Review of Electric Circuits

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IV

Review of Electric Circuits

Rev 20 June 2007



Current


Electrons are negatively charged particles. The movement of electrons constitutes the flow of
electric current
. The movement of a negative charge in one direction can be thought of as the
movement o
f a positive charged in the opposite direction. The direction of the positive charge is
the direction of
current flow
. You can think of an electric charge as water and of an electric
current as water flowing in a river or a pipe. The water flows in a ce
rtain direction


from the
higher ground to the lower ground. The measure of current is
ampere
(A)
.

An instrument called
ammeter

is used to measure the current that must pass through it.



Battery and Voltage


You can think of an electric battery as a ge
nerator of a positive charge. Once an electric load is
connected to the battery, the current flows from its positive terminal (+) to its negative terminal
(

) through the connecting wires and the load. The batteries come with different voltage ratings
ex
pressed in
Volts

(V)
. For instance, a car battery is a 12
-
volt battery. The higher the voltage,
the more current will flow from the battery when you connect an electric load to it. You can
think of a battery as a pump, which pumps the water from the low
er ground (usually referenced
as zero) to a higher ground. You can think of a battery’s voltage as the height of the elevated
water level. The battery is often called a
voltage source
. Voltages are measured by an
instrument called
voltmeter
. To measure

the voltage, the voltage must be connected between two
points where there is a certain voltage difference (like between the + and the


terminals of the
battery). The reference point used to measure voltages is called
ground
. For example, one of the
ter
minals of the car battery is connected to the car’s body and is considered an electric ground.
Drawings of batteries and associated symbols are shown in
Figure
IV
-
1
.



Figure
IV
-
1
:

Batteries and ground.

Resistors


The basic element of electrical and electronic circuits is the
resistor. The resistor obeys
Ohm’s Law




V = IR




[
IV
-
1

]


where
V

is the voltage drop across the resistor,
I

is the current
through the resistor, and
R

is the resistance of the resistor (see
Figure
IV
-
2
).

Resistors dissipate energy as heat (this is known as
Joule Heating
) according to the
formu
la



P = IV = I
2
R = V
2
/R
.


[
IV
-
2

]



Resistors are rated by how much heat they can dissipate before they melt, e.g. ¼W.



Colour Code


Resistors come in a wide variety of standard resistances. The value of the r
esistance is give by
the colour bands on the resistance. One band isolated from the other three gives the tolerance or
accuracy of the resistance value. The other three bands encode the value of the resistance. The
colour code is given in the
diagram
Figure
IV
-
3

below.

Colour

Value

Multiplier

Black

0

10
0

Brown

1

10
1

Red

2

10
2

Orange

3

10
3

Yellow

4

10
4

Green

5

10
5

Blue

6

10
6

Violet

7

10
7

Gray

8

10
8

White

9

10
9



Colour

Tolerance

Gold


5%

Silver

10%

N
one

20%




Value (×10)

Value (×1)

Multiplier

Tolerance

Schematic Symbol

Figure
IV
-
3
:
Symbol and colour code for

resistors.

I

R

V

Figure
IV
-
2
: Ohm's Law.

As an example, a resistor with red, green, and blue bands in sequenced separated from a gold
band would have a resistance of R = (2 × 10 + 5 × 1) × 10
6



± 5% = 25 × 10
6



± 5%. What is
the resistance of a resistor with yellow, violet, brown an
d silver bands?



Series/Parallel Connections


Resistors can be connected by conducting wires (or paths) in series or in parallel as shown in
Figure
IV
-
4

(a) and (b). In
Figure
IV
-
4
(b) we use the symbols

designated for battery and
resistors.



Figure
IV
-
4
:
(a) Series and (b) parallel connections.


The equivalent resistors will draw the same current
I

from the battery. Their value can be
calculated as:



R
s
eries

= R
1

+ R
2

[
IV
-
3

]




[
IV
-
4

]


Note that series resistors always carry the same current.



Short
ing


One consequence of Eq
n

[

IV
-
4

]

, is that the equivalent resistance of two resistors in parallel is always less than

either of the two resistors. Since parallel resistors always have the same voltage drop, V
1

= V
2
. Ohm’s Law, applied to this result,

means I
1
R
1

= I
2
R
2
. So the smaller resistor carries the
bigger current. These results reach their extreme conclusion when a resistor is
shorted
. This
means a wire of negligible resistance is connected across a resistor as shown in … below.
Having no resist
ance all the current flows through the wire and no current flows through R
2
. The
equivalent resistance of R
2

and the wire is zero, so they can be replaced by the wire itself as
shown in
Figure
IV
-
5
. Shorting resistors unintentiona
lly can cause big problems. If R
1

is a small
resistor, the current produced by the battery would be damaging high and the Joule heating
could
be large.






BreadBoard


A breadboard is a convenient way to assemble
circuits where the parts are small s
uch as
microchips, diodes, and resistors. Pinholes in
the front of the breadboard let you securely
connect circuit elements to it. Sets of pinholes
are connected by conducting bars in two
different patterns, short horizontal and long
vertical, under the pl
astic cover. See
Figure
IV
-
6
. This setup allows parallel and series
arrangements of elements to be put together
quickly. When using wire connectors make
sure to keep them short and neat!
R
1

R
2



a

b

R
1



a

b

Figure
IV
-
5
: A short circuit.

Figure
IV
-
6
:
Breadboard connectors.

Pins

Connector

bars

Kirchhoff’s Rules


More
complicated circui
ts are usually analyzed using
Kirchhoff
’s

Rules
. Consider the circuit in
Figure
IV
-
7

below. Points
b

and
e
, where three or more wires meet, are known as
nodes

or
junctions
. A
branch

is a path from one node to another. In
Figure
IV
-
7

there are three branches,
be
,
bcde
, and
bafe
. These branches are said to be parallel since the start and end nodes are the
same. Each branch carries a single current. Elements in the same branch are said to be in series,
so the

5


and 15


resistors in branch
bcde

are in series. At a node,
Kirch
h
off’s Current Rule

is
obeyed
,



,

[
IV
-
5

]


the sum of the currents entering a node equal the sum of the currents leavi
ng a node.
At point
b

we have
I
1

+ I
2

+ I
3

= 0

and
0 = I
1

+ I
2

+ I
3

at point
e
.
Kirchhoff’s Current Rule
is simply
conservation of charge


electrons are neither being created not destroyed in the circuit.


A
loop

is a closed path from a point back to itse
lf. Where you start is arbitrary
. Several loops in
the circuit in
Figure
IV
-
7

are
fabef

and
dcbed
. Around a loop
Kirch
h
off’s
Loop

Rule
,




,

[
IV
-
6

]


is obeyed
wh
ere
V
i

is the voltage drop across each circuit element in the loop. The orientation of
the voltmeter must be kept the same as you go around the elements of the loop.

For loop
fabef

we
get
(12 V)


(10

)I
1

+ (20

)I
3

= 0

and for loop
dcbed

we find
(10 V)


(15

)I
2

+ (20

)I
3



(5

)I
2
= 0
.

Note that battery voltages are positive when we travel from the negative to the
positive side and vice versa. The voltage drop across a resistor is negative if we travel around the
resistor in the direction or the curren
t through it and positive if the current is in the opposite
direction.


Kirchhoff’s
Loop

Rule
is simply conservation of energy since voltage is energy per charge.

Figure
IV
-
7
: A complicated circuit.

5


㈰2


ㄲ⁖

㄰⁖

ㄵ1


a

b

c

d

e

f

I
1

I
2

I
3

Electronics Style Circuit Diagrams


In electronics, batteries and other power supplies are s
eldom shown. A small circle indicates
where one end of the power supply goes. The other end is assumed to be connect the ground
symbol
.

There may be more than one ground symbol in a diagram. This indicates that a
wire connecting the two points has not be

drawn.
The diagram below
Figure
IV
-
8

illustrates how
a stand
ard

circui
t would be drawn in this style.



Voltage Divider


Shown in
Figure
IV
-
9
(a) is a simple circuit called
voltage divider
.



Figure
IV
-
9
:
Voltage divider.

The current flowing through both series resistors is given by
I

=
E
/(
R
1

+
R
2
). This current,
following Ohm’s law, will cause a voltage drop across the resistor
R
2
. This voltage drop is g
iven
as:



[
IV
-
7

]

R
1

R
3

12 V

R
2

R
1

R
3

R
2

+12

Figure
IV
-
8
. Standard and electronics style circuit diagrams.

By selecting a proper ratio of
R
1
/
R
2
, we can obtain any voltage V
0

between zero and
E
.


Figure
IV
-
9

(b) shows
the electronics style
of drawing t
he circuit shown
in
Figure
IV
-
9
(a). Here

we use the symbol “ground” to indicate common connection (or common node). Any voltage
indicated without additional reference information is taken with respect to the grou
nd.



Potentiometer


A voltage divider can be conveniently built
using a variable resistor called a
potentiometer

(or
pot

for short). A
potentiometer has the sliding electrode (a
wiper), the position of which can be manually
adjusted at a desired locatio
n resulting in a
variable resistance. A voltage divider using a
potentiometer is shown symbolically in
Figure
IV
-
10
.




The total resistance of the potentiometer is
R

=
R
1

+
R
2
. A potentiometer usually has the form of
a round dr
um with a central rotating shaft that controls the position of the wiper as shown in
Figure
IV
-
11
.



Figure
IV
-
11
:
Potentiometer



Diodes


A diode is electrical “valve” that conduc
ts current only in one direction


from Anode to Cathode
as shown in
Figure
IV
-
12
. A special diode called Light Emitting Diode (LED) emits light
whenever there is current flowing through it. An LED is often used as an indicator
of the voltage
presence (LED is “on”) or absence (LED is “off”).


Figure
IV
-
10
:
Potentiometer


Figure
IV
-
12
:
Diodes


A simple circuit with LED is shown in
Figure
IV
-
13
.



Figure
IV
-
13
:
LED in action.


A small

voltage
V
forward
, of between 0.5 and 2.0 V, must first be applied to a diode before a
current can pass through it. In the forward, current
-
carrying direction, a diode has a resistance of
about
1
0

. In the r
everse, current
-
blocking direction, the resistance is in the millions of ohms.



Capacitors


A capacitor
, in its simplest form, is just a pair of parallel metal plates. When equal and charges
+q and

q are placed on the plates there is a voltage differenc
e that depends on the capacitance of
the particular configuration



V = q/C
.

[
IV
-
8

]


The symbol for a capacitor is shown
below
in
Figure
IV
-
14
.


As you put
charge on a capacitor, it ge
ts harder and
harder and takes longer and longer to add more
charge. The diagram below shows a capacitor
charging circuit and a plot of the voltage across the
capacitor plates as a function of time.


T
he equation for a capacitor being charged from zero is

given by the formula


.

[
IV
-
9

]

The quantity RC is called the time constant


of the circuit. When


is small the capacitor charges
quickly and when it is large the capacitor charges slowl
y. After 5

, the capacitor is over 99%
charged.


A convenient way of thinking of a capacitor is that it acts as a closed switch the instant it starts
charging, current is maximum, and an open switch, current is zero, when it is fully charged.



When disc
onnect
ed

from a battery, a capacitor can hold its charge for minutes or even hours
before it leaks away. Providing a path between the plates discharges the capacitor much more
quickly. The discharging circuit and the V
C
-
t graph are shown in
Figure
IV
-
16

below.

The equation for a capacitor being charged from zero is given by the formula


I

C

V

+q


q

Figure
IV
-
14
: Capacitor symbol.

R

C

V

V
C

V
C

t

V
C





Figure
IV
-
15
: Charging a capacitor.

V
C

t

V
C

V
1

V
2

t
1

t
2

R

C

V
C

Figure
IV
-
16
: Discharging a capacitor.


.

[
IV
-
10

]


When the time constant


is small the capacitor discharges quickly

and when it is large the
capacitor discharges slowly. After 5

, the capacitor is over 99% discharged. The discharge of a
capacitor can be used for timing if you can measure voltage much faster than

. Consider the two
voltages V
2

and V
1

shown on the graph

in
Figure
IV
-
16
, using Eqn

.

[
IV
-
10

]

the time difference between the voltages is


.


[
IV
-
11

]


Capacitors also come in h
andy when you want to measure a varying voltage
.

An ordinary DC
voltmeter reading the signal would fluctuate.

A “sample and hold” circuit can be built

using

a
very fast switch
and
a
series
resistor and
capacitor

whose

time constant


= RC much smaller that

the time

variation of the voltage sign
al
. The switch is closed long enough for the capacitor to
charge
nearly completely then opened. Since a voltmeter has a huge resistance, connecting it
across the charged capacitor creates a circuit with a large time c
onstant thus allowing for a more
leisurely measurement.



Transducers


Transducers

are electronic devices whose physical properties, usually voltage output or
resistance, changes with variations in its surroundings such as temperature, pressure, ambient
li
ght, and the like. For example you may have learned that the resistance of a resistor depends on
its temperature. However this effect is weak when the temperature changes are only a few
degrees and it would be impractical to use a resistor as a thermometer
. A thermocouple, on the
other hand, is very sensitive to temperature and is often used as a high temperature thermometer.
A thermocouple utilizes the Seebeck Effect in which two dissimilar metals in contact with one
another develop a potential difference.

This potential difference is strongly dependent on
temperature. In general the relationship between the external effect one wishes to measure and
the physics property of the transducer is nonlinear or at best approximately linear over some
range
.
As well
if the voltage output is small, the result may have to be amplified.