Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.
-
Upload
adeline-golly -
Category
Documents
-
view
227 -
download
0
Transcript of Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.
![Page 1: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/1.jpg)
Passive Circuit Elements in the Frequency Domain
Section 9.4-9.6
![Page 2: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/2.jpg)
Outline
• I-V relationship for a capacitor• I-V relationship for an inductor
![Page 3: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/3.jpg)
![Page 4: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/4.jpg)
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
![Page 5: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/5.jpg)
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.
![Page 6: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/6.jpg)
The rate of change of a sine wave
Determine the slope by putting a ball on the curve.
![Page 7: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/7.jpg)
Time Domain Interpretation
IC=C ∆V/ ∆t
![Page 8: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/8.jpg)
Phasor Interpretation
Z=1/(jωC)=-j/(ωC)VC=ICZVC=IC[-j/(ωC)]
![Page 9: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/9.jpg)
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
![Page 10: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/10.jpg)
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.
![Page 11: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/11.jpg)
Similarity to resistance
• ADD impedance of series capacitors – ZTC=ZC1+ZC2+ZC3
• Calculate Impedance of parallel capacitors like parallel resistors.– ZTC=ZC1ZC2/(ZC1+ZC2)
![Page 12: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/12.jpg)
Example
![Page 13: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/13.jpg)
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
![Page 14: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/14.jpg)
![Page 15: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/15.jpg)
Capacitive Voltage Divider
• Vx=(XCx/Xc,tot) Vs
• This is similar to the formula for voltage divider
![Page 16: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/16.jpg)
Power in a capacitor
• Instantaneous Power• True Power• Reactive Power
![Page 17: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/17.jpg)
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.
![Page 18: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/18.jpg)
Inductor
![Page 19: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/19.jpg)
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.)
![Page 20: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/20.jpg)
Phasor Interpretation
Z=jωLVC=IC(jωL)IC=VC [-j/(ωL)]
![Page 21: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/21.jpg)
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.
![Page 22: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/22.jpg)
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
![Page 23: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/23.jpg)
Example 13-13
![Page 24: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/24.jpg)
Ohm’s law
I=VS/ZL
![Page 25: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/25.jpg)
Example 13-14
![Page 26: Passive Circuit Elements in the Frequency Domain Section 9.4-9.6.](https://reader035.fdocuments.us/reader035/viewer/2022081512/56649c745503460f94927303/html5/thumbnails/26.jpg)
Power in an inductor