0maxdiV sin t LdtD w - =orSolve the equation for i ( )maxVdi sin t dtLDw=( ) ( )2max maxmaxV Vi di sin t dt cos t I sin tL LD Dpw w ww� �= = =- = -� �� �� �or with 2maxmax maxVi I sin t , ILDpww� �= - =� �� �Inductors in an AC circuit, voltage leads currentExamining the formulas for voltage over and current through the inductor:( )L maxv v V sin tD D D w= =Voltage leads the current by ¼ of a period (T/4 or 90° or π/2) . Or in a phasor diagram, the rotating current vector is 90° behind the voltage vector.2maxi I sin tpw� �= -� �� �PLAYACTIVE FIGUREInductive Reactance, the “resistance” the inductor offers in the circuit.Examining the formulas for voltage over and current through the inductor again:( )L maxv V sin tD D w= with 2maxmax maxVi I sin t , ILDpww� �= - =� �� �This time pay attention to the relationship between the maximum values of the current and the voltage: maxmaxVILDw=This could be Ohm’s Law if we define a “resistance” for the inductor to be: LX Lw=And this is called the inductive reactance. Remember, it is the product of the inductance, and the angular frequency of the AC source. I guess that this is the reason for it to be called a “reactance” instead of a passive “resistance”. The following formulas may be useful: and max L max rms L rmsV X I , V X ID D= =( )L max Lv I X sin tD w=Capacitors in an AC circuit, voltage and currentThe voltage over the capacitor is( )C maxv v V sin tD D D w= =To find the current i through the capacitor, we start with Kirchhoff’s loop rule:0Cv vD D+ =( )0 with maxq dqV sin t , iC dtD w - = =orSolve the first equation for q, and take the derivative for i ( )maxq C V sin tD w=( )2max maxdqi C V cos t I sin tdtpw D w w� �= = = +� �� �or( )1 with 2maxmax maxVi I sin t , ICDpww-� �= + =� �� �Here I still like to keep the Ohm’s Law type of formula for voltage, current and a type of resistance.Capacitors in an AC circuit, current leads the voltageExamining the formulas for voltage over and current through the capacitor:( )C maxv V sin tD D w=2maxi I sin tpw� �= +� �� �Current leads the voltage by ¼ of a period (T/4 or 90° or π/2) . Or in
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