ETURE 9 A Generators A ircuits Start by considering simple circuits with one element (R,, or ) in addition to the driving emf. It will lead to Oscillations and Driven R circuits Alternating urrent Generators (DEMO) (N = 2 for this coil) 3/23/27 2 Phasors for R Power Dissipated in a Resistor Peak value V in phase with I Average value I R R V R RI R 3/23/27 7 3/23/27 8
Standard Alternating Voltage in the US + peak Using rms values: summary Using rms values of current and voltage allows you to use the familiar dc formulas, such as V = IR and P = I 2 R. - peak One ac ampere is said to flow in a circuit if it produces the same joule heating as one ampere of dc current under the same conditions. 3/23/27 At your house the peak voltage will be 7 V 3/23/27 2 Inductors in A ircuits Relationship between I rms & V rms V V I I peak,peak sin t cost Since sin cos, I Ipeak cos t 9 I rms V, rms X where X is the inductive reactance.. X is similar to R in I rms V R, rms R. 2. SI unit for X : (ohm) 3. Average power delivered to an inductor in an ac circuit is zero. Potential drop, V (t), leads the current, I(t) by 9 3/23/27 6 3/23/27 7 2
Phasors for V leads I by 9 apacitors in A ircuits V di dt I sin t 9 cost V V I I,peak peak cost sin t Since sin cos, I Ipeak cos t 9 Potential drop, V (t), lags the current, I(t), by 9 3/23/27 8 3/23/27 2 Relationship between I rms & V,rms I rms V, rms X where X is the capacitive reactance. X is similar to R in I rms V R, rms R. 2. SI unit for X : (ohm) 3. Average power delivered to a capacitor in an ac circuit is zero. Phasors for V lags I by 9 I m sin t 9 m cos t V Q 3/23/27 22 3/23/27 23 3
Summary Impedances for,, R Symbol R Reactance X R R is resistance X is capacitive reactance For high, X goes to zero, acts like a wire. For low, X grows larger and at D, acts like an open switch I I current leads V current lags V X = is inductive Reactance For high, X grows large and acts like an open switch. For low, X grows small and at D, acts like a conducting wire. 3/23/27 24 3/23/27 25 Phasors for R Phasors for I R R V R RI R V in phase with I I m sin t 9 m cos t V Q V lags I by 9 3/23/27 27 3/23/27 28 4
Phasors for 3/23/27 V leads 29 I by 9 A Power Distribution A power can travel at high voltages Nikola Tesla and low amps, therefore smaller power loss Tesla liked 6 Hz and 24 V Standard in Europe was defined by a German company AEG ( monopoly) V di who chose 5Hz (2% less efficient in generation, -5% less efficient in dt m I transmission) m sint 9 Originally Europe was also V, but they changed to reduce power loss and voltage drop for the same copper m cost diameter http://www.teslasociety.com/ 3/23/27 32 The Power Grid Example If 735 kv line is used to transmit electric energy km. I = 5 A and R =.22 /km Energy is supplied at a rate of Energy dissipated from resistance of wires If you doubled the current and halved the voltage, energy dissipated 3/23/27 33 3/23/27 34 5
ircuits onsider the and R series circuits shown: ++++ Suppose that at t= the ---- capacitor is charged to a value of Q. R ++++ ---- Oscillations Kirchoff s loop rule di Q VV dt ++++ ---- I Is there is a qualitative difference in the time development of the currents produced in these two cases. Why?? 35 36 3/23/27 3/23/27 Q Q Q cos( t) V Oscillations Example t= t=t + + Q = Q o Q = - - I di dt I Q sin( t) di 2 oqcos( t) dt V t At t=, the capacitor in the circuit shown has a total charge Q. At t = t, the capacitor is uncharged. What is the value of V ab, the voltage across the inductor at time t? (a) V ab < (b) V ab = (c) V ab > 3/23/27 t 37 3/23/27 38 6
Example 2 At t= the capacitor has charge Q ; the resulting oscillations have frequency. The maximum current in the circuit during these oscillations has value I. What is the relation between I and I 2, the maximum current in the circuit when the initial charge = 2Q? t= + + Q = Q o - - Example 3 At t= the capacitor has charge Q ; the resulting oscillations have frequency. The maximum current in the circuit during these oscillations has value I. What is the relation between and 2, the frequency of oscillations when the initial charge = 2Q? t= + + Q = Q o - - (a) I 2 = I (b) I 2 = 2I (c) I 2 = 4I (a) 2 = /2 (b) 2 = (c) 2 = 2 39 4 3/23/27 3/23/27 7