Chapter 6: Alternating Current

Transcription

1 hapter 6: Alternating urrent 6. Alternating urrent.o 6.. Define alternating current (A) An alternating current (A) is the electrical current which varies periodically with time in direction and magnitude. The usual circuit-diagram symbol for an A source is. Extra Knowledge: Alternating current can be used to create a changing magnetic field, and changing magnetic fields can be used to create alternating current. This relationship between alternating current and magnetic fields makes three important devices possible: alternator, motor and transformer..o.o 6.. Sketch and interpret sinusoidal A wform 6..3 Use sinusoidal voltage and current equations o sint o sint Phase (Phase denotes the particular point in the cycle of a wform, measured as an angle in degrees/ radian) BP3 FYS

2 Terminology in A Peak (maximum) current ( ) Definition: Magnitude of the maximum current. Peak (maximum) voltage ( ) Definition: Magnitude of the maximum voltage. Frequency ( f ) Definition: Number of complete cycle in one second. Unit: Hertz (Hz) or s - Period ( T ) Definition: Time taken for one complete cycle. Unit: second (s) Angular frequency ( ) Unit: radian per second (rad s - ) Example Equation: f or T Question Solution Figure shows the variation of voltage with time for a sinusoidal A current. Determine a. the frequency b. the phase angle c. the peak-to-peak voltage and d. write the expression for the graph The current in an A circuit is given by the expression: 5mA sin 5 t Sketch a -t graph for the A circuit. Determine the current when t = 5 s. BP3 FYS

3 6. oot Mean Square ().O 6.. Define root mean square () current and voltage for A source.o 6.. Use voltage and current equations power dc rage power ac oot mean square current ( ) is defined as the effective value of A which produces the same power (mean/rage power) as the steady d.c. when the current passes through the same resistor. oot mean square voltage/ p.d ( ) is defined as the value of the steady direct voltage which when applied across a resistor, produces the same power as the mean (rage) power produced by the alternating voltage across the same resistor. **Equations and are valid only for a sinusoidal alternating current and voltage** Average value for one complete cycle is zero The rage power: elationship between P and P P BP3 3 FYS P P P

4 Example Question A sinusoidal, 6. Hz, ac voltage is read to be by an ordinary voltmeter. Determine Solution a. the maximum value the voltage takes on during a cycle b. the equation for the voltage The alternating potential difference shown above is connected across a resistor of k. alculate a. the current, b. the frequency, c. the mean power dissipated in the resistor. An A source = sin t is connected across a resistor of. alculate a. the current in the resistor. b. the peak current. c. the mean power. Exercise Question A voltage = 6 sin πt is applied across a Ω resistor. Determine a. the reading of the A ammeter in series with the resistor b. the peak current and mean power Answer: An A current is given as = 5 sin (t) where the clockwise direction of the current is positive. Find a. the peak current b. the current when t = / s c. the frequency and period of the oscillation. Answer: 5 A; 4.55 A; 3.88 Hz;.34 s BP3 4 FYS

5 6.3 esistance, eactance and mpedance.o.o 6.3. Sketch and use phasor diagram and sinusoidal wform to show the phase relationship between current and voltage for a single component circuit consisting of pure resistor, pure capacitor and pure inductor Define and use capacitive reactance, inductive reactance, impedance and phase angle Phasor Diagram Phasor is defined as a vector that rotates anticlockwise about its axis with constant angular velocity. A diagram containing phasor is called phasor diagram. t is used to represent a sinusoidally varying quantity such as alternating current (A) and alternating voltage. t also being used to determine the phase angle (is defined as the phase difference between current and voltage in A circuit). nstantaneous value θ sin ON OP Phasor diagram (not including the circle) Sinusoidal Wform esistance, reactance and impedance Key Term/Ω esistance, eactance, X apacitive reactance, X nductive reactance, X mpedance, Z Meaning Opposition to current flow in purely resistive circuit. Opposition to current flow resulting from inductance or capacitance in A circuit. Opposition of a capacitor to A. Opposition of an inductor to A. Total opposition to A. (esistance and reactance combine to form impedance) BP3 5 FYS

6 Pure resistor in the A circuit The current flows in the resistor is The voltage across the resistor at any instant is The phase difference between and is n pure resistor, the voltage is in phase with the current and constant with time (the current and the voltage reach their maximum values at the same time). The resistance in a pure resistor is The instantaneous power P sinωt t sinωt sinωt ωt ωt P P sint sint sin and :Supply voltage The rage power P P A resistor in A circuit dissipates energy in the form of heat BP3 6 FYS

7 Extra knowledge Pure capacitor in the A circuit d dt hapter 6 ( > ) ( < ) ( > ) ( < ) = when the plate is neutral BP3 7 FYS

8 Pure capacitor in the A circuit Pure capacitor means that no resistance and self-inductance effect in the A circuit. The voltage across the capacitor The voltage on a capacitor depends on the amount of charge you store on its plates. The charge accumulates on the plates of the capacitor is Q Q sint The current flows in the ac circuit is The phase difference between and is dq dt d dt d dt sint sint cos( ωt) and sinωt sinωt Δ t ωt Δ n pure capacitor, the voltage lags behind the current by / radians or the current leads the voltage by / radians. BP3 8 FYS

9 The capacitive reactance in a pure capacitor is The instantaneous power P The rage power X X P P cost sint P P sin t sin t f Definition P For the first half cycle where the power is positive, the capacitor is saving the power (in electric field). For the second half of the cycle where the power is negative, the power is returned to the circuit. The relationship between capacitive reactance X and frequency f Z X in pure capacitor circuit BP3 9 FYS

10 Extra knowledge Pure inductor in the A circuit d dt hapter BP3 FYS

11 Pure inductor in the A circuit Pure inductor means that no resistance and capacitance effect in the A circuit. The current flows in the ac circuit is sinωt When the current flows in the inductor, the back emf caused by the self-induction is produced and given by εb B d dt d dt sin ωt B ωcosωt At each instant the supply voltage must be equal to the back e.m.f B (voltage across the inductor) but the back e.m.f always oppose the supply voltage. Hence, the magnitude of and B : B The phase difference between and is sinωt ωcosωt ωsinωt Δ ωt ωt Δ n pure inductor, the voltage leads the current by / radians or the current lags behind the voltage by / radians. and BP3 FYS

12 The inductive reactance in a pure inductor is X X f Definition The instantaneous power P P sint cost P P P The rage power sin t sin t P For the first half of the cycle where the power is positive, the inductor is saving the power. For the second half cycle where the power is negative, the power is returned to the circuit. The relationship between inductive reactance X and the frequency f Z X in pure inductor circuit BP3 FYS

13 .O 6.3. Use phasor diagram to analyze voltage, current, and impedance of series circuit of, and ircuit leads by ϕ The total p.d (supply voltage), across and is equal to the vector sum of and as shown in the phasor diagram. X X Z From the phasor diagrams, the current leads the supply voltage by ϕ radians where Graph of Z against f X tan or tan X The resistance, is independent of frequency BP3 3 FYS

14 ircuit leads by ϕ The total p.d (supply voltage), across and is equal to the vector sum of and as shown in the phasor diagram. X X Z From the phasor diagram, the supply voltage leads the current by ϕ radians where Graph of Z against f X tan or tan X The resistance, is independent of frequency BP3 4 FYS

15 ircuit ( > ) leads by ϕ The total p.d (supply voltage), across, and is equal to the vector sum of, and as shown in the phasor diagram. > X X ( ( X ) X ) Z X X From the phasor diagram, the supply voltage leads the current by ϕ radians where tan Graph of Z against f <? =? or tan X X BP3 5 FYS

16 .O Explain graphically the dependence of, X, X and Z on f and relate it to resonance esonance is defined as the phenomenon that occurs when the frequency of the applied voltage is equal to the frequency of the series circuit. The graph shows that: at low frequency, impedance Z is large because /ω is large. at high frequency, impedance Z is high because ω is large. at resonance frequency, impedance Z is minimum (Z = ; X = X ) and is maximum esonance frequency X X f r f f r r Z Z min Z min X X Z, Z Z min, max BP3 6 FYS

17 Example Question A capacitor with = 47 pf is connected to an A supply with r.m.s. voltage of 4 and frequency of 5 Hz. alculate Solution a. the capacitive reactance. b. the peak current in the circuit. A 4 supply with a frequency of 5 Hz causes an current of 3. A to flow through an inductor which can be taken to h zero resistance. alculate a. the reactance of the inductor. b. the inductance of the inductor. An alternating current of angular frequency of. 4 rad s- flows through a k resistor and a. F capacitor which are connected in series. alculate the voltage across the capacitor if the voltage across the resistor is. BP3 7 FYS

18 Question Solution Based on the series circuit in figure above, the voltages across, and are shown. a. With the aid of the phasor diagram, determine the applied voltage and the phase angle of the circuit. alculate: b. the current flows in the circuit if the resistance of the resistor is 6, c. the inductance and capacitance if the frequency of the A source is 5 Hz, d. the resonant frequency. Exercise Question A voltage of. with a frequency of. khz is applied to a.9 mh inductor. a. What is the current in the circuit? b. Determine the peak current for a frequency of.5 khz. Answer: 6.7A; 3.78 A A F capacitor and a resistor are placed in series with an alternating voltage source of and frequency of 5 Hz. alculate a. the current flowing, b. the voltage across the capacitor, c. the phase angle of the circuit. Answer: A;. ; 57.9 or. rad A series circuit has a resistance of 5. Ω, a capacitance of 5. μf, and an inductance of.3 H. f the circuit is driven by a, 6 Hz source, calculate a. The total impedance of the circuit b. The current in the circuit c. The phase angle between the voltage and the current. Answer: 64.9 Ω,.85 A, 67.3 o BP3 8 FYS

19 6.4 Power and Power Factor.O 6.4. Apply rage power, instantaneous power and power factor equations in A circuit consisting of, and in series n an A circuit, the power is only dissipated by a resistance; none is dissipated by inductance or capacitance. Therefore, the real power (P r ) that is used or gone is equal to that dissipated from the resistor and given by the rage power (P ) P across resistor For circuit: P P r () Power used by the load From the diagrams above, cos and cos Z () Substitute () into (): P P or Z cos cos Papparent P a Power delivered to the load earranging cos P P a The term cos ϕ is called the power factor. The power factor (cos ϕ ) can vary from a maximum of + (or %) to a minimum of. When ϕ = o (cos ϕ = +),the circuit is completely resistive or when the circuit is in resonance (). When ϕ = +9 o (cos ϕ = ),the circuit is completely inductive. When ϕ = -9 o (cos ϕ = ),the circuit is completely capacitive. BP3 9 FYS

20 Example Question Solution A F capacitor, a. H inductor and a resistor are connected in series with an alternating source given by the equation below : 3sin3t alculate : a. the frequency of the source. b. the capacitive reactance and inductive reactance. c. the impedance of the circuit. d. the maximum (peak) current in the circuit. e. the phase angle. f. the mean power of the circuit. An oscillator set for 5 Hz puts out a sinusoidal voltage of effective. A 4. Ω resistor, a.μf capacitor, and a 5. mh inductor in series are wired across the terminals of the oscillator. a. What will an ammeter in the circuit read? b. What will a voltmeter read across each element? c. What is the real power dissipated in the circuit? Exercise Question A coil having inductance.4 H and resistance of is connected to an alternating source, 5 Hz. alculate a. the current flows in the coil. b. the phase angle between the current and supply voltage. c. the power factor of the circuit. d. the rage power loss in the coil. Answer: 4.4 A, 6.3 o,.48,.3 kw A series circuit contains a 5. μf capacitor and a generator whose voltage is.. At a resonant frequency of.3 khz the power dissipated in the circuit is 5. W. alculate a. the inductance b. the resistance c. the power factor when the generator frequency is.3 khz. Answer:.94 x -3 H, 4.84 Ω,.63 BP3 FYS