Chapter 22: Alternating current. What will we learn in this chapter?

Transcription

1 Chapter 22: Alternating current What will we learn in this chapter? Contents: Phasors and alternating currents Resistance and reactance Series R L C circuit Power in ac-circuits Series resonance Parallel resonance In more general terms: vintage radio How do R, L and C behave in ac currents/voltages? Technological relevance of R L C circuits. Notation: angular frequencies are in s - 1, frequencies in Hz.

2 Phasors and alternating currents Reminder: Alternators (ac generators) produce a sinusoidally-alternating current. L C circuits can produce alternating currents of variable frequency. Nomenclature: We use the term ac source for any device that supplies a harmonically-varying potential difference v or current i: v = V cos ωt i = I cos ωt V [I] is the maximum potential difference or voltage amplitude [current amplitude], v [i] is the instantaneous potential difference [current], and ω =2πf is the angular frequency f = cycles/second ac sources contd. Because the current and voltage in the circuit oscillate, the direction of the current in the emf sources changes continuously with time. For time-dependent quantities we use lower-case letters.

3 Representing time-dependent currents Phasors: Sinusoidally-varying voltages/currents are best represented with vector diagrams (see harmonic motion in physics 201). Elements of a phasor: Length = maximal current I. The projection onto the horizontal axis corresponds to the instantaneous value of the current: i = I cos ωt The rotation frequency is ω =2πf At t = 0 i = I, hence cosine. phasor diagram Phasors contd. Why the phasors? Phasors are not physical quantities. They are merely a convenient geometric construct. Adding harmonic currents/voltages with phase differences becomes a matter of vector addition. The truth behind it... Any harmonic motion can be represented as a complex number in the plane: x = Re + iim = re iφ = r(cos φ + i sin φ) i = 1 r = Re 2 + Im 2 Doing the calculus with complex numbers makes life much simpler. But we are not allowed to in this class

4 Characterizing currents/voltages Two options: Root-mean-square of the current Irms (1: graph I t diagram, 2: square, 3: average, 4: take root). Maximum of the current Imax. Note: Irms is always positive. Calculation: i 2 = I 2 cos 2 ωt (1 + cos 2ωt) 2 Recall: [cos x] ave =0 = I2 We obtain: [i 2 ] ave = I2 2 I = 3A Characterizing currents/voltages contd. rms value of harmonic currents/voltages: The rms current/voltage for a harmonic current/voltage with amplitudes I (resp. V) is I rms = I 2 V rms = V 2 Notes: Currents and voltages in power-distribution systems are always described in terms of their rms values. Examples: A 120V power supply has a peak voltage of ~170V. A 0.5 A lightbulb has an amplitude of 0.71A. Next: Derive voltage current relations for individual resistors, inductors, and capacitors in a harmonic current.

5 Resistor in an ac circuit Setup: Resitor R in a sinusoidal current i = I cos ωt. Use Ohm s law: v R = ir = IRcos ωt v R = V R cos ωt with V R = IR. in phase means crests overlap Because the voltage and the current are both proportional to cos ωt, the voltage is in step or in phase with the current. Phasors rotate together. Inductor in an ac circuit Setup: inductor L in an ac circuit. i = I cos ωt Recall: E = L i/ t v L = L i/ t = i/ t The inductor is proportional to the rate of change of the current. This corresponds to the derivative: v L = IωL sin ωt

6 Inductor in an ac circuit contd. Current and voltage are now out of phase by a quarter cycle. Because the voltage peak occurs a quarter cycle before the current peak, the current leads the voltage by 90º. Using trigonometry we obtain: v L = IωL cos(ωt + π/2) The phasor diagram shows the situation. Note: In general we describe the voltage with respect to the current, i.e., if i = I cos ωt then v = V cos(ωt + φ) where φ is the phase angle. Inductor in an ac circuit contd. Voltage amplitude for an inductor: v L = IωL cos(ωt + π/2) = X L i V L = IωL Inductive reactance: We define the inductive reactance XL of an inductor as the product of inductance and frequency, i.e., X L = ωl V L = IX L Note: Because V = IXL like for a resistor, the unit of the reactance is also Ohm (!). Since the reactance behaves like a resistance and is directly proportional to the frequency ω, it is used to filter high frequencies out (also called a choke): XL big when ω big.

7 Capacitors in a ac circuit Setup: Capacitor C in a harmonic!! current i = I cos ωt. Goal: Compute the voltage between a and b. The rate of change of the voltage is given by v C t Performing an integral v C = = 1 C I ωc q t = i C sin ωt v C t = i C Thus the voltage is again sinusoidal, and the current is largest when the capacitor is charging or discharging. The voltage and current are out of phase by a quarter cycle, the voltage lags the current by 90º. Capacitors in a ac circuit contd. Graphic representation of the voltage lag. The voltage runs behind the current. Corresponding phasor diagram.

8 Capacitors in a ac circuit contd. Voltage lag: The maximum voltage VC is given by V C = I ωc Note: v C = I ωc cos(ωt π/2) Capacitive reactance: We define the capacitive reactance XC of a capacitor as the inverse product of capacitance and frequency, i.e., X C = 1 V C = IX C ωc The capacitive reactance is inversely proportional to the frequency, i.e., low frequencies and dc currents are filtered. Summary Note: The figure shows how resistance and reactances vary with frequency. When the frequency is zero (dc), there is no current trough a capacitor and there is no inductive effect X C ω 0 X L ω 0 0

9 Application: Speakers Question:! There is one wire connecting speakers. How are low!!!! and high frequencies filtered out to woofer and!!!! tweeter? The capacitor in the tweeter blocks the low frequencies The inductor in the woofer blocks the high frequencies. The series R L C circuit Many electronic systems consist of R, L and C in series with i = I cos ωt. Analysis: Following Kirchhoff s rules, the total voltage is the same for all components. The phasor representing this total voltage is the vector sum of the voltage phasors of all components. Because the elements are connected in series, the current at any instant can be represented by a single phasor I, representing all elements of the circuit. Nomenclature: vn, n = R, L, C is used for instantaneous values e.g., vc = vcd. Vn, n = R, L, C is used for the maximum values.

10 The series R L C circuit: Phasors Resistor (in phase): V R = IR Inductor (leads by "/2): V L = IX L Capacitor (lags by "/2): V C = IX C One can now compute the source voltage phasor V as the vector sum of VR, VL, VC. After some algebra The series R L C circuit contd. After performing the vector sum we obtain: V = I R 2 +(X L X C ) 2 The quantity XL - XC is called the reactance X of the circuit. The impedance of the circuit is defined as: It follows: Z = R 2 +(X L X C ) 2 = R 2 + X 2 V = IZ Impedance of a series R L C circuit: The impedance is always a ratio of voltage to current. The SI unit is the Ohm (!): Z = R 2 + X 2 = R 2 +(X L X C ) 2 = R 2 +[ωl (1/ωC)] 2

11 The series R L C circuit contd. Note: The equation Z = R 2 +[ωl (1/ωC)] 2 only gives the impedance of a series R L C circuit. Via V = IZ the impedance of any circuit can be computed. The angle φ between current I and voltage V is given by ( ) ( ) ωl 1/ωC XL X C φ = arctan = arctan R R If XL > XC the source voltage leads the current by an angle between 0 and 90º and v = V cos(ωt + φ). If XL < XC the vector V lies on the opposite side of the current vector I and the voltage lags the current. In this case the angle lies between 90 and 0º. If R/C/L are missing, drop them in the equations. Maximum vs rms So far, we have only dealt with maximal values of current and voltage (their amplitudes). General rule: For any harmonic quantity the rms value is always times the amplitude. 1/ 2 Example: Note: V rms = I rms Z In all previous discussions we have described the steady state of the system. Transient effects have not been considered.

12 Power in ac circuits (resistance only) Motivation:! Because ac circuits are generally used for power!!!! transfer and generation, we need to understand how!!!! the power behaves in a harmonic environment. Assumption: i = I cos ωt Resistance-only circuit (pure resistor): i and v are in phase. In general, P = VI. Because i and v are in phase, p > 0. Average power: P = 1 2 VI = V 2 I 2 = V rms I rms BecauseV rms = I rms R P = I 2 rmsr = V 2 rms R only for resistors! = V rmsi rms Power in ac circuits (inductor/capacitor) the average power is zero! Because for inductors/capacitors the current and voltage are out of phase, the product oscillates around zero. It follows that the net energy transfer over one cycle is zero!

13 Power in ac circuits (general case) In general: i = I cos ωt v = V cos(ωt + φ) Power: p = vi =[V cos(ωt + φ)][i cos ωt] using trigonometry relations p = VIcos φ cos 2 ωt VIsin φ cos ωt sin ωt average over time and recall that to obtain P = 1 2 VIcos φ = I rmsv rms cos φ [cos ωt] ave =0 [cos 2 ωt] ave =1/2 When v and i are in phase, P > 0, when the phase is 90º, P = 0. Power in ac circuits (general case) contd. For an R-L-C circuit V cos φ is the voltage amplitude for the resistor, hence P = I rms V rms cos φ represents the power dissipated in the resistor. The power dissipated in the capacitor and the inductor is zero. The factor cos φ is called the power factor of the circuit: For a resistance cos φ =1 For a capacitor/inductance cos φ =0 For a R-L-C circuit cos φ = R/Z. Note: a large lead/lag angle is undesirable since for a given potential a large current is needed to supply a given amount of power.

14 Series resonance In a R-L-C circuit, the impedance depends on the frequency as follows: Z = R 2 + X 2 = R 2 +(X L X C ) 2 = R 2 +[ωl (1/ωC)] 2 For XC = XL the impedance Z as a function of the frequency has a minimum which only depends on R. We can now tune the frequency of a potential source trough this minimum and see what happens Series resonance contd. Connect a potential source with fixed V and variable frequency to the system. current peaks at the frequency where Z is least As the frequency changes, the current amplitude I has a maximum when Z has a minimum. This is called a resonance. The angular frequency ω 0 at which this occurs is called the resonance frequency. This is the case when X C = X L ω 0 = 1 LC At the resonance, the voltage across the L C part of the circuit is zero (180º phase). The resonance frequency can be tunes with L and C (radio ).

15 Series resonance contd. At the resonance, the impedance is minimal and the current I is maximal. ω Example response curve: 0 = 1000 rad/s V = 100V R = 500! L = 2.0H C = 0.5µF The height of the peak is determined by the value of R. Note: in the old days the shape of the resonance (response) curve was crucial for radio tuning. Parallel resonance A different kind of resonance occurs when R, L and C are connected in parallel. The system also shows resonant behavior, but the roles of the voltage and current are reversed. In this case, the potential difference is the same for all elements v = V cos ωt but the current is different. The current ir is in phase with the source and has a peak value I R = V/R. The current il lags by "/2 and has a peak value I L = V/X L = V/ωL. The current ic leads by "/2 and has a peak value I C = V/X C = V ωc.

16 Parallel resonance contd. At any frequency the inductor current and the capacitor current phase differs by " and cancel. Again, when XL = XC we have a resonance with ω 0 = 1 LC This is the same as before, except that here the total current reaches a minimum at the resonance. For the parallel case 1 Z = hence Z has a maximum when Z = R. 1 R 2 + ( ωc 1 ) 2 ωl