Name Section
In analyzing circuits, it is generally the current that is of interest. You have seen how Ohm's Law can be used to analyze very simple circuits consisting of an EMF and a single resistance. This method can be extended to include circuits with more than one resistance, provided the resistances are in series or parallel arrangements that can be replaced with an equivalent resistance.
However, there are more complex circuits in which Ohm's Law is inadequate for a full analysis. The resistances may not be in simple series or parallel arrangements, for example. Or, EMF 's and currents may be shared between different loops in the circuit. This is where Kirchhoff's Rules are applied.
First, some terminology – reference the circuit below.
Junction
A point at which three or more wires connect; i.e., where three or more currents meet. Points A and B are junctions.
Branch
A single path connecting two junctions; there may be any number of circuit elements in the branch. There are three branches connecting junctions A and B – the left branch containing V
1
and R containing V
2
and R
2
, and the right branch containing R
3
1
, the center branch
.
Loop
A closed path of two or more branches. There are three here – two inside loops labeled 1 and 2 , and the loop all the way around the outside of the circuit.
Kirchhoff's two rules are:
Junction rule
The algebraic sum of the currents at any junction is zero; this is a consequence of the conservation of charge. Sign convention is that any current into the junction is positive and any current out of the junction is negative. You assign a direction to the current in each branch when you start – it is arbitrary. In solving the subsequent system of equations for the currents in the circuit, a negative value just means that the actual direction of the current in that branch is opposite of the direction you chose – the magnitude is the same.
Loop rule
The algebraic sum of the changes in potential around any closed loop is zero; this is a consequence of the conservation of energy. Sign convention depends on the direction of the current in a branch as well as the direction taken around the loop
(they do not have to be the same). Any time you traverse an EMF from negative to positive (up the potential "hill"), the potential is taken as positive; if going from positive to negative across the EMF the potential is negative (a "drop"). For resistances, when you traverse one in the direction of the current, the change in potential is negative. If you go across the resistance opposite the direction of the current, it is positive.
You use these rules to write equations involving the unknowns in the circuit (the current in each branch). You will need the same number of independent equations as you have unknowns; these may then be solved simultaneously to obtain the unknown currents.
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1.5V Batteries, DMM, Connecting Wires, Resistors, Switches.
Note: There are switches in each of these circuits – put them there and use them. Close the switches only when measuring potentials or currents – do not drain the batteries while thinking about what to do next.
Testing Kirchhoff's Rules via Direct Calculation
Use the DMM to measure the resistances of the resistors and record these values in Table 1.
Resistor
R
1
R
2
R
3
Table 1
Resistance ( Ω )
Wire the circuit below, and label each of the resistors in the circuit diagram with the values above. Measure the effective potential of each of the sources (switches closed ) and label these as well (notice that one of the sources is a single cell, while the other is two cells in series).
Now, assign a unique variable name ( I
1
, I
2
, …) to each of the unknown currents in the circuit and label these (you will also need to assign a direction for each – indicate the direction chosen for each as well). Using these resistances and potentials, set up and solve a system of equations to obtain the unknown currents. These are your theoretical values. Use the DMM to measure the experimental current in each branch for comparison. Record these experimental currents in Table 2.
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1. Display the system of equations you came up with here:
2. Solve these equations for the currents here (also place these values in Table 2):
Sp07
Current
I
1
I
2
I
3
Experimental Value
Table 2 from DMM
(A)
Theoretical Value from Equations
(A)
Percent
Difference
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Name Section
1. What is a junction ?
2. What is a branch ?
3. What is a loop ?
4. How many different currents are in the circuit shown in the Theory section?
5. Does the direction you assign to each current in a circuit matter when you are setting up the Kirchhoff equations?
Why or why not?
6. Solve the following system of equations:
4x − 3y  z = 11
2x  y − 4z =− 1 x  2y − 2z = 1
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