Week4 - Physics 1C Chapter 31 notes Winter Quarter 2009 AC circuits W Gekelman Consider a capacitor This is an element of an AC circuit The charge

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1 Physics 1C Winter Quarter 2009 AC circuits Chapter 31 notes© W. Gekelman Consider a capacitor. This is an element of an AC circuit. The charge is related to the voltage by Q = CV . Since I = dQ dt then I = C dV dt . This is the current through the capacitor. We will use complex notation to express all quantities. Let V = V 0 e i ! t dV dt = i V 0 e i t . All quantities in this way of looking at things are complex therefore the current is written as I=I 0 e i t . Then I = C dV dt becomes I 0 = Ci V 0 . This is in complex notation and the i really means that the current and voltage are out of phase. I 0 = Ci V 0 ; V 0 = I 0 i C = iI 0 C = I 0 e i " 2 C . This means that the current and the voltage are out of phase by 90 degrees in the capacitor. The voltage LAGs the current by 90 degrees. For example if the Voltage is V = Re V 0 e i t ( ) then the current is I = Re CV 0 e i t e " i # 2 $ % & ( ) = CV 0 cos t " 2 $ % & ( ) This is the essence of equation 31.16 in the text. This is summarized in a figure 31.9 from the book.
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3 Next let us generalize the concept of Resistance. For a simple resistor the current and voltage are in phase V=IR. For a capacitor (instead of resistance) let us define a quantity called the capacitive part of the impedance. Z c ! V 0 I 0 = 1 i " C . Thus instead of R we have Z C , but it is complex. Why are we doing this? The goal is that if we have a bunch of elements, R’s, C’s and L’s we can calculate a quantity made out of R, Z C and an Z L and then relate the current to the voltage across the component. Now let us consider an inductor only V = L di dt = L d dt i 0 e i ! t = i Li 0 e i t = V 0 e i t
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4 We are interested in the ratio of voltage to current to find the inductive reactance V 0 i 0 = i ! L " Z L As shown in the book: Here the voltage LEADs the current Let us use this to re-examine the undriven series LRC circuit. The Kirchoff equation was LC d 2 I dt 2 + RC dI dt + I = 0 . Using the complex current method take I t ( ) = I 0 e i t . When we are finished we will take the real part of the solution. But since damping will occur because of the resistor the angular frequency must be complex as well.
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This note was uploaded on 01/21/2010 for the course PHYS 1C taught by Professor Whitten during the Winter '07 term at UCLA.

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Week4 - Physics 1C Chapter 31 notes Winter Quarter 2009 AC circuits W Gekelman Consider a capacitor This is an element of an AC circuit The charge

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