DC circuits - Rose-Hulman

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4
-
10








DC CIRCUITS
-

OHM'S AND KIRCHHOFF'S LAWS




References

Crummett and Western,
Physics: Models and Applications
, Sec. 27.1,2,3

Halliday, Resnick, and Walker,
Fundamentals of Physics

(5th ed.), Sec. 27
-
4,5; 28
-
3,4,5,6

Tipler,
Physics for Scientists and

Engineers

(3rd ed.), Sec. 22
-
2, 23
-
1



Introduction

The most basic electrical property of matter is
resistance
. The electrical resistance of a piece of
material determines its behavior in a DC circuit. (Other properties must also be considered in circui
ts in
which signals vary with time.) A material whose resistance is independent of the current in it obeys
Ohm's law
:


(1)

where the resistance
R

is a constant. In (1),
V

is the potential difference across
R
, and
I

is the current
th
rough it.

The behavior of resistances and voltage sour
ces con
nected
in complex circuits can be analyzed by Kirchhoff's laws. The first
of these, the
junction theorem
, expresses the fact that electric
charge is conserved:





For example, in the circuit in Figure 1, currents
I
1

and
I
2

flow
toward

point B, while current
I
3

flows
away from B
. Therefore


Kirchhoff's second rule, the
loop theorem
, expresses t
he fact that the potential at any point in a
steady
-
state circuit has a definite value:





In Figure 1, as an example, consider pat
h ABCD. Starting at A, we first encounter a potential drop
I
1
R
1

(by Ohm's law) in resistor
R
1
, then another drop
I
3
R
3

in resistor
R
3
, and finally a potential increase of
amount _
1

as we pass through the emf from negative to positive.


Thus


The net current flowing into any

junc
-
tion in the circuit must be zero.

The algebraic sum of all the potential differences encountered in going around any
closed path in the circuit must be zero.

Figure 1

A DC circuit

DC CIRCUITS: OHM’S AND KIRCHHOFF’S LAWS

4
-
11









or




If we know the emf's and resistances in a circuit, Kirchhoff's two laws always provide exactly enough
independent equations to solve for all the branch currents.

The familiar rules for series and parallel combinations of r
esistances follow from these ideas. If two
resistances
R
1

and
R
2

are connected in series, the voltage drop
V

across the combina
tion is the sum of
the voltage drops across each, while the same current is flowing in each:


or




(2)

If the two resistors are connected in parallel, on the other hand, the potential difference across each is
the same, but the current divides; the total current is

the sum of the two individual currents:


or




(3)

In this experiment, you'll check the application of Ohm's law, the formula (3) for parallel combination
of resistances, Kirchhoff's laws, and (if there is time) the superposition theorem of elementary network
theory, in some simple circuits
.



Equipment

DC circuit patchboard

regulated variable DC power supply

Digital multimeter(s)

leads set including probe leads



Procedure

The circuit of the "patchboard" is shown in Figure 2 (top of the next page).


By making different
connections on it, y
ou can set up several different simple cir
cuits.


In this experiment, you'll make current
measurements by measuring the voltage drop across precision resistors, using a digital voltmeter.


In the
circuit of Figure 2, 200
Ω
, 1% tolerance res
istors were
used in making the patchboards; "tolerance" is
several standard deviations, so you can consider that each of the resistance values has a stan
dard
devia
tion of around 0.3%.

Use the multimeter to verify the resistance values. Given precision resistors,
u
s
ing the voltage drop across the resistor

for current measurement is

much more accurate than direct
current
measurement with an analogue meter would be, so you'll determine

currents just by measuring
the voltage drop across the already known resistors.



4
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12

DC CIRCUITS: OHM’S AND KIRCHHOFF’S LAWS








Figure 2

Patch board for DC circuit experiments



In applying Ohm's and Kirchhoff's rules, keep the following conventions in mind:


1.

The positive terminal of a battery or power supply is at a higher poten
tial than the negative terminal.


2.

Current in a
resistance

flows from higher to lower p
otential.


3.

A voltmeter reads the difference of potential between the two points to which its terminals are
connect
ed.


When the a volt
me
ter indi
cates a positive value, the lead con
nected to the (
+
) terminal
(often red) of the voltme
ter is con
nect
ed to a point of the circuit higher in potential than is the lead

connected to the (
-
) (usually black) terminal.


4.

In what follows,
V
ab

=
V
a

-

V
b

stands for the difference of potential between points a and b. If point a
is at the higher potential, then
V
ab

is positive.



A.


Ohm's law and resistors in series

(1)

Arrange the circuit as shown in Fig. 3, using the DC power
supply as voltage source
. (The positive terminal of the
power supply is red, the negative terminal black.)


S
et your
digital meter to the 10 V DC scale, and connect it from point A
(+) to point B (
-
), so that it measures the output voltage of the
power supply.


Turn on the power supply and the voltage

difference

between the terminals should be close to

6.0

V.


Le
ave the controls at this posi
tion for the remainder of the ex
-
periment.


Now disconnect the digital voltme
ter from A and B
without

changing any
thing else in the circuit.












Warning
:


A test lead with a poor connection in it can easily add several unsus
pected ohms into a
circuit.


It's a good idea to check your test leads on the digital multimeter's resistance scale before you
get down to work.


A "
good" lead should show no more than 0.1
Ω

or so.


Figure 3

Circuit for part A

DC CIRCUITS: OHM’S AND KIRCHHOFF’S LAWS

4
-
13










(2)

Measure and record
V
AC
.


Calcu
late the cur
r
ent in
R
1

from your measured value of
V
AC

and the
known value of
R
1

(100 ohms).


(3)

Measure the potential difference
V
CD

between points C and D. Knowing that the cur
rent you
calculated in step 1 is also the cur
rent in
R
3
, calculate the resistance of
R
3
.


(4)

Measure
V
AD
.


Compare this value with the sum of
V
AC

and
V
CD

measured in steps 2 and 3.


Calcu
-
late the percentage difference between these two values. How do they compare? Is the difference
reasonable in light of your measurement uncertainties?



(5)

Turn off

the power supply. It is possible, by connecting certain points of the patch board and
connecting the power supply to appropriate points, to achieve a circuit just like that of Fig. 3
except

that resistor
R
4

is located in the position occupie
d by
R
3

in the figure.


Diagram this, showing which
points of the patch board you connected to achieve the de
sired circuit.


Wire up the circuit,
double
-
check it, turn the power supply on, and repeat the procedure of steps (l) through (4).


De
-
termine the

value of
R
4
.


B.

Resistors in parallel

(6)

Connect the variable DC power supply be
-
tween points
A (+) and B.


Make the circuit shown in Figure 4, with
resis
tors
R
3

and
R
4

con
nect
ed in par
allel.


(7)

Connect the DVM between points A and B.


Turn on
the power supply and set the sup
ply voltage
= 5.00
V.


Measure the potential drop from A to C, and from C
to D. Using the known value of 100
Ω

for
R
1
, calculate
from
V
AC

the current being provided by the power
supply. This current
flows through the parallel
combination of
R
3

and
R
4

and returns to the supply.


The effective re
si
stance of the parallel
combination can therefore be found from


Calculate this value.


It should agree with what you calculate from Equa
tion (3), using the known
values of
R
3

= 400
Ω

and
R
4

= 200
Ω
, within experimental error.


Does it?



C.


Kirchhoff's rules


(8) Turn off the power supply. On the patchboard, wire the circuit shown in Figure 5. The DC


Figure 4

C
ircuit for part B

4
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DC CIRCUITS: OHM’S AND KIRCHHOFF’S LAWS








power supply is
, while the dry cell
mounted on the pa
tchboard is
.


Connect the DVM between A and B, and
adjust the pow
er supply for a reading of
ex
actly 5.00 V.


(9)

Without changing the supply voltage,
measure directly the ter
minal voltage
of
the dry cell in t
he circuit. Next, mea
sure
and record the potential drop across each
of the five re
sistors, with sign. Calculate
the cur
rent in each of the five resis
tors.
Figure out and record the direction of the
cur
rent in each branch of the circuit (re
-
member t
hat current flows through a
resistance from the hig
her to the lower
poten
tial). At each of the circuit's three
junc
tions

(C, E, and D/F), add the currents flowing into the junction. (Add them
with sign
; that is, a current
flow
ing away from the juncti
on is taken as negative.)


Do the cur
rents add to zero (within
experimental error), as they should according to Kirchhoff's first rule?


(10) Look at the loop B
-
A
-
C
-
(D/F)
-
B in the circuit of Figure 5. According to Kirchhoff's second rule,


that is,




or


Remember that the
V
's have signs!


That is, if the poten
tial at point A were lower than that at point C,
then
V
AC

would be negative.


From your measure
ments, does the sum (with appropriate s
igns) of
V
AC

and
V
CD

agree (within experimental error) with
V
AB
?


(11) Repeat step (10) above for the loops G
-
E
-
(D/F)
-
H
-
G

and C
-
E
-
(D/F)
-
C
.



Figure 5

Circuit for part C

DC CIRCUITS: OHM’S AND KIRCHHOFF’S LAWS

4
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15








Superposition


(this part is optional: consult your instructor)

In DC network theory, the
superposition theorem

s
tates that the current in any branch of a network
containing multiple sources of emf is the simple
sum

of the currents that would be produced in that
branch by each
of the emf's acting alone.


By

(say) "acting alone", we mean with a
ll other emf's set to
zero (that is, short
-
cir
cuited).


(12) In step (9), you found the current in each branch of the circuit of F
ig. 5 due to

and

together.

On the patchboard, discon
nect the dry cell from G

and H, and use a lead to connect G and H
together.


(Do not connect G' and H' togeth
er!)


You have now "set

= 0" and can me
a
sure the
currents produced by

alone.


Use the DVM to measure the voltage drop acro
ss each of the five
resistors, and divide by the known resistance values to determine the five branch currents (with sign)
in this case.


(13) Now disconnect the lead between points G and H, and recon
nect point G to G' and H to H'.


Turn

off the DC power supply and disconnect it from points A and B, and use a lead to connect A and B
together.


You have now "set
= 0" in the circuit of Figure 5, and can measure the currents due to

acting alone.


Use the DVM to measure the voltage drop across each of the five resistors, and
divide by the known resistance values to determine the five branch currents (with sign) in this case.


(14) In each of the five branches, the sum (with signs) of the current du
e to

alone and that due to

alone should equal, within experimental error, the total current in that branch which you measured in
part (9).


Does it?


If not, can you suggest where the discrepancy might come fro
m?















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