lesson34 - Lesson 34 AC Circuit Analysis II (Sections 8-2...

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Lesson 34 – AC Circuit Analysis II (Sections 8-2 and 8-3) (CLO 8-3) This lesson begins to apply all of the theorems learned back in Chapters 2 and 3 to ac circuits. But, before we start we bring in one very important concept involving impedance. It is very important that students understand the relationship between frequency and impedance. Once they do the whole idea of how RLC elements behave and why such things as filters work becomes clear. They may not really fully appreciate this concept now but it will be a useful landmark to return to often in subsequent lectures. The figure on the left shows how the magnitude of the impedance of each element varies with frequency. A resistor is not affected (in the ideal) by frequency, hence a 1-k resistor at dc is a 1-k Ω Ω resistor at 10 krad/s and at 1 Grad/s or whatever. This is not so for the other two elements. At dc, an inductor has 0 – a short circuit. We used this concept in Chapter 7 when Ω we found the initial and final conditions of a circuit excited by a step. An inductor, however, changes its character as frequency is increased from a short at dc to an open as . A capacitor has the opposite reaction. At ω→ ∞ dc it behaves like an open circuit or . As the frequency increases its ∞Ω impedance decreases so that as , | ω→ ∞ Z C | 0 , a short circuit. → Ω In a circuit with inductors and capacitors there could exist a frequency where the impedance of the inductor equals that of the capacitor, at that frequency, the reactance of the circuit is zero. When that occurs we say that the circuit is in resonance and the frequency at which that occurs is called the resonant frequency ω 0 . We are now ready to start ac circuit analysis. Start with an RC voltage divider circuit. Do this with two purposes in mind. First is to simply show that voltage division works quite well in the Phasor domain and second to demonstrate how the circuit’s behavior changes as we change the frequency of the source. Do this latter analysis qualitatively explaining how the impedance of the capacitor changes from an open to a short as we increase the frequency. The focus here is not filters but rather that the capacitor and inductor have an ac “resistance” or impedance that depends on the frequency of the source. True it will serve us well when we discuss filters in later discussions but for now the frequency-dependence nature of these new devices is what is important. Now apply some values to the various parameters – say R =100 k Ω , C =10 μF, ° = 0 10 i V ~ V and ϖ = 2 rad/s, and solve for O V ~ and i I ~ . Repeat for ω = 0 (dc) and for ω = 20 rad/s. ϖ O V ~ 0(dc) S V ~ 1/ RC 2 S V ~ 0 AC v O (t) C R v i (t) + _ RC j V RC CR j R C j C j V 1 1 1 1 1 1 i O + ϖ = + ϖ = + ϖ ϖ = ~ ) ( ~ ( 29 ( 29 A CR j C R C j V I μ ° = ° ° × = ° × = + ϖ ° × ° × = + ϖ ° × ° ϖ = + ϖ = - - - ϖ - - 6 26 4 89 4 63 236 2 90 10 2 5 90 10 2 1 0 10 90 10 2 1 0 10
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This note was uploaded on 10/12/2010 for the course MAE 140 taught by Professor Mauriciodeoliveria during the Spring '08 term at UCSD.

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lesson34 - Lesson 34 AC Circuit Analysis II (Sections 8-2...

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