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=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

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 VC vs time plot.

The rate of change of a sine wave

Determine the slope by putting a ball on the curve.

Time Domain Interpretation

IC=C ∆V/ ∆t

Phasor Interpretation

Z=1/(jωC)=-j/(ωC)VC=ICZVC=IC[-j/(ωC)]

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 ZC, the Impedance, is inversely proportional to f.

IC=C ∆V/ ∆t

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 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.

Similarity to resistance

• ADD impedance of series capacitors – ZTC=ZC1+ZC2+ZC3

• Calculate Impedance of parallel capacitors like parallel resistors.– ZTC=ZC1ZC2/(ZC1+ZC2)

Example

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=Vs/XC

Capacitive Voltage Divider

• Vx=(XCx/Xc,tot) Vs

• This is similar to the formula for voltage divider

Power in a capacitor

• Instantaneous Power• True Power• Reactive Power

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.

Inductor

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.)

Phasor Interpretation

Z=jωLVC=IC(jωL)IC=VC [-j/(ωL)]

Understanding ZL

• 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.

Inductive reactance formula

• In general:– XL=2ΩfL=ωL

• For series inductors:– XLT=XL1+XL2+XL3….

• For parallel inductors:– 1/XLT=1/XL1+1/XL2+1/XL3

Example 13-13

Ohm’s law

I=VS/ZL

Example 13-14

Power in an inductor