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Resistance or Impedance
What we do day to day varies from engineer to engineer. The impression from non-engineers is that we stand in front of a large blackboard covered in equations, and while that is sometimes the case, more often then not we do simple arithmetic or 9th grade algebra.
One of the first real circuits facing a new electrical engineering student is the voltage divider. This is a spectacular circuit because it reinforces the basic understanding of ohms law, but it is complex enough, when you first see it, that the answer is not immediately obvious.
In the voltage divider circuit on the left, we have two series resistors between a voltage difference. In this case we have two 1000-ohm resistors between 3.3 volts and ground. We want to know the voltage between them, Vout1. Most of us can do this in our heads, half of 3.3V is 1.65V. Let’s look a bit closer at how we got there.
The voltage divider formula we all memorized is Vout1 is Vin (3.3V) times (R2/ (R1 +R2)). Simple enough, but do we really think much about that equation. Simple algebra allows us to swap R2 and Vin. Then we have:
Vout1 = R2 * Vin/(R1+R2)
This is very interesting because voltage divide by resistance is current and Vin divided by R1 + R2 is the current that is drawn by the circuit when it is powered, in this case 1.65mA. So, Vout1 is R2 times the system current, the current in R2. This is nothing more than ohms law; V=R*I.
Fine, fine, but what about AC
This is all fine and good. Childs play. But when someone asks the same question about a simple AC circuit, add a few caps and a resistor and run the thing at 1000 hz, well then … It’s not that much harder, but it does burn a bit more pencil lead.
In the right circuit above, we have two capacitors, a resistor and want to have a 50V source at 1000 hz power the circuit. It’s really the same circuit as on the left. We now have Z1 and Z2 instead of R1 and R2. And Z2 is a parallel combination of a cap and a resistor. If you do this all the time, you have probably stopped reading by now, otherwise it looks a bit daunting. You will have to pull out a book to remind yourself how to do algebra on complex numbers. You will get a real and an imaginary part which you need to change to polar coordinates.
When I worked this out, I got Vout2 = 140mA at -89.84 degrees. Let me know if you agree.
Sean O’Leary is sometimes known as the Celtic Engineer. He was involved in putting two missions on the space shuttle. He has worked at the Smelter’s Biproducts department of Kennecott Utah Copper. Has helped design ballistic guidance systems for Northrop Grumman. Worked on various DARPA projects, an anti-RPG system known as Iron-Curtain and has been involved with the downhole oil and gas industry. He currently is the owner of Celtic Engineering Solutions a consulting Engineering Company in West Jordan and Murray Utah.
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