 ece240l_labmanual_3.pdf

21

EXPERIMENT THREE DC CIRCUITS

EQUIPMENT NEEDED:

1) DC Power Supply 2) DMM 3) Resistors 4) ELVIS

THEORY Kirchhoff's Laws: Kirchhoff's Voltage Law: The algebraic sum of the voltages around any closed path is zero.

N

i iv

1

1.30

Kirchhoff's Current Law: The algebraic sum of the currents at any node is zero.

N

i ii

1

2.30

22

Series Circuits: In a series circuit the current is the same through all the elements.

Figure 3. 1

The total series resistance RS is given by

3.3121 NNS RRRRR

and

4.3SS IRV

The Kirchhoff's voltage law indicates that:

5.3121 NNS VVVVV

V

R

+ V1 -

R

+ V2 -

R

- VN + - VN-1 +

RN-

I

23

The voltages across resistors can be obtained by multiplying the current by the corresponding resistors.

6.3

1 1

2 2

1 1

11

22

11

S S

N N

S S

N N

S S

S S

S

S

NN

NN

V R RV

V R RV

V R RV

V R RV

R VI

IRV

IRV

IRV

IRV

The last expressions of equation 3.6 are known as voltage division.

24

Parallel Circuits: In a parallel circuit the voltage is the same across all the elements.

Figure 3. 2

The total parallel resistance, Rp is given by

7.3 11111

121 NNP RRRRR

and

8.3PPP RIV

Kirchhoff's current law states:

9.3121 NNP IIIII

I RI RI RN-IN- RI PV P

+

-

25

The current through the branch resistors can be obtained by dividing the terminal voltage PV

by the corresponding branch resistance, R. therefore:

10.3

1 1

2 2

1 1

1 1

2 2

1 1

P N

P N

P N

P N

P P

P P

PPP

N

P N

N

P N

P

P

I R RI

I R RI

I R RI

I R RI

RIV

R VI

R VI

R VI

R VI

The last expressions of equation 3.10 are known as current division.

The reciprocal of resistance is known as conductance. It is expressed in the following equa- tions:

11.31 R

G

and

12.3 1

P P R G

26

This expression can be used to simplify equations 3.12 as shown below.

13.3

1 1

2 2

1 1

11

22

11

P P

N N

P P

N N

P P

P P

P

P P

PNN

PNN

P

P

I G GI

I G GI

I G GI

I G GI

G IV

VGI

VGI

VGI

VGI

where

14.31111

121 NN P RRRR G

27

If only two resistors make up the network, as shown next

Figure 3.3

then the current in branches 1 and 2 can be calculated as follows:

18.31

17.311

16.31

15.3

21

2 1

21

21

1 1

21

21

21

1 1

1 1

PP

PP

P P

I RR

RII RR RR

R I

RR RRG

RR G

and

R G

I G GI

In a similar fashion it can be shown that

19.3 21

1 2 PIRR

RI

(Note how the current in one branch depends on the resistance in the opposite branch)

R1 I1

R2 I2

IP

28

But, if the network consists of more than two resistors - say four

Figure 3.4 Then the calculation or branch currents using individual resistance becomes complex as demonstrated next, e.g.,

21.3111111

20.3

4321

321421431432

4321

3 3

RRRR RRRRRRRRRRRR

RRRRRR

I R RI

PP

P P

so that

22.31

321421431432

4321

3 3 PIRRRRRRRRRRRR

RRRR R

I

and

23.3 321421431432

421 3 PIRRRRRRRRRRRR

RRRI

R3

I3

R4

I4

IP R1

I1

R2

I2

29

By using conductances, the above is simplified to

24.31111

1

4321

3 3 PI

RRRR

RI

and is easily accomplished with a hand calculator. As the above demonstrates, when using current division, always use conductances and avoid using resistances in the calculation for all parallel networks with more than two resistors.

Series - Parallel Circuits

The analysis of series -parallel circuits is based on what has already been discussed. The solution of a series-parallel circuit with one single source usually requires the computation of total resistance, application of Ohm's law, Kirchhoff's voltage law, Kirchhoff's current law, vol- tage and current divider rules. Preliminary Calculations:

Be sure to show all necessary calculations. l. The resistors used in this lab all have 5% tolerances. This is denoted by the gold band. Calculate the minimum and maximum values for resistances with nominal values of 1kΩ and 2.7kΩ. Enter the values in Table 3.1. 2. Assume that the two resistors of problem 1 are used in the circuit of Figure 3.5. Calculate v1, v2, and when R1 and R2 take on their minimum and maximum values and enter in Table

3.2.

30

Figure 3.5

3. From your calculations in 2, record the maximum and the minimum possible values of , v1, and v2 that you should see in the circuit in Table 3.3. Also, calculate and record the value of these variables when R1 and R2 are at the nominal values. What is the maximum % error in

each of the variables possible due to the resistor tolerances? 4. For the circuit of Figure 3.6 calculate the resistance between nodes: a. a and b (Ra-b) b. a and c (Ra-c) c. c and d (Rc-d)

Enter your results in Table 3.4 Hint: Part c cannot immediately be reduced using series and parallel combinations.

R2

2.7kΩ

R1 1kΩ

10V

I + V1 -

+ V2 -

31

Figure 3.6

5. Use voltage division to calculate V1 and V2 for the circuit in Figure 3.7. Enter your re-

sults in Table 3.5.

Figure 3.7

3.3kΩ 2.7kΩ

1kΩ

15V

4.7kΩ

+ V1 -

+ V2 -

3.3kΩ

1kΩ 1kΩ

2.7kΩ

2.7kΩ

d

a b

c

32

6. For the circuit in Figure 3.8, if R = 1k ohm, calculate Use current division to calculate

R. Enter your results in Table 3.6. Repeat for R = 2.7k and 3.3k ohms.

Figure 3.8

7. For the circuit in Figure 3.9, calculate each of the variables listed in Table 3.7.

Figure 3.9

Procedure

3.3kΩ 2.7kΩ

1kΩ

10V

4.7kΩ

1kΩ

I1

I3

I2 I4

I5

+ V1 - + V2 -

- V5 +

+ V3 -

+ V4 -

R 1kΩ

100kΩ

15V

I IR

33

l. Place a wire between the two measuring terminals of the ohmmeter and adjust the mea- surement reading to zero ohms. Obtain a 1kΩ and 2.7kΩ resistor and measure their values with the ohmmeter. What is the % error as compared to their nominal values? Enter your re- sults in Table 3.1. 2. Construct the circuit in Figure 3.5. Measure V1 and V2 using the DMM only. Calculate

from your measurements. What is the % error as compared to their nominal values? Enter your results in Table 3.3. 3. Construct the circuit of Figure 3.6. Use an ohmmeter to measure the resistances listed in Table 3.4. Calculate the % error. 4. Construct the circuit of Figure 3.7. Measure V1 and V2 using the DMM only. Calculate

the % error. Enter your results in Table 3.5. 5. Construct the circuit of Figure 3.8. Find and R for R = 1kΩ, 2.7 kΩ, and 3.3 kΩ by mea-

suring the appropriate voltages using the DMM only and applying Ohm's Law. Enter your re- sults in Table 3.6. Note that is approximately constant. Why? 6. Construct the circuit of Figure 3.9. Using the DMM, measure each of the variables listed in Table 3.7, and calculate the % error for each. Verify that KVL holds for each of the 3 loops in the circuit. Verify that KCL holds at each node. What can be said about 2+ 3 and 1?

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Table 3.1

Rnominal Rmin Rmax Rmeas % error 1kΩ

2.7kΩ Table 3.2

R1,min R2, min

R1, max R2, min

R1, min R2, max

R1, max R2,max

I V1 V2

Table 3.3

max min nom max % error

meas % error

V1 V2 I

Table 3.4

Resistance Calculated Measured % error Rab Rac Rcd

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Table 3.5 Calculated Measured % error

V1 V2

Table 3.6

R I, calc IR, calc I, meas IR, meas 1kΩ

2.7kΩ 3.3kΩ

Table 3.7 PARAMETER CALCULATED MEASURED % ERR

V1 V2 V3 V4 V5 I1 I2 I3 I4 I5