Lab Report #1 CIRCUIT THEOREMS

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Oct 10, 2013 (3 years and 10 months ago)

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ELEC 250 : Linear Circuits I


Lab Report #1

CIRCUIT THEOREMS


Toshio Ouchi

Joe Reimer

Gabrielle Odowichuk



















1.0 OBJECTIVE



2

To verify and become familiar with the following linear
circuit theorems:



1. Kirchhoff’s Current a
nd Voltage Laws



2. Linearity and Superposition Theorems



3. Thevenin’s and Norton’s Theorems


3

2.0 RESULTS


2.1 Kirchhoff’s Current Law


Figure 2.1


The circuit in Figure 2.1 was connected to demonstrate
Kirchhoff’s Current and Voltage Laws. The voltag
e sources
and resistors had the following magnitudes:





V
a

= 6V

R
1

= 2.2 k



R
3

= 1 k







V
b

= 3V

R
2

= 2.2 k



R
L

= 1 k



Currents at each of the four points in the system were
measured with a Digital Multimeter. The meter was
connected in series with

the circuit, and measured with the
DMM in current measuring mode in the direction of the sign
convention indicated by the arrows in Figure 2.1.


I
1

= 1.433 mA

I
2

= 0.054 mA

I
3

= 1.487 mA

I
L

= 1.487 mA


2.2 Kirchhoff’s Voltage Law



Using the connected cir
cuit, voltages across each of
the four resistors were measured in parallel with a digital
multimeter in voltage measuring mode. The direction on the
voltages is indicated by figure 2.1.


V
1

= 3.11 V

V
2

= 0.117 V

V
3

= 1.477 V

V
L

= 1.477 V





4

2.3 Linearity

Theorem


Figure 2.2


Voltage source V
b

was replaced with a short circuit and
V
a

was varied as seen in the table below to develop
different values of I
L

and V
L
. A multimeter was used to
measure both the current I
L

and the voltage V
L
.


The measured value
s of V
L

and I
L

as V
b

varies are as
follows:















2.4 Superposition Theorem


With the circuit still set up as prev
iously, V
a

was set to
6V. The load voltage and the load current across R
L

due to
V
a

alone was measured.


V
a

= 6 V

V
L

= 0.9901 V

I
L

= 0.985 mA






V
a

V
L

(V)

I
L

(mA)

7

1.1402

1.144

6

0.9901

0.985

5

0.8173

0.826

4

0.6628

0.654

3

0.4938

0.5

2

0.3414

0.344

1

0.177

0.168


5

2.5 Thevenin’s Theorem


Figure 2.3


Resistor R
L

from figure 2.1 was disconnected leaving
branch AB open
-
ci
rcuited. A resultant Thevenin voltage, V
T
,
was measured. While RL was still disconnected, both voltage
sources V
a

and V
b

were disabled and a Thevenin resistance R
T

was measured. The resistors in the circuit were replaced
with a circuit box of equivalent r
esistance and a single
voltage source of V
T

was used. V
L
' and I
L
' were measured
across the open resistor R
L
.


V
T

= 4.572 V

R
T

= 2.661 k

V
L
' = 1.458 V

I
L
' = 1.458 mA


2.6 Norton’s Theorem


Figure 2.4


Resistor RL from the circuit in figure 2.1 was replaced

with a straight wire creating a short circuit in the
system. A Norton current, I
N
, was measured across the short.
The circuit was modified as per figure 2.4 with the
measured I
N

and the resistance box of R
T
. V
L
'' and I
L
'' were
measured across the short.


I
N

= 2.17 mA


6

V
L
'' = 1.503 V

I
L
'' = 1.497 mA

3.0 DISCUSSION


3.1 Kirchhoff’s Current Law


3.1.1 Theoretical Calculations


Using Mesh analysis,



Mesh 1









(3.1)


Mesh 2





(3.2)


Using Kirchhoff’s Current

Law,



Node A








(3.3)


Node B








(3.4)


Node C








(3.5)


Rearranging (3.1),










(3.6)


Rearranging (3.2) and substituting for I
L
(3.4),











(3.7)


Combining (3.6) and (3.7),









(3.8)


7


Solving for (3.8),

I
3

= I
L

= 1.45 mA

Solving for (3.7),

I
1

= 1.41 mA

Solving for (3.3),

I
2

= 0.043 mA

3.1.2 Comparison


Measured

Theoretical

I
1

= 1.433 mA

I
1

=

1.41 mA

I
2

= 0.054 mA

I
2

= 0.043 mA

I
3

= 1.487 mA

I
3

= 1.45 mA

I
L

= 1.487 mA

I
L

= 1.45 mA


Percent difference for I
1
:




Percent difference for I
2
:




Percent difference for I
3
:




P
ercent difference for I
L
:





The current values measured from the system were quite
close to the theoretical values for the most part. The
value for I
2

differed much more than the other values. This
may have been due number roundi
ng in the digital multimeter
and the small size of the current. When the size of the
current is smaller, slight errors will affect the final
precision of the value to a greater extent.


3.1.3 Verifying Kirchhoff’s Current Law



8

Using our measured values
for I
1
, I
2
, I
3
, and I
4
, we can
verify Kirchhoff’s Current Law.



At Node A, KCL tells us that

I
1

+ I
2

-

I
3

= 0

Substituting in our measured values,

1.433 mA + 0.054 mA
-

1.487 mA = 0



At Node B, KCL tells us that

I
3

-

I
L

= 0


Substituting in our measured
values,

1.487 mA
-

1.487 mA = 0






At Node C, KCL tells us that

I
L

-

I
2

-

I
1

= 0


Substituting in our measured values,

1.487
-

0.054
-

1.433 = 0



3.2 Kirchhoff’s Voltage Law


3.2.1 Theoretical Calculations



Using Ohm’s Law,

V
1

= R
1
I
1

= 3.10 V

V
2

= R
2
I
2

= 0.0937 V

V
3

= R
3
I
3

= 1.45 V

V
L

= R
L
I
L

= 1.45 V


3.2.2 Comparison


Measured

Theoretical

V
1

= 3.11 V

V
1

= 3.10 V

V
2

= 0.117 V

V
2

= 0.0937 V

V
3

= 1.477 V

V
3

= 1.45 V

V
L

= 1.477 V

V
L

= 1.45 V


Percent difference for V
1
:


Percent

difference for V
2
:


Percent difference for V
3
:


9


Percent difference for V
L
:



The voltage measurements were close to the theoretical
values with the exception of V
2
. Similarly to with th
e
current measurement, this may have been sure to errors in
the digital multimeter that would affect smaller values
more than larger values.


3.2.3 Verifying Kirchhoff’s Voltage Law



In Loop 1, KVL tells us that

V
a

-

V
1

+ V
2

-

V
b

= 0


Substituting in our
measured values,

6 V
-

3.11 V + 0.117 V
-

3V = 0.07


0



In Loop 2, KVL tells us that

V
b

-

V
2

-

V
3

-

V
L

= 0


Substituting in our measured values,

3 V
-

0.117 V
-

1.477 V
-

1.477 V =
-
0.071


0


3.3 Linearity Theorem


The equations of the circuit are:


I
1

= I
2

+ I
3

V
a



R
1

I
1



R
2

I
2

= 0

V
a



R
1

I
1



R
3

I
3

= 0


Where V
a

is the source voltage.














Measured





Theoretical


V
a

V
L

(V)

I
L

(mA)

7

1.1402

1.144

6

0.9901

0.985

5

0.8173

0.826

4

0.6628

0.654

3

0.4938

0.5

2

0.3414

0.344

1

0.177

0.168

V
a

V
L

(V)

I
L

(mA)

7

1.129

1.129

6

0.
968

0.968

5

0.806

0.806

4

0.645

0.645

3

0.484

0.484

2

0.323

0.323

1

0.161

0.161


10







When the experimental values are plotted, the result
is the following graphs:



y = 0.1611x + 0.016




11



y = 0.162x + 0.0121

The theoretical and measured values for K1 and K2 are seen
in the tables:


Theoretical


Measured

V
a

K1

K2

V
a

K1

K2

7

0.161

0.000161

7

0.163

0.000163

6

0.161

0.000161

6

0.165

0.000164

5

0.161

0.000161

5

0.163

0.000165

4

0.161

0.000161

4

0.166

0.000164

3

0.161

0.000161

3

0.165

0.000167

2

0.161

0.000161

2

0.171

0.000172

1

0.161

0.000161

1

0.177

0.000168


3.4 Superposition Theorem



With the circuit still set up as previously, Va was
set to 6V.

The load voltage and the load current across RL
due to Va alone was measured and compared with theoretical
calculations.



Va




= 6 V


Measured load voltage

= 0.9901 V


Measured load current

= 0.985 mA


Theoretical load voltage

= 0.968 V


Theoretical

load current

= 0.968 mA


12



The voltage source V
b

was set to 3V while the other
source V
a

was short circuited. Both theoretical and
measured values of the load voltage and load current are
shown in the table below:


V
b

= 3V

Theoretical

Measured

Load volt
age (V)

0.4839

0.4965

Load current (mA)

0.4839

0.4990



These theoretical values were determined from the
equations:


I2 = I1 + I3


3


2200 I2


2200 I1 = 0


3


2200 I2


2000 I3 = 0



For this circuit, the superposition theorem holds true
because:

V
L

= V
La

+ V
Lb

I
L

= I
La


+I
Lb


1.48 V


0.9901 V + 0.4965 V = 1.4866 V (V
a

= 6V and V
b

=
3V)

and

1.487 mA


0.985 mA + 0.499 mA = 1.484 mA

3.5 Thevenin’s Theorem


Measured

Calculated

V
T

= 4.572 V

V
T

= 4.5 V

R
T
=2.166 k


R
T

= 2.1k



V
L
' = 1.458 V

V
L
= 1.
48 V

I
L
' = 1.458 mA

I
L
= 1.487 mA


Percent Difference for V
T

:



Percent Difference for R
T
:



The values for V
T

and R
T
calculated are close to the

13

theoretical values. Any error can be attributed to slight
ins
trument error.


Percent Difference for V
L

:



Percent Difference I
L
:



The values for V
L

and I
L

calculated in this step are also
quite close to the values obtained earlier in the
experiment. The small error c
an be attributed to the
digital multimeter.


3.6 Norton’s Theorem


V
L
'' = 1.503 V

I
L
'' = 1.497 mA

I
N

Measured = 2.17 mA

I
N

Calculated = 2.143 mA





Percent Difference for V
L
'':



Percent difference for I
L
'' :



Percent Difference for I
N

:




The resultant load voltage and current values, VL'' and
IL'' , are extremely close to the original values, VL and
IL. This is expected since using Norton's and Thevenin's
theorems we get equivalent c
ircuits with fewer components.
We assume that the errors are due to internal resistances
of the circuit's components.


14


4.0 CONCLUSIONS








Our circuit analysis proved that the theorems studied in
this lab ar
e accurate. Each theorem was proved through
experiment with fairly precise data.


Sources of error can be attributed to small inconsistencies
in the lab equipment. The resistors and voltage sources
used in the lab are an approximation of the numbers us
ed
for the theoretical calculations, as is the valued recorded
from the digital multimeter.