ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5,...
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Transcript of ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5,...
![Page 1: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/1.jpg)
ECE201 Lect-5 1
Single-Node-Pair Circuits (2.4); Sinusoids (7.1);
Dr. S. M. Goodnick
September 5, 2002
![Page 2: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/2.jpg)
ECE201 Lect-5 2
Example: 3 Light Bulbs in Parallel
How do we find I1, I2, and I3?
I R2 V
+
–
R1
I1 I2
R3
I3
![Page 3: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/3.jpg)
ECE201 Lect-5 3
Apply KCL at the Top Node
I= I1 + I2 + I3
11 R
VI
22 R
VI
33 R
VI
![Page 4: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/4.jpg)
ECE201 Lect-5 4
Solve for V
321321
111
RRRV
R
V
R
V
R
VI
321
1111
RRR
IV
![Page 5: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/5.jpg)
ECE201 Lect-5 5
Req
321
1111
RRR
Req
iMpar RRRRR
11111
21
Which is the familiar equation for parallel resistors:
![Page 6: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/6.jpg)
ECE201 Lect-5 6
Current Divider
• This leads to a current divider equation for three or more parallel resistors.
• For 2 parallel resistors, it reduces to a simple form.
• Note this equation’s similarity to the voltage divider equation.
j
parSR R
RII
j
![Page 7: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/7.jpg)
ECE201 Lect-5 7
Is2 VR1 R2
+
–
I1 I2
Example: More Than One Source
How do we find I1 or I2?
Is1
![Page 8: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/8.jpg)
ECE201 Lect-5 8
Apply KCL at the Top Node
I1 + I2 = Is1 - Is2
212121
11
RRV
R
V
R
VII ss
21
2121 RR
RRIIV ss
![Page 9: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/9.jpg)
ECE201 Lect-5 9
Multiple Current Sources
• We find an equivalent current source by algebraically summing current sources.
• As before, we find an equivalent resistance.
• We find V as equivalent I times equivalent R.
• We then find any necessary currents using Ohm’s law.
![Page 10: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/10.jpg)
ECE201 Lect-5 10
In General: Current Division
Consider N resistors in parallel:
Special Case (2 resistors in parallel)
iNpar
j
parSR
RRRRR
R
Rtiti
kj
11111
)()(
21
21
2)()(1 RR
Rtiti SR
![Page 11: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/11.jpg)
ECE201 Lect-5 11
Class Examples
• Learning Extension E2.11
![Page 12: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/12.jpg)
ECE201 Lect-5 12
Sinusoids: Introduction
• Any steady-state voltage or current in a linear circuit with a sinusoidal source is a sinusoid.– This is a consequence of the nature of
particular solutions for sinusoidal forcing functions.
– All steady-state voltages and currents have the same frequency as the source.
![Page 13: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/13.jpg)
ECE201 Lect-5 13
Introduction (cont.)
• In order to find a steady-state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency).
• Usually, an AC steady-state voltage or current is given by the particular solution to a differential equation.
![Page 14: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/14.jpg)
ECE201 Lect-5 14
The Good News!
• We do not have to find this differential equation from the circuit, nor do we have to solve it.
• Instead, we use the concepts of phasors and complex impedances.
• Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problems.
![Page 15: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/15.jpg)
ECE201 Lect-5 15
Phasors
• A phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or current.
• Remember, for AC steady-state analysis, this is all we need---we already know the frequency of any voltage or current.
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ECE201 Lect-5 16
Complex Impedance
• Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor).
• Impedance is a complex number.
• Impedance depends on frequency.
![Page 17: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/17.jpg)
ECE201 Lect-5 17
Complex Impedance (cont.)
• Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage, and voltage from current.
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ECE201 Lect-5 18
Sinusoids
• Period: T– Time necessary to go through one cycle
• Frequency: f = 1/T– Cycles per second (Hz)
• Angular frequency (rads/sec): = 2 f
• Amplitude: VM
tVtv M cos)(
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ECE201 Lect-5 19
Example
What is the amplitude, period, frequency, and angular (radian) frequency of this sinusoid?
-8
-6
-4
-2
0
2
4
6
8
0 0.01 0.02 0.03 0.04 0.05
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ECE201 Lect-5 20
Phase
-8
-6
-4
-2
0
2
4
6
8
0 0.01 0.02 0.03 0.04 0.05
![Page 21: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/21.jpg)
ECE201 Lect-5 21
Leading and Lagging Phase
x1(t) leads x2(t) by -x2(t) lags x1(t) by -
On the preceding plot, which signals lead and which signals lag?
tXtx M cos)(11
tXtx M cos)(22
![Page 22: ECE201 Lect-51 Single-Node-Pair Circuits (2.4); Sinusoids (7.1); Dr. S. M. Goodnick September 5, 2002.](https://reader030.fdocuments.us/reader030/viewer/2022032607/56649ec15503460f94bcd8c2/html5/thumbnails/22.jpg)
ECE201 Lect-5 22
Class Examples
• Learning Extension E7.1
• Learning Extension E7.2