Parallel DC Circuits
Basic Electricity
Don’t just sit there! Build something!! 
Learning to mathematically analyze circuits requires much study and practice. Typically, students practice by working through lots of sample problems and checking their answers against those provided by the textbook or the instructor. While this is good, there is a much better way.
You will learn much more by actually building and analyzing real circuits, letting your test equipment provide the “answers” instead of a book or another person. For successful circuitbuilding exercises, follow these steps:
 Carefully measure and record all component values prior to circuit construction.
 Draw the schematic diagram for the circuit to be analyzed.
 Carefully build this circuit on a breadboard or other convenient medium.
 Check the accuracy of the circuit’s construction, following each wire to each connection point, and verifying these elements onebyone on the diagram.
 Mathematically analyze the circuit, solving for all values of voltage, current, etc.
 Carefully measure those quantities, to verify the accuracy of your analysis.
 If there are any substantial errors (greater than a few percent), carefully check your circuit’s construction against the diagram, then carefully recalculate the values and remeasure.
Avoid very high and very low resistor values, to avoid measurement errors caused by meter “loading”. I recommend resistors between 1 kΩ and 100 kΩ, unless, of course, the purpose of the circuit is to illustrate the effects of meter loading!
One way you can save time and reduce the possibility of error is to begin with a very simple circuit and incrementally add components to increase its complexity after each analysis, rather than building a whole new circuit for each practice problem. Another timesaving technique is to reuse the same components in a variety of different circuit configurations. This way, you won’t have to measure any component’s value more than once.
The formula for calculating total resistance of three parallelconnected resistors is as follows:

Algebraically manipulate this equation to solve for one of the parallel resistances (R_{1}) in terms of the other two parallel resistances (R_{2} and R_{3}) and the total resistance (R). In other words, write a formula that solves for R_{1} in terms of all the other variables.
Suppose I connect two resistors in parallel with one another, like this:

How much electrical resistance would you expect an ohmmeter to indicate if it were connected across the combination of these two parallelconnected resistors?

Explain the reasoning behind your answer, and try to formulate a generalization for all combinations of parallel resistances.
A quantity often useful in electric circuit analysis is conductance, defined as the reciprocal of resistance:

In a series circuit, resistance increases and conductance decreases with the addition of more resistors:

Describe what happens to total resistance and total conductance with the addition of parallel resistors:

In a parallel circuit, certain general rules may be stated with regard to quantities of voltage, current, resistance, and power. Express these rules, using your own words:
“In a parallel circuit, voltage . . .”
“In a parallel circuit, current . . .”
“In a parallel circuit, resistance . . .”
“In a parallel circuit, power . . .”
For each of these rules, explain why it is true.
What will happen in this circuit as the switches are sequentially turned on, starting with switch number 1 and ending with switch number 3?

Describe how the successive closure of these three switches will impact:
 • The voltage drop across each resistor
 • The current through each resistor
 • The total amount of current drawn from the battery
 • The total amount of circuit resistance “seen” by the battery
The circuit shown here is commonly referred to as a current divider. Calculate the voltage dropped across each resistor, the current drawn by each resistor, and the total amount of electrical resistance ßeen” by the 9volt battery:

 • Current through the 2 kΩ resistor =
 • Current through the 3 kΩ resistor =
 • Current through the 5 kΩ resistor =
 • Voltage across each resistor =
 • R_{total} =
Can you think of any practical applications for a circuit such as this?
There is a simple equation that gives the equivalent resistance of two resistances connected in parallel. Write this equation.
Secondly, apply this tworesistance equation to the solution for total resistance in this threeresistor network:

No, this is not a “trick” question! There is a way to apply a tworesistance equation to solve for three resistances connected in parallel.
Suppose you needed a resistance equal to precisely 235 Ω for the construction of a precision electrical meter circuit. The only resistors available to you are two 1 kΩ resistors, one 500 Ω resistor, and a rheostat variable between 600 and 1000 ohms. Design a parallel resistor network using any combination of these components that will yield a total resistance of 235 Ω. If you use the rheostat in your design, specify its resistance setting.
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