### Experiment 5

#### Kirchhoff’s Rules

Objective:

The objective of this experiment is to verify Kirchhoff’s laws applied to a two-loop circuit.

Equipment:

A few ceramic resistors (200 to 500 ohms), two dc power sources (0 to 20 volts), 1 to 5 multi-meters, a calculator, and a few connecting wires with alligator clips

Theory:

Ohm’s law alone is not sufficient to solve for unknown currents in multi-loop circuits.  In such circuits, Kirchhoff’s rules are used to solve for the unknowns.  There are two rules:

a.                Kirchhoff’s loop rule (KLR)

b.               Kirchhoff’s junction rule (KJR)

KLR:

KLR simply states that the algebraic sum of voltage jumps and drops across the elements of a closed loop is zero.  To apply this law to a selected closed loop, a point must be selected, and then moving from that point in one direction, voltage ups and downs must be written with a plus sign for a jump and minus sign for a drop, until that point is arrived at again.  The sum then must be set equal to zero.  The reason is that if both terminals of a voltmeter are placed at the same point, the voltmeter will show a potential difference (voltage) of zero.  There are a few points that should be considered when applying KLR.

1)                  Assume a direction for current in the loop, either CW or CCW.

2)                  If moving with the current, then the voltage will drop when current passes through a resistor (for example, if the assumed current is I1, and the resistance is R1, then there will be a voltage drop of  (-V1 = -R1 I1)).

If moving against the assumed current, then the voltage will increase as the current goes through a resistor, and the voltage jump will be (+V1 = +R1 I1).

3)                  For a battery (inside of it), going from (-) to (+) is associated with a voltage jump, and from (+) to (-) with a voltage drop.  In Fig. 1, the directions for both currents I1 and I2 are assumed to be clockwise (CW).

KLR for loop efabe is written as follows:

- V1 - I1R2 - I1R1 – I1R3 + I2R3  = 0                                                                             (1)

Note that in branch be, I1 goes down and I2 goes up.  In fact, the current in branch be is

(I2 – I1), and the voltage across resistor 3 is R3 (I2 – I1).

Fig. 1

KLR for loop ebcde may be written as:

- I2R3 + I1R3 + V2 – I2R4 = 0                                                                                        (2)

KJR:

A junction is a point in a circuit where more than two wires are connected.

Points b and e are the two junctions in the above diagram (Fig. 1).

KJR states that the algebraic sum of currents going toward and away from a junction is zero.  If the currents going toward the junction are taken to be positive (+), then the currents going away from the junction will be negative (-).

The current in branch be can be named I3.  (The current I3 is equal to I2 – I1.Referring to

Fig. 2, the KJR at junctions e and b may be written as

At junction e:    I2 – I1 – I3 = 0     or      I3 = I2 – I1

At junction b:    I3 + I1 – I2 = 0     or      I3 = I2 – I1

Fig. 2

A two-loop circuit has three branches.  For each branch, the current must be determined; therefore, there are three unknownsThree equations are needed to solve for three unknownsTwo KLRs and one KJR will provide the three equations.

Procedure:

With R1, R2, and R3 chosen between 200 Ω and 500 Ω, construct a two-loop circuit as shown below (Fig. 3).  Set V1 = 8 V and V2 = 5 V.

Fig. 3

When the circuit is complete, write down the values of V1, V2, I1, I2, and I3 as read from the five multi-meters.  Use V1 and V2 in conjunction with R1, R2, and R3 (previously measured by one of the multi-meters), as data (given).  Use I1, I2, and I3 measured here as data (measured).   The measurements can be made with one multi-meter.  Consult your lab instructor.

Data: Given:

R1 = ? Ω                      R1, R2, and R3 should be measured by an ohmmeter

R2 = ? Ω                       for more-accurate values.  Use these in your calculations.

R3 = ? Ω

V1 = 8 V                      V1, and V2 are read from the voltmeters when the circuit is on.

V2 = 5 V                      Use these in your calculations.

Measured:

I1 = ?

I2 = ?                            These are measured values.

I3 = ?                            Do not use these in your calculations.

Calculations:

With the given values of V1, V2, R1, R2, and R3, apply KLR and KJR to solve for I1, I2, and I3, and use these calculated values as accepted values.

Comparison of the Results:

Corresponding to every measured value, there is an accepted value.  Calculate a percent error using the same percent error equation as in Experiment 2.

Conclusion:            To be explained by students

Discussion:             To be explained by students