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Chapter 4 Alternating Current Circuits 9-12-2015 FCI- F. Univ. 1 Chapter 4: 4-1 AC Sources 4.2 Resistors in an AC Circuit 4.3 Inductors in an AC Circuit 4.4 Capacitors in an AC Circuit 4.5 The RLC Series Circuit 9-12-2015 4.6 Power in an AC Circuit 4.7 Resonance in a Series RLC Circuit 4.8 The Transformer and Power Transmission 4.9 Rectifiers and Filters FCI- F. Univ. 2 Objecties: The students should be able to: Describe the sinusoidal variation in ac current and voltage, and calculate their effective values. Write and apply equations for calculating the inductive and capacitive reactance for inductors and capacitors in an ac circuit. Describe, with diagrams and equations, the phase relationships for circuits containing resistance, capacitance, and inductance. 9-12-2015 FCI- F. Univ. 3 Write and apply equations for calculating the impedance, the phase angle, the effective current, the average power, and the resonant frequency for a series ac circuit. Describe the basic operation of a step up and a step-down transformer. Write and apply the transformer equation and determine the efficiency of a transformer. 9-12-2015 FCI- F. Univ. 4 4-1 AC Circuits An AC circuit consists of a combination of circuit elements and a power source The power source provides an alternative voltage, Dv Notation Note 9-12-2015 Lower case symbols will indicate instantaneous values Capital letters will indicate fixed values FCI- F. Univ. 5 In Last Lec. The output of an AC power source is sinusoidal and varies with time according to the following equation: Δv = Vmax sin ωt Δv is the instantaneous voltage Vmax is the maximum output voltage of the source 9-12-2015 Also called the voltage amplitude ω is the angular frequency of the AC voltage FCI- F. Univ. 6 Resistors in an AC Circuit ΔvR = Imax R sin ωt i= Imax sin ωt The current and the voltage are said to be in phase 9-12-2015 FCI- F. Univ. 7 rms Current and Voltage The average current in one cycle is zero The rms current is the average of importance in an AC circuit rms stands for root mean square Irms Imax 0.707 Imax 2 Alternating voltages can also be discussed in terms of rms values 9-12-2015 DVrms DVmax 0.707 DVmax 2 FCI- F. Univ. 8 Power The rate at which electrical energy is dissipated in the circuit is given by 9-12-2015 P=i2R i is the instantaneous current The heating effect produced by an AC current with a maximum value of Imax is not the same as that of a DC current of the same value The maximum current occurs for a small amount of time FCI- F. Univ. 9 Power, cont. The average power delivered to a resistor that carries an alternating current is Pav I 9-12-2015 2 rms R FCI- F. Univ. 10 Notes About rms Values rms values are used when discussing alternating currents and voltages because AC ammeters and voltmeters are designed to read rms values Many of the equations that will be used have the same form as their DC counterparts 9-12-2015 FCI- F. Univ. 11 Example 1: Solution: Comparing this expression The voltage output of an AC for voltage output with the source is given by the general form expression ∆v = ∆ Vmax sinωt, we see ∆v = (200 V) sin ωt. that ∆ Vmax = 200 V. Thus, the Find the rms current in the rms voltage is circuit when this source is connected to a 100 Ohm resistor. 9-12-2015 FCI- F. Univ. 12 - Current in an Inductor The equation obtained from Kirchhoff's loop rule can be solved for the current DVmax DVmax iL sin ωt dt cos ωt L ωL DVmax π DVmax iL sin ωt I max ωL 2 ωL This shows that the instantaneous current iL in the inductor and the instantaneous voltage ΔvL across the inductor are out of phase by (p/2) rad = 90o 9-12-2015 FCI- F. Univ. 13 Phasor Diagram for an Inductor This represents the phase difference between the current and voltage The current lags behind the voltage by 90o 9-12-2015 FCI- F. Univ. 14 Inductive Reactance Current can be expressed in terms of the inductive reactance DV DV Imax max XL or Irms rms XL The factor is the inductive reactance and is given by: XL = ωL As the frequency increases, the inductive reactance increases 9-12-2015 FCI- F. Univ. 15 Example 33.2 A Purely Inductive AC Circuit In a purely inductive AC circuit, L = 25.0 mH and the rms voltage is 150 V. Calculate the inductive reactance and rms current in the circuit if the frequency is 60.0 Hz. 9-12-2015 FCI- F. Univ. 16 9-12-2015 FCI- F. Univ. 17 Capacitors in an AC Circuit dq iC ωC DVmax cos ωt dt π or iC ωC DVmax sin ωt 2 The current is p/2 rad = 90o out of phase with the voltage 9-12-2015 FCI- F. Univ. 18 Phasor Diagram for Capacitor The phasor diagram shows that for a sinusoidally applied voltage, the current always leads the voltage across a capacitor by 90o 9-12-2015 FCI- F. Univ. 19 Capacitive Reactance The maximum current in the circuit occurs at cos ωt = 1 which gives Imax ωCDVmax DVmax (1 / ωC ) The impeding effect of a capacitor on the current in an AC circuit is called the capacitive reactance and is given by 1 XC ωC 9-12-2015 which gives Imax FCI- F. Univ. DVmax XC 20 Voltage Across a Capacitor The instantaneous voltage across the capacitor can be written as ΔvC = ΔVmax sin ωt = Imax XC sin ωt As the frequency of the voltage source increases, the capacitive reactance decreases and the maximum current increases 9-12-2015 FCI- F. Univ. 21 Example.3 A Purely Capacitive AC Circuit ω=2 πf =377 s-1 9-12-2015 FCI- F. Univ. 22 4-5 The RLC Series Circuit The resistor, inductor, and capacitor can be combined in a circuit The current and the voltage in the circuit vary sinusoidally with time 9-12-2015 FCI- F. Univ. 23 The RLC Series Circuit, cont. The instantaneous voltage would be given by Δv = ΔVmax sin ωt The instantaneous current would be given by i = Imax sin (ωt - φ) φ is the phase angle between the current and the applied voltage Since the elements are in series, the current at all points in the circuit has the same amplitude and phase 9-12-2015 FCI- F. Univ. 24 i and v Phase Relationships – Graphical View The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the inductor leads the current by 90° The instantaneous voltage across the capacitor lags the current by 90° 9-12-2015 FCI- F. Univ. 25 i and v Phase Relationships – Equations The instantaneous voltage across each of the three circuit elements can be expressed as Dv R Imax R sin ωt DVR sin ωt π Dv L Imax X L sin ωt DVL cos ωt 2 π DvC Imax XC sin ωt DVC cos ωt 2 9-12-2015 FCI- F. Univ. 26 More About Voltage in RLC Circuits ΔVR is the maximum voltage across the resistor and ΔVR = ImaxR ΔVL is the maximum voltage across the inductor and ΔVL = ImaxXL ΔVC is the maximum voltage across the capacitor and ΔVC = ImaxXC The sum of these voltages must equal the voltage from the AC source Because of the different phase relationships with the current, they cannot be added directly 9-12-2015 FCI- F. Univ. 27 Phasor Diagrams To account for the different phases of the voltage drops, vector techniques are used Remember the phasors are rotating vectors The phasors for the individual elements are shown 9-12-2015 FCI- F. Univ. 28 Resulting Phasor Diagram The individual phasor diagrams can be combined Here a single phasor Imax is used to represent the current in each element 9-12-2015 In series, the current is the same in each element FCI- F. Univ. 29 Vector Addition of the Phasor Diagram Vector addition is used to combine the voltage phasors ΔVL and ΔVC are in opposite directions, so they can be combined Their resultant is perpendicular to ΔVR 9-12-2015 FCI- F. Univ. 30 Total Voltage in RLC Circuits From the vector diagram, ΔVmax can be calculated DVmax DV DVL DVC 2 R 2 ( Imax R ) Imax X L Imax XC 2 DVmax Imax R X L XC 2 9-12-2015 FCI- F. Univ. 2 2 31 Impedance The current in an RLC circuit is Imax DVmax R X L XC 2 2 DVmax Z Z is called the impedance of the circuit and it plays the role of resistance in the circuit, where Z R X L XC 2 9-12-2015 2 Impedance has units of ohms FCI- F. Univ. 32 Phase Angle The right triangle in the phasor diagram can be used to find the phase angle, φ X L XC φ tan R 1 The phase angle can be positive or negative and determines the nature of the circuit 9-12-2015 FCI- F. Univ. 33 Determining the Nature of the Circuit If f is positive If f is negative XL> XC (which occurs at high frequencies) The current lags the applied voltage The circuit is more inductive than capacitive XL< XC (which occurs at low frequencies) The current leads the applied voltage The circuit is more capacitive than inductive If f is zero 9-12-2015 XL= XC The circuit is purely resistive FCI- F. Univ. 34 9-12-2015 FCI- F. Univ. 35 4-6 Power in an AC Circuit The average power delivered by the AC source is converted to internal energy in the resistor av = ½ Imax ΔVmax cos f = IrmsΔVrms cos f cos f is called the power factor of the circuit We can also find the average power in terms of R 9-12-2015 av = I2rmsR FCI- F. Univ. 36 Power in an AC Circuit, cont. The average power delivered by the source is converted to internal energy in the resistor No power losses are associated with pure capacitors and pure inductors in an AC circuit 9-12-2015 In a capacitor, during one-half of a cycle, energy is stored and during the other half the energy is returned to the circuit and no power losses occur in the capacitor In an inductor, the source does work against the back emf of the inductor and energy is stored in the inductor, but when the current begins to decrease in the circuit, the energy is returned to the circuit FCI- F. Univ. 37 Power and Phase The power delivered by an AC circuit depends on the phase Some applications include using capacitors to shift the phase to heavy motors or other inductive loads so that excessively high voltages are not needed 9-12-2015 FCI- F. Univ. 38 4-7 Resonance in an AC Circuit Resonance occurs at the frequency ωo where the current has its maximum value To achieve maximum current, the impedance must have a minimum value This occurs when XL = XC Solving for the frequency gives ωo 1 LC The resonance frequency also corresponds to the natural frequency of oscillation of an LC circuit 9-12-2015 FCI- F. Univ. 39 Resonance, cont. Resonance occurs at the same frequency regardless of the value of R As R decreases, the curve becomes narrower and taller Theoretically, if R = 0 the current would be infinite at resonance Real circuits always have some resistance 9-12-2015 FCI- F. Univ. 40 Power as a Function of Frequency Power can be expressed as a function of frequency in an RLC circuit av DVrms 2 Rω 2 R ω L ω ω 2 2 2 2 2 o 2 This shows that at resonance, the average power is a maximum 9-12-2015 FCI- F. Univ. 41 Quality Factor The sharpness of the resonance curve is usually described by a dimensionless parameter known as the quality factor, Q Q = ωo / Δω Δω is the width of the curve, measured between the two values of ω for which avg has half its maximum value 9-12-2015 These points are called the half-power points FCI- F. Univ. 42 Quality Factor, cont. A high-Q circuit responds only to a narrow range of frequencies Narrow peak A low-Q circuit can detect a much broader range of frequencies 9-12-2015 FCI- F. Univ. 43 Example 33.7 A Resonating Series RLC Circuit 9-12-2015 FCI- F. Univ. 44