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Objective:__

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

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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

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__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 I_{1},
and the resistance is R_{1}, then there will be a voltage drop of (-V_{1}
= -R_{1} I_{1})).

If moving against the assumed
current, then the voltage will increase as the current goes through a resistor,
and the voltage jump will be (+V_{1} = +R_{1} I_{1}).

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 I_{1} and I_{2} are assumed to be
clockwise (CW).

KLR for loop **efabe** is written as follows:

- V_{1} - I_{1}R_{2}
- I_{1}R_{1} – I_{1}R_{3} + I_{2}R_{3
} = 0
(1)

Note that in branch **be**, I_{1} goes down and I_{2} goes
up. In fact, the current in branch **be** is

(I_{2} – I_{1}), and the voltage across resistor 3 is R_{3}
(I_{2} – I_{1}).

Fig. 1

KLR for loop **ebcde** may be written as:

- I_{2}R_{3 }+ I_{1}R_{3}
+ V_{2} – I_{2}R_{4} =
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 I_{3}. (The current I_{3
}is equal to I_{2} – I_{1.}) **Referring to **

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

At junction **e**:
I_{2} – I_{1} – I_{3} = 0 or I_{3} =
I_{2} – I_{1}

At junction **b**: I_{3}
+ I_{1} – I_{2} = 0 or I_{3} = I_{2} –
I_{1}

Fig. 2

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

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Procedure:__

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With R_{1}, R_{2}, and R_{3}
chosen between 200 Ω and 500 Ω, construct a two-loop circuit as shown below
(Fig. 3). Set V_{1} = 8 V and V_{2} = 5 V.

Fig. 3

When the circuit is complete, write down the values of V_{1},
V_{2}, I_{1}, I_{2}, and I_{3 }as read from the
five multi-meters. Use V_{1} and V_{2} in conjunction with R_{1},
R_{2}, and R_{3} (previously measured by one of the
multi-meters), as data (given). Use I_{1}, I_{2}, and I_{3}
measured here as data (measured). The measurements
can be made with one multi-meter. Consult your lab instructor.

Data:**
**__Given:__

R_{1} = ? Ω
R_{1}, R_{2}, and R_{3} should be
measured by an ohmmeter

R_{2 }= ? Ω for
more-accurate values. Use these in your calculations.

R_{3} = ? Ω

V_{1} = 8
V V_{1}, and V_{2} are read from the
voltmeters when the circuit is on.

V_{2} = 5
V Use these in your calculations.

__Measured:__

I_{1} = ?

I_{2} = ?
These are **measured values.**

I_{3} = ? Do
not use these in your calculations.

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Calculations:__

With the given values of V_{1}, V_{2}, R_{1},
R_{2}, and R_{3}, apply KLR and KJR to solve for I_{1},
I_{2}, and I_{3, }and use these calculated values as **accepted
values**.

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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.

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Conclusion:__ To be explained by students

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Discussion:__ To be explained by students