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# Passive Circuit Elements in the Frequency Domain - PowerPoint PPT Presentation

Passive Circuit Elements in the Frequency Domain. Section 9.4-9.6. Outline. I-V relationship for a capacitor I-V relationship for an inductor. Current and Voltage Relationship. Q=CV (at t=t 1 ) (Current is 0)

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### Passive Circuit Elements in the Frequency Domain

Section 9.4-9.6

• I-V relationship for a capacitor

• I-V relationship for an inductor

• Q=CV (at t=t1) (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 t1+ ∆t

• We are interested only in the incremental change of charges. ∆Q/ ∆t=C ∆V/ ∆t=i

• 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 VC vs time plot.

Determine the slope by putting a ball on the curve.

IC=C ∆V/ ∆t

Phasor Interpretation

Z=1/(jωC)=-j/(ωC)

VC=ICZ

VC=IC[-j/(ωC)]

IC=C ∆V/ ∆t

The faster the voltage changes, the higher the frequency,

the greater the current, and hence lower the Impedance.

So ZC, the Impedance, is inversely proportional to f.

• 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 VC 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.

• ADD impedance of series capacitors

• ZTC=ZC1+ZC2+ZC3

• Calculate Impedance of parallel capacitors like parallel resistors.

• ZTC=ZC1ZC2/(ZC1+ZC2)

• When applying Ohm’s law in AC circuits, you must express both the current and the voltage in rms, peak,…and so on.

• I=Vs/XC

• Vx=(XCx/Xc,tot) Vs

• This is similar to the formula for voltage divider

• Instantaneous Power

• True Power

• Reactive Power

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.

(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.)

Phasor Interpretation

Z=jωL

VC=IC(jωL)

IC=VC [-j/(ωL)]

• IC=VC [-j/(ωL)]

• An inductor has a natural tendency to resist change in current. Therefore, as the frequency of VC 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.

• In general:

• XL=2ΩfL=ωL

• For series inductors:

• XLT=XL1+XL2+XL3….

• For parallel inductors:

• 1/XLT=1/XL1+1/XL2+1/XL3

I=VS/ZL