Transcript Slide 1
Talk about the application of capacitors in flashbulbs or windshield wipers. RVOTD • http://www.youtube.com/watch?v=syCpfNu1Hqc • (no volume) Time constant (Tau) • The time required to charge a capacitor to 63.2% of maximum voltage, or the time to discharge a capacitor to 36.8% of its final voltage. = R∙C Seconds • Similarly for an inductor, it is the time required to change the amount of current through an inductor to 63.2% of max current (or reduce it to 36.8% of final amount) = L/R Seconds 95% 86.5% 98.2% 99.3% Every time constant, the voltage rises 63% of what is remaining. 63.2% See table 21.1 in your book for these values. Determining the time constant • What is the time constant of a 0.01uF capacitor in series with a 2kΩ resistor? • 20us • What is the time constant of a 10uF capacitor in series with a 100kΩ resistor? • 1 sec • What is the time constant for a 200mH inductor with a 2Ω resistor? • 100ms • What value of resistance is needed to cause a of 1.2ms with a 4.7uF capacitor? • 255 Ω Not quite so Random VOTD • http://www.youtube.com/watch?v=coW1RHUsf_I&feature=yo utube_gdata_player • Only watch til 2:45min A 50kΩ resistor is connected in series with a 40uF capacitor. With a DC source of 50V, what is the charge across the capacitor after 6 sec? Assume VC=0 at t=0 = RC = 50k x 40u = 2sec A 10kΩ resistor is connected in series with a .01uF capacitor. With a DC source of 20V, what is the charge across the capacitor after 200us? Assume VC=0 at t=0 = RC = 10k x .01u = 100us Do the following 2 problems on your own. • What is the voltage across a 5uF capacitor connected in series with a 22kΩ resistor after 330ms with a 30V DC source voltage? (Assume 0V for start up) • For the problem above, what is the voltage across the resistor after 440ms? Try one more… • What is the voltage across a 20uF capacitor connected in series with a 100kΩ resistor after 3s if the source voltage is 10V? (Assume 0V for start up) • Hint: It is not 74.85% or 7.485V • When the amount of time does not fall exactly on an even number of time constants, such as 1, 2, etc. then we use the following equation 21.2: 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 • • • • 𝑡 − 𝜏 VC is the voltage across the capacitor VS is the DC source voltage t is the amount of time elapsed is the time constant ) So what the heck is e in the𝑡 equation: 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 − 𝜏 ) • Everyone think of a large number. Something larger than 1000. • Now plug that number into the formula, where N is your number: • (1+1/N)N • With the help of magic I bet your number is: • 2.718… • Find the “e” on your calculator and press enter. • (There are 2 buttons, a green one and ex) Going back to our original problem • What is the voltage across a 20uF capacitor connected in series with a 100kΩ resistor after 3s if the source voltage is 10V? (Assume 0V for start up) • Hint: It is not 74.85% 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 • • • • VC = ? VS = 10V t=3 = 2sec 𝑉𝐶 = 10 (1 − 𝑒 𝑉𝐶 = 7.76𝑉 𝑡 − 𝜏 3 − 2 ) ) • What is the voltage across a .002uF capacitor connected in series with a 22kΩ resistor after 160us if the source voltage is 12V? (Assume 0V for start up) • • • • VC = ? VS = 12V t = 160us = 44us 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 𝑉𝐶 = 12 (1 − 𝑒 𝑡 − 𝜏 ) 160𝑢 − 44𝑢 ) 𝑉𝐶 = 11.68𝑉 Does this answer make sense? How many time constants have passed? Calculators are about to become very important in this class. • What is the voltage across a .05uF capacitor connected in series with a 500Ω resistor after 75us if the source voltage is 100V? (Assume 0V for start up) • • • • VC = ? VS = 100V t = 75us = 25us 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 𝑡 − 𝜏 𝑉𝐶 = 100 (1 − 𝑒 ) 75𝑢 − 25𝑢 ) 𝑉𝐶 = 95𝑉 Does this answer make sense? How many time constants have passed? • What is the voltage across a .05uF capacitor connected in series with a 500Ω resistor after 2.2 if the source voltage is 100V? (Assume 0V for start up) Doing it 2 different ways: 𝑉𝐶 = 100 (1 − 𝑒 𝑉𝐶 = 100 (1 − 𝑒 𝑉𝐶 = 88.9𝑉 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 − 2.2 ) −2.2 ) 𝑡 − 𝜏 𝑉𝐶 = 100 (1 − 𝑒 ) − 2.2 = 25us − ) 2.2(25𝑢) 25𝑢 𝑉𝐶 = 100 (1 − 𝑒 𝑉𝐶 = 100 (1 − 𝑒 −2.2 ) 𝑉𝐶 = 88.9𝑉 ) Solving for t: 𝑉𝐶 = 𝑉𝑆 (1 − 𝑒 − 𝑡 𝜏 ) • How much time does it take to charge a 4uF capacitor to 5V if there is a 10V DC source and a 10kΩ resistor in series? 𝑉𝐶 𝑡 = −𝜏 ∙ ln(1 − ) 𝑉𝑆 5 𝑡 = −40𝑚𝑠 ∙ ln(1 − ) 10 𝑡 = 27.7𝑚𝑠 Shall we have another… • How many time constants does it take to charge a capacitor to 25% of being fully charged? 𝑉𝐶 𝑡 = −𝜏 ∙ ln(1 − ) 𝑉𝑆 𝑡 = −𝜏 ∙ ln(1 − .25) 𝑡 = 0.288𝜏 Get to here before lab 30. Calculating Current in RC circuits • Since we know 𝑉𝐶 = 𝑉𝑆 (1 • To calculate VR it would be 𝑉 𝑡 − 𝜏 −𝑒 ) 𝑅 = 𝑉𝑆 − 𝑉𝐶 = 𝑉𝑆 − 𝑉𝑆 (1 − 𝑒 = 𝑉𝑆 (1 − 1 + 𝑒 = 𝑉𝑆 (𝑒 𝑡 −𝜏 • Using Ohms Law IR would be 𝑉𝑆 (𝑒 𝐼𝑅 = 𝑅 − ) 𝑡 𝜏 ) = 𝐼𝐶 𝑡 𝜏 𝑡 − 𝜏 − ) ) Thus… • 𝐼𝐶 = 𝑡 −𝜏 𝑉𝑆 (𝑒 𝑅 ) = 𝐼𝑅 𝑉𝑅 = 𝑅 What is the current through a .002uF capacitor connected in series with a 22kΩ resistor after 160us if the source voltage is 12V? (Assume 0V for start up) − 𝑉𝑆 (𝑒 𝐼𝐶 = 𝑅 𝑡 𝜏) = 12 (𝑒 − 160𝑢 44𝑢 ) 22000 = 14.4𝑢𝐴 Another Capacitor current problem • 𝐼𝐶 = 𝑡 −𝜏 𝑉𝑆 (𝑒 𝑅 ) = 𝐼𝑅 𝑉𝑅 = 𝑅 What is the current through a 1kΩ resistor with a .2uF capacitor connected in series with a after 280us if the source voltage is 18V? (Assume 0V for start up) − 𝑉𝑆 (𝑒 𝐼𝐶 = 𝑅 𝑡 𝜏) = 18 (𝑒 − 280𝑢 200𝑢 ) 1000 = 4.4𝑚𝐴 Voltage and current in LR circuits • Recall = 𝐿 (This is how long it takes to get 63.2% of max 𝑅 current through an inductor) • It turns out the equations for Voltage across an Inductor and Current through an inductor are as follows: 𝑉𝑆 (1 − 𝑒 𝐼𝐿 = 𝑅 − 𝑡 𝜏) 𝑉𝐿 = 𝑡 − 𝑉𝑆 (𝑒 𝜏 ) Inductor Problems 𝑡 𝑉𝑆 (1 − 𝑒 −𝜏 ) 𝐼𝐿 = 𝑅 𝑉𝐿 = 𝑉𝑆 (𝑒 − 𝑡 𝜏) • An 8H inductor and 1kOhm resistor are connected in series to a 10V source. Calculate the inductor current at t = 6ms. • Calculate the inductor voltage at this same time. • Calculate the resistor voltage at this same time. • Calculate the resistor current at this time. Another Inductor Problem 𝑡 𝑉𝑆 (1 − 𝑒 −𝜏 ) 𝐼𝐿 = 𝑅 𝑉𝐿 = 𝑉𝑆 (𝑒 − 𝑡 𝜏) • Calculate the inductor current at t = 5us after the switch is turned on for a 5mH inductor and a 2.2kOhm series resistance if the source voltage is 24V. • Calculate the inductor voltage at this same time. • Calculate the resistor voltage at this same time.