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Passive Circuit Elements in the Frequency Domain Section 9.4-9.6

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Outline I-V relationship for a capacitor I-V relationship for an inductor

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Current and Voltage Relationship Q=CV (at t=t 1 ) (Current is 0) If you increase voltage by ∆V, then more charges will be shoved to the capacitor. (Q+ ∆Q) Here is what we have. Q+ ∆Q=C(V+ ∆V) Charges can not be moved instantaneously. The accumulation of charges will take place between t1 and t 1 + ∆t We are interested only in the incremental change of charges. ∆Q/ ∆t=C ∆V/ ∆t=i

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Current and Voltage Relationship i=C ∆V/ ∆t – ∆V/ ∆t represent the rate of change of voltage across a capacitor. – The faster the rate of change, the greater the current. – ∆V/ ∆t is the slope V C vs time plot.

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The rate of change of a sine wave Determine the slope by putting a ball on the curve.

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Time Domain Interpretation I C =C ∆V/ ∆t

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Phasor Interpretation Z=1/(jωC)=-j/(ωC) V C =I C Z V C =I C [-j/(ωC)]

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Capacitive Impedance as a function of frequency The faster the voltage changes, the higher the frequency, the greater the current, and hence lower the Impedance. So Z C, the Impedance, is inversely proportional to f. I C =C ∆V/ ∆t

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Impedance as a function of Capacitor i=C ∆V/ ∆t Assume that ∆V/ ∆t is constant, the larger the C, the greater the current. In other words, ∆V/ ∆t represent changes in the voltage across the capacitor. The changes in V C can not happen without the changes in Q. A larger the capacitance will require more charges for the same ∆V/ ∆t. So it will require more current. Reactance is inversely proportional to capacitance.

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Similarity to resistance ADD impedance of series capacitors – Z TC =Z C1 +Z C2 +Z C3 Calculate Impedance of parallel capacitors like parallel resistors. – Z TC =Z C1 Z C2 /(Z C1 +Z C2 )

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Example

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Ohm’s law When applying Ohm’s law in AC circuits, you must express both the current and the voltage in rms, peak,…and so on. I=V s /X C

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Capacitive Voltage Divider Vx=(X Cx /X c,tot ) Vs This is similar to the formula for voltage divider

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Power in a capacitor Instantaneous Power True Power Reactive Power

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Power curve Instantaneous power fluctuates as twice the frequency of voltage and current. Ideally all the energy stored by a capacitor during the positive power cycle is returned to the source during the negative portion Note that the average power is 0.

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Inductor

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Time Domain Interpretation (An inductor resists change in current. At t=0, voltage is maximum, but current is 0. It takes time for the current to catch up to voltage.)

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Phasor Interpretation Z=jωL V C =I C (jωL) I C =V C [-j/(ωL)]

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Understanding Z L I C =V C [-j/(ωL)] An inductor has a natural tendency to resist change in current. Therefore, as the frequency of V C increases, it will not be able to keep up with changes. At sufficiently high frequencies, the current will cease to track the voltage, and begins to behave as an open circuit.

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Inductive reactance formula In general: – X L =2ΩfL=ωL For series inductors: – X LT =X L1 +X L2 +X L3 …. For parallel inductors: – 1/X LT =1/X L1 +1/X L2 +1/X L3

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Example 13-13

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Ohm’s law I=V S /Z L

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Example 13-14

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Power in an inductor

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