Tuesday, April 27, 1999
Announcements:
Recitation and Lab Grades:

• Recitation overall score (HW and quizzes) will be normalized to an average of 75% for each instructor. The lowest quiz and homework assignment will be dropped.
• Lab reports will be normalized to an average of 85%
• Lab quizzes will not be normalized. The lowest lab quiz and report will be dropped.

• Eric Wertz:              May 2        1:00 - 2:30        117 Osmond
• Seiji Takemore        May 2        2:30 - 4:00        101 Osmond
• Yuri Zadorozhny     May 2        6:00 - 7:30        117 Osmond
• Steve Minnick         May 3        10:30 - 12:00    117 Osmond
• Karl Nelson            ????            ?????                ??????

• Professor Samarth:        May 3, 1-3 pm  and May 4, 1-3 pm
• Professor Chan             May 4, 10 am-12 pm and 3-5 pm

• When? Tuesday, May 4,  6:50 to 8:40 pm
• Where?
• Sections 1,2,5,16,12,13,15             111 Forum
• Sections 3,4,7,11,17,20                  108 Forum
• Sections 6,8,19,9,10,14,18             105 Forum
• The exam will have 25 multiple choice questions of which 15 will be on new material
• You may bring a 8.5 x 11" sheet of paper will writing on both sides.

• Let's measure the voltage across R in an RLC circuit; also across L; across C
• Note: x = v_R + v_L + v_C    at all times
• What are the phase relations between  v_R and i? They are in phase.
• between v_L and i? v_L leads i by 90^o
• between v_C and i? v_C lags behind i by 90^o
• See figure 33-11 on page 824 of the text for pictures of the phasor diagrams.
• Note: x_m² = V_R² + (V_L -V_C)² = (IR)² + (IX_L - IX_C)² = I²[R² + (X_L - X_C)²]
• or similarly I = x_m / [R² + (X_L - X_C)²]^1/2
• Impedance of RLC circuit Z = [R² + (X_L - X_C)²]^1/2 = [R² + (wL - (1/wC)²]^1/2 ,   so I= x_m / Z
• If wL = 1/wC, then I has a maximum. For a fixed L and C, we can change w so that wL = 1/wC, therefore the current is a maximum. This is called tuning the RLC circuit to resonant condition. w² = 1/LC
• See figure 33-13 on page 826 of the text for a picture of Resonance curves

• x = x_m sin wt, then i = Isin (wt - phase constant)
• In an RLC circuit, energy supplied by a generator is stored in electric field in C and in magnetic field in L. Average energy stored in C and L is constant. The transfer of energy is from the generator G to the resistor.  (from electromagnetic energy to heat).
• P = i²R = [ I sin (wt - phase constant)]²R = I²Rsin² (wt - phase constant)
• P is never negative
• P_average = I²R/ 2 = (I/(2)^1/2)²R
• See figure 33-34 on page 827 of the text.
• I/(2)^1/2 = the root mean square or rms value of i
• I/(2)^1/2 = I_rms or P_average = I_rms ²R
• Similarly V_rms = V/ 2^1/2, x_rms = x_m / 2^1/2 , then I_rms = x_rms /Z = x_rms /[R² + (X_L - X_C)²]^1/2
• V = 2^1/2 V_rms , where V is the amplitude of the voltage and I = 2^1/2 I_rms
• P_average = I_rms ²R =  (x_rms / Z)  I_rms  R = x_rms  I_rms  R/Z = x_rms  I_rms  cosq where the cosq is the power factor
• In order to maximize the the rate energy is supplied to a resistive load, need to keep cosq close to 1 or q = 0
• How? cosq = R /[R² + (X_L - X_C)²]^1/2 ; match X_L to X_C

• In transmission of electrical energy from a power station to towns or cities, there must be current flowing in wires (with resistance); in order to minimize loss in the transmission cables, current must be kept low. but since total power supplied by the generator is P = xI,  x must be kept high.
• Typically, the voltage of a long distance transmission line is kept at about 500 kV and the power generator runs at 200 megawatts. Let's assume this power station is 1000 km from a city and the resistance of the cable is 0.2 W / km.
• P = 300 MW = xI,  I = 600 amps
• The power lost in the transmission = I²R = 72 MW
• What if the transmission line voltage is kept at 250 kV?
• xI= 300 MW and I = 1200 amps
• Power lost in the transmission is 288 MW.
• Therefore, potential needs to be kept high in transmission, but high voltage is dangerous for household appliances. There are transformers to step down the voltage just before power is fed into houses.