W03 Analysis of DC Circuits

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Oct 7, 2013 (3 years and 8 months ago)

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Yrd. Doç. Dr. Aytaç Gören
W03 Analysis of
D
C Circuits
ELK 2018
-
Contents
W01 Basic Concepts in Electronics
W02 AC to DC Conversion
W03 Analysis of
D
C Circuits (self and condenser)
W04 Transistor
s
and Applications
(H
-
Bridge)
W05 Op
Amps and Applications
W06 Sensors and Measurement (1/2)
W07 Sensors and Measurement (2/2)
W08 Midterm
W09 Basic Concepts in Digital Electronics
(Boolean Algebra, Decimal to binary,
gates)
W10 Digital Logic Circuits
(Gates and Flip Flops)
W11 PLC’s
W12 Microprocessors
W13 Data Acquisition, D/A and A/D Converters.
2
Yrd. Doç. Dr. Aytaç Gören
ELK 2018

W01 Contents
1.
Kirchoffs
Circuit
Law
2.
Basic
definitions
for
circuit
analysis
3.
Circuit
Analysis
1.
Mesh
Current
Analysis
2.
Nodal
Voltage
Analysis
4.
Thevenins
Theorem
5.
Nortons
Theorem
6.
Transient
Analysis
7.
Transient
Analysis

Capacitor
8.
Transient
Analysis
-
Inductor
3

Yrd. Doç. Dr. Aytaç Gören
Extra Reference for this week:
http://www.electronics
-
tutorials.ws
Yrd. Doç. Dr. Aytaç Gören
Reminder

Reminder

Yrd. Doç. Dr. Aytaç Gören
Parameter
Symbol
Measuring
Unit
Description
Voltage
Volt
V
or
E
Unit of Electrical Potential
V
=
I
×
R
Current
Ampere
I
or
i
Unit of Electrical Current
I
=
V
÷
R
Resistance
Ohm
R or
Ω
Unit of DC Resistance
R
=
V
÷
I
Conductance
Siemen
G or

Reciprocal of Resistance
G
=
1
÷
R
Capacitance
Farad
C
Unit of Capacitance
C
=
Q
÷
V
Charge
Coulomb
Q
Unit of Electrical Charge
Q
=
C
×
V
Inductance
Henry
L or H
Unit of Inductance
V
L
=
-
L(
di
/
dt
)
Power
Watts
W
Unit of Power
P
=
V
×
I
or
I
2
×
R
Impedance
Ohm
Z
Unit of AC Resistance
Z
2
=
R
2
+
X
2
Frequency
Hertz
Hz
Unit of Frequency
ƒ
=
1
÷
T
Reminder

Yrd. Doç. Dr. Aytaç Gören
Prefix
Symbol
Multiplier
Power
of Ten
Terra
T
1,000,000,000,000
10
12
Giga
G
1,000,000,000
10
9
Mega
M
1,000,000
10
6
kilo
k
1,000
10
3
none
none
1
10
0
centi
c
1/100
10
-
2
milli
m
1/1,000
10
-
3
micro
µ
1/1,000,000
10
-
6
nano
n
1/1,000,000,000
10
-
9
pico
p
1/1,000,000,000,00
0
10
-
12
Yrd. Doç. Dr. Aytaç Gören
Kirchoffs
Circuit
Law
In complex circuits such as bridge or T networks, we can not simply use
Ohm's Law alone to find the voltages or currents circulating within the
circuit.
Kirchoff
developed a pair or set of rules which deal with the conservation
of current and energy within electrical circuits. The rules are commonly
known as:
Kirchoffs
Circuit Laws
with one of these laws dealing with
current flow around a closed circuit,
Kirchoffs
Current Law, (KCL)
and
the other which deals with the voltage around a closed circuit,
Kirchoffs
Voltage Law, (KVL)
.
Kirchoffs
Circuit
Law
Kirchoffs
Current
Law
or
KCL,
states
that
the
"
total
current
or
charge
entering
a
junction
or
node
is
exactly
equal
to
the
charge
leaving
the
node
as it has no
other
place
to
go
except
to
leave
, as no
charge
is
lost
within
the
node
". (
Conservation
of
Charge
)
The
term
Node
in an
electrical
circuit
generally
refers
to
a
connection
or
junction
of
two
or
more
current
carrying
paths
or
elements
such
as
cables
and
components
Kirchoffs
Circuit
Law
Kirchoffs
Voltage
Law
or
KVL,
states
that
"
in
any
closed
loop
network,
the
total
voltage
around
the
loop
is
equal
to
the
sum
of
all
the
voltage
drops
within
the
same
loop
"
which
is
also
equal
to
zero
.
In
other
words
the
algebraic
sum
of
all
voltages
within
the
loop
must
be
equal
to
zero
.
Basic
definitions
for
circuit
analysis
Th
e
terms
,
used frequently in circuit analysis
:

Path
-
a
line
of
connecting
elements
or
sources
with
no
elements
or
sources
included
more
than
once
.

Node
-
a
node
is a
junction
,
connection
or
terminal
within
a
circuit
were
two
or
more
circuit
elements
are
connected
or
joined
together
giving
a
connection
point
between
two
or
more
branches
.
A
node
is
indicated
by
a
dot
.

Branch
-
a
branch
is a
single
or
group
of
components
such
as
resistors
or
a
source
which
are
connected
between
two
nodes
.

Loop
-
a
loop
is a
simple
closed
path
in a
circuit
in
which
no
circuit
element
or
node
is
encountered
more
than
once
.

Mesh
-
a mesh is a
single
open
loop
that
does
not
have
a
closed
path
.
Example
Using
Kirchoffs Voltage Law
,
KVL
the equations are given as;

Loop 1 is given as
:
10
=
R
1
x
I
1
+
R
3
x
I
3
=
10I
1
+
40I
3

Loop 2 is given as
:
20
=
R
2
x
I
2
+
R3
x
I3
=
20I
2
+
40I
3

Loop 3 is given as
:
10
-
20
=
10I
1
-
20I
2
As
I
3
is the sum of
I1
+
I2
we can rewrite the equations as;

10
=
10I
1
+
40(I
1
+
I
2
)
=
50I
1
+
40I
2

20
=
20I
2
+
40(I
1
+
I
2
)
=
40I
1
+
60I
2
Using
Kirchoffs
Current
Law
(
KCL)
the
equations
are
given
as:

At
node
A
:
I
1
+
I
2
=
I
3

At
node
B
:
I
3
=
I
1
+
I
2
Example
Using
Kirchoff's
Circuit
Laws
is as
follows
:
1.
Assume
all
voltage
sources
and
resistances
are
given
.
2.
Label
each
branch
with
a
branch
current
.
3.
Find
Kirchoff's
first
law
equations
for
each
node
.
4.
Find
Kirchoff's
second
law
equations
for
each
of
the
independent
loops
of
the
circuit
.
5.
Use
Linear
simultaneous
equations
as
required
to
find
the
unknown
currents
.

I
1
=
-
0.143
Amps
(
Wrong
Direction
)

I
2
=
+0.429
Amps

I
3
=
I
1
+
I
2

At
node
B
:
I
3
=
I
1
+
I
2
-
0.143
+
0.429
=
0.286
Amps
Circuit
Analysis
While
Kirchoff
´
s
Laws
give
us
the
basic
method
for
analysing
any
complex
electrical
circuit
,
there
are
different
ways
of
improving
upon
this
method
by
using

Mesh
Current
Analysis

Nodal
Voltage
Analysis
that
results
in a
lessening
of
the
math's
involved
and
when
large
networks
are
involved
this
reduction
in
maths
can be a
big
advantage
.
Circuit
Analysis
One
simple
method
of
reducing
the
amount
of
math's
involved
is
to
analyse
the
circuit
using
Kirchoff's
Current
Law
equations
to
determine
the
currents,
I
1
and
I
2
flowing
in
the
two
resistors
.
Then
there
is
no
need
to
calculate
the
current
I
3
as
its
just
the
sum
of
I
1
and
I
2
.
So
Kirchoff's
second
voltage
law
simply
becomes
:
Equation
No
1
:
10
=
50
I
1
+
40
I
2
Equation
No
2
:
20
=
40
I
1
+
60
I
2
therefore,
one
line
of
math's
calculation
have
been
saved
.
Mesh
Current
Analysis
A more easier method of solving the above circuit is by using
Mesh
Current Analysis
or
Loop Analysis
which is also sometimes
called
Maxwell
´
s Circulating Currents
method.
Instead of labelling the branch currents we need to label each
"closed loop" with a circulating current.
As a general rule of thumb, only label inside loops in a clockwise
direction with circulating currents as the aim is to cover all the
elements of the circuit at least once
Mesh
Current
Analysis
Kirchoff's
voltage
law
equation
can be
applied
in
the
same
way
as
before
to
solve
them
but
the
advantage
of
this
method
is
that
it
ensures
that
the
information
obtained
from
the
circuit
equations
is
the
minimum
required
to
solve
the
circuit
as
the
information
is
more
general
and
can
easily
be put
into
a
matrix
form.
Mesh
Current
Analysis
[ V ]
gives
the
total
battery
voltage
for
loop
1
and
then
loop
2.
[ I ]
states
the
names
of
the
loop
currents
[ R ]
is
called
the
resistance
matrix
.
Mesh
Current
Analysis
T
he
basic
procedure
for
solving
Mesh
Current
Analysis
equations
is
as
follows
:
1
.
Label
all
the
internal
loops
with
circulating
currents
.
(
I
1
,
I
2
,
...
I
L
etc
)
2
.
Write
the
[
L
x
1
]
column
matrix
[
V
]
giving
the
sum
of
all
voltage
sources
in
each
loop
.
3
.
Write
the
[
L
x
L
]
matrix
,
[
R
]
for
all
the
resistances
in
the
circuit
as
follows
;
o
R
11
=
the
total
resistance
in
the
first
loop
.
o
R
nn
=
the
total
resistance
in
the
n
th
loop
.
Nodal
Voltage
Analysis
Nodal Voltage Analysis
uses the "Nodal" equations of Kirchoff's first
law to find the voltage potentials around the circuit. By adding
together all these nodal voltages the net result will be equal to
zero.
For each node we apply Kirchoff's first law equation, that is: "
the
currents entering a node are exactly equal in value to the currents
leaving the node
" then express each current in terms of the
voltage across the branch.
Nodal
Voltage
Analysis
As
V
a
=
10v
and
V
c
=
20v
Thevenins
Theorem
Thevenins
Theorem
states
that
"
Any
linear
circuit
containing
several
voltages
and
resistances
can be
replaced
by
just
a
Single
Voltage
in
series
with
a
Single
Resistor
".
it is
possible
to
simplify
any
"
Linear
"
circuit
,
to
an
equivalent
circuit
with
just
a
single
voltage
source
in
series
with
a
resistance
connected
to
a
load
as
shown
below
.
Thevenins
Theorem
is
especially
useful
in
analyzing
power
or
battery
systems
and
other
interconnected
circuits
where
it
will
have
an
effect
on
the
adjoining
part
of
the
circuit
.
the
value
of
the
voltage
required
Vs
is
the
total
voltage
across
terminals
A
and
B
with
an
open
circuit
and
no
load
resistor
Rs
connected
.
The
value
of
resistor
Rs
is
found
by
calculating
the
total
resistance
at
the
terminals
A
and
B
with
all
the
emf
´
s
removed
Thevenins
Theorem
Example
Thevenins
Theorem
Example
The Equivalent Resistance (Rs)
The Equivalent Voltage (Vs)
Thevenins
Theorem
Example
Thevenins
Equivalent
circuit
is
shown
below
with
the
40Ω
resistor
connected
.
The
basic
procedure
for
solving
a
circuit
using
Thevenins
Theorem
is
as
follows
:

1
.
Remove
the
load
resistor
R
L
or
component
concerned
.

2
.
Find
R
S
by
shorting
all
voltage
sources
or
by
open
circuiting
all
the
current
sources
.

3
.
Find
V
S
by
the
usual
circuit
analysis
methods
.

4
.
Find
the
current
flowing
through
the
load
resistor
R
L
.
Nortons
Theorem
Nortons
Theorem
states
that
"
Any
linear
circuit
containing
several
energy
sources
and
resistances
can be
replaced
by
a
single
Constant
Current
generator
in
parallel
with
a
Single
Resistor
".
The
value
of
this
"
constant
current
" is
one
which
would
flow
if
the
two
output
terminals
where
shorted
together
while
the
source
resistance
would
be
measured
looking
back
into
the
terminals
,
(
the
same
as
Thevenin
).
Nortons
Theorem
To find the Nortons equivalent of the above circuit we firstly have to
remove the centre
40Ω
load resistor and short out the
terminals
A
and
B
to give us the following circuit.
Nortons
Theorem
When the terminals
A
and
B
are shorted together the two resistors
are connected in parallel across their two respective voltage
sources and the currents flowing through each resistor as well as
the total short circuit current can now be calculated as:
Nortons
Theorem
The
value
of
the
internal
resistor
Rs
is
found
by
calculating
the
total
resistance
at
the
terminals
A
and
B
giving
us
the
following
circuit
.
Nortons
Theorem
Nortons
equivalent
circuit
.
The
basic
procedure
for
solving
a
circuit
using
Nortons
Theorem
is
as
follows
:
1
.
Remove
the
load
resistor
R
L
or
component
concerned
.
2
.
Find
R
S
by
shorting
all
voltage
sources
or
by
open
circuiting
all
the
current
sources
.
3
.
Find
I
S
by
placing
a
shorting
link
on
the
output
terminals
A
and
B
.
4
.
Find
the
current
flowing
through
the
load
resistor
R
L
.
Transient
Analysis
The time constant
Electrical or Electronic circuits or systems suffer from some form of
"time
-
delay" between its input and output, when a signal or
voltage, either continuous, (DC) or alternating (AC) is firstly
applied to it.
This delay is generally known as the
Time Constant
of the circuit
and it is the time response of the circuit when a step voltage or
signal is firstly applied.
The resultant time constant of any circuit or system will mainly
depend upon the reactive components either capacitive or
inductive.
Transient
Analysis
-
Capacitor
When an increasing DC voltage is applied to a
discharged
Capacitor
the capacitor draws a charging current and
"charges up", and when the voltage is reduced, the capacitor
discharges in the opposite direction.
Because capacitors are able to store electrical energy they act like
small batteries and can store or release the energy as required.
The charge on the plates of the capacitor is given as:
Q = CV.
This charging (storage) and discharging (release) of a capacitors
energy is never instant but takes a certain amount of time to occur
with the time taken for the capacitor to charge or discharge to
within a certain percentage of its maximum supply value being
known as its
Time Constant
(τ).
Transient
Analysis
-
Capacitor
RC Charging Circuit
The figure below shows a capacitor, (C) in series with a resistor, (R)
forming a
RC Charging Circuit
connected across a DC battery
supply (Vs) via a mechanical switch. When the switch is closed, the
capacitor will gradually charge up through the resistor until the
voltage across it reaches the supply voltage of the battery.
Transient
Analysis
-
Capacitor
RC
Charging
Circuit
If
C
is
fully
"
discharged
"
and
the
switch
(S) is
fully
open
,
these
are
the
initial
conditions
of
the
circuit
,
then
t = 0,
i = 0
and
q = 0.
When
the
switch
is
closed
the
time
begins
at
t = 0
and
current
begins
to
flow
into
the
capacitor
via
the
resistor
. Since
the
initial
voltage
across
the
capacitor
is
zero
, (
V
c
= 0)
the
capacitor
appears
to
be
a
short
circuit
and
the
maximum
current
flows
through
the
circuit
restricted
only
by
the
resistor
R.
Transient
Analysis
-
Capacitor
RC
Charging
Circuit
The
current
no
flowing
around
the
circuit
is
called
the
Charging
Current
and
is
found
by
using
Ohms
law
as:i = V
R
/R.
Transient
Analysis
-
Capacitor
RC Charging Circuit
The capacitor now starts to charge up as
shown, with the rise in the RC charging
curve steeper at the beginning because
the charging rate is fastest at the start
and then slow down as the capacitor
takes on additional charge at a slower
rate.
As the capacitor charges up, the voltage
difference between
Vs
and
Vc
reduces, so
to does the circuit current,
i.
Then at the final condition,
t =
∞,
i = 0,
q = Q
= CV. Then at infinity the current
diminishes to zero, the capacitor acts like
an
open circuit condition
therefore, the
voltage drop is entirely across the
capacitor
.
Transient
Analysis
-
Capacitor
RC
Charging
Circuit
As
the
capacitor
charges
the
potential
difference
across
its
plates
increases
with
the
actual
time
taken
for
the
charge
on
the
capacitor
to
reach
63%
of
its
maximum
possible
voltage
, in
our
curve
0.63Vs
being
known
as
the
Time
Constant
, (T)
of
the
circuit
.
V is related to charge on a capacitor given
by the equation,
Vc
= Q/C, the voltage
across the value of the voltage across
the capacitor, (
Vc
) at any instant in time
during the charging period is given as:
Transient
Analysis
-
Capacitor
RC
Charging
Circuit
Vc
is the voltage across the capacitor
Vs is the supply voltage
RC is the time constant of the RC charging circuit
After
a
period
equivalent
to
4 time
constants
, (4T)
the
capacitor
in
this
RC
charging
circuit
is
virtually
fully
charged
and
the
voltage
across
the
capacitor
is
approx
0.99Vs.
The
time
period
taken
for
the
capacitor
to
reach
this
4T
point
is
known
as
the
Transient
Period
.
As
the
capacitor
is
fully
charged
no
more
current
flows
in
the
circuit
.
The
time
period
after
this
5T
point
is
known
as
the
Steady
State
Period
.
Transient
Analysis
-
Inductor
An
inductor
could
not
change
instantaneously
, but
would
increase
at
a
constant
rate
determined
by
the
self
-
induced
emf
in
the
inductor
.
In
other
words
, an
inductor
in a
circuit
opposes
the
flow
of
current
, (
i
)
through
it.
An
LR
Series
Circuit
consists
basically
of an
inductor
of
inductance
L
connected
in
series
with
a
resistor
of
resistance
R.
The
resistance
R
is
the
DC
resistive
value
of
the
wire
turns
or
loops
that
goes
into
making
up
the
inductors
coil
.
Transient
Analysis
-
Inductor
The above
LR series circuit
is connected across a constant voltage
source, (the battery) and a switch. Assume that the switch,
S
is
open until it is closed at a time
t
=
0, and then remains
permanently closed producing a "step response" type voltage
input.
The current,
i
begins to flow through the circuit but does not rise
rapidly to its maximum value of
I
max
as determined by the ratio
of
V
/
R
(Ohms Law). This limiting factor is due to the presence of
the self induced emf within the inductor as a result of the growth
of magnetic flux, (Lenz's Law).
Transient
Analysis
-
Inductor
After
a time
the
voltage
source
neutralizes
the
effect
of
the
self
induced
emf
,
the
current
flow
becomes
constant
and
the
induced
current
and
field
are
reduced
to
zero
.
Kirchoffs
Voltage
Law
, (KVL)
to
define
the
individual
voltage
drops
:
Transient
Analysis
-
Inductor
Expression
for
the
Current
in an LR
Series
Circuit
Since
the
voltage
drop
across
the
resistor
,
V
R
is
equal
to
IxR
(
Ohms
Law
), it
will
have
the
same
exponential
growth
and
shape
as
the
current
.
However
,
the
voltage
drop
across
the
inductor
,
V
L
will
have
a
value
equal
to
:
Ve
(
-
Rt
/L)
.
42

Yrd. Doç. Dr. Aytaç Gören
Ref
for
this
week
:
http://www.electronics
-
tutorials.ws