| bonfire | 17-04-2011 12:36 AM |
Example of Network Theorems
Example:
In the circuit below, http://fourier.eng.hmc.edu/e84/lectures/ch2/img182.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img183.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img51.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img184.png. Find the value of current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png when http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png is http://fourier.eng.hmc.edu/e84/lectures/ch2/img186.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img187.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img188.png. Moreover, find the value for http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png for the desired current http://fourier.eng.hmc.edu/e84/lectures/ch2/img47.png. http://fourier.eng.hmc.edu/e84/lectu...ge_example.gif Method 1, http://fourier.eng.hmc.edu/e84/lectures/ch2/img189.png conversion
Find http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png when http://fourier.eng.hmc.edu/e84/lectures/ch2/img190.png. First convert the http://fourier.eng.hmc.edu/e84/lectures/ch2/img155.png composed of http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png into a http://fourier.eng.hmc.edu/e84/lectures/ch2/img156.png composed of http://fourier.eng.hmc.edu/e84/lectures/ch2/img162.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img163.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img164.png: http://fourier.eng.hmc.edu/e84/lectures/ch2/img191.png
Find overall resistance: http://fourier.eng.hmc.edu/e84/lectures/ch2/img192.png
Find overall current: http://fourier.eng.hmc.edu/e84/lectures/ch2/img193.png
Find currents through http://fourier.eng.hmc.edu/e84/lectures/ch2/img194.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img195.png (current divider): http://fourier.eng.hmc.edu/e84/lectures/ch2/img196.png http://fourier.eng.hmc.edu/e84/lectures/ch2/img197.png
Find voltage at points http://fourier.eng.hmc.edu/e84/lectures/ch2/img28.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img23.png (assuming negative end of voltage source is ground): http://fourier.eng.hmc.edu/e84/lectures/ch2/img198.png
Find current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png through http://fourier.eng.hmc.edu/e84/lectures/ch2/img190.png: http://fourier.eng.hmc.edu/e84/lectures/ch2/img199.png
The same steps can be repeated for http://fourier.eng.hmc.edu/e84/lectures/ch2/img200.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img201.png. But it is hard to find a value of http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png given the require current http://fourier.eng.hmc.edu/e84/lectures/ch2/img202.png. Method 2, Thevenin's theorem
Solve the problem using Thevenin's theorem by the following steps: Here, we remove http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png as the load of a network composed of all other resistors http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img194.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img195.png and the voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img182.png, then apply Thevenin's theorem to find the open-circuit voltage between the two terminals a and b: http://fourier.eng.hmc.edu/e84/lectures/ch2/img203.png
and the internal resistance between a and b (with voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img138.png short circuit): http://fourier.eng.hmc.edu/e84/lectures/ch2/img204.png
Now find current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png for different http://fourier.eng.hmc.edu/e84/lectures/ch2/img185.png and when http://fourier.eng.hmc.edu/e84/lectures/ch2/img202.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img211.png Example 2: The circuit below, often used in some control system, is composed of two voltages, two potentiometers, and a load resistor. Assume http://fourier.eng.hmc.edu/e84/lectures/ch2/img212.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img213.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img214.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img215.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img216.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img217.png. Find the current http://fourier.eng.hmc.edu/e84/lectures/ch2/img218.png through the load resistor http://fourier.eng.hmc.edu/e84/lectures/ch2/img219.png. http://fourier.eng.hmc.edu/e84/lectu...ntiometers.gif Method 1, Superposition theorem
Find http://fourier.eng.hmc.edu/e84/lectures/ch2/img220.png caused by voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img221.png, and then http://fourier.eng.hmc.edu/e84/lectures/ch2/img222.png caused by voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img223.png, then get http://fourier.eng.hmc.edu/e84/lectures/ch2/img224.png. - Short circuit http://fourier.eng.hmc.edu/e84/lectures/ch2/img64.png. Assume http://fourier.eng.hmc.edu/e84/lectures/ch2/img225.png, so that currents through http://fourier.eng.hmc.edu/e84/lectures/ch2/img194.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img195.png are, respectively, http://fourier.eng.hmc.edu/e84/lectures/ch2/img226.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img227.png (current divider), and http://fourier.eng.hmc.edu/e84/lectures/ch2/img228.png.
- http://fourier.eng.hmc.edu/e84/lectures/ch2/img229.png, current through http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png is http://fourier.eng.hmc.edu/e84/lectures/ch2/img230.png, current through http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png is http://fourier.eng.hmc.edu/e84/lectures/ch2/img231.png.
- http://fourier.eng.hmc.edu/e84/lectures/ch2/img232.png. But http://fourier.eng.hmc.edu/e84/lectures/ch2/img221.png, we get scaling factor http://fourier.eng.hmc.edu/e84/lectures/ch2/img233.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img234.png.
- Short circuit http://fourier.eng.hmc.edu/e84/lectures/ch2/img63.png. Assume http://fourier.eng.hmc.edu/e84/lectures/ch2/img235.png, so that currents through http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png are, respectively, http://fourier.eng.hmc.edu/e84/lectures/ch2/img236.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img237.png (current divider), and http://fourier.eng.hmc.edu/e84/lectures/ch2/img238.png.
- http://fourier.eng.hmc.edu/e84/lectures/ch2/img239.png, current through http://fourier.eng.hmc.edu/e84/lectures/ch2/img194.png is http://fourier.eng.hmc.edu/e84/lectures/ch2/img240.png, current through http://fourier.eng.hmc.edu/e84/lectures/ch2/img195.png is http://fourier.eng.hmc.edu/e84/lectures/ch2/img241.png.
- http://fourier.eng.hmc.edu/e84/lectures/ch2/img242.png. But http://fourier.eng.hmc.edu/e84/lectures/ch2/img223.png, we get scaling factor http://fourier.eng.hmc.edu/e84/lectures/ch2/img243.png, and http://fourier.eng.hmc.edu/e84/lectures/ch2/img244.png.
- Finally, we get the load current:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img245.png
Method 2, Thevenin's theorem
Remove http://fourier.eng.hmc.edu/e84/lectures/ch2/img246.png, find open-circuit voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img247.png and equivalent resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img246.png, then find http://fourier.eng.hmc.edu/e84/lectures/ch2/img248.png. |
