Series Circuits

1
Series Circuits
The circuit below is the simplest possible
electrical circuit. Current flows out of the
batterys positive terminal, into the upper
terminal of the resistor and out the lower one,
and returns to the battery through the negative
terminal.
How many nodes does this circuit have?
I


E
R
IR
-
-
2
Series Circuits
There are two nodes in this circuit, designated a
and b. KCL says the current into a must equal
the current out of a, and the current into b must
equal the current out of b. Since there is only
one path into and one path out of each node, the
same current (designated I) must flow at any
point in the loop. Ohms law tells us the
voltage across the resistor is IR, with the
indicated polarity.
a
KVL tells us the sum of the voltage rises equals
the sum of the voltage drops, or
I
I


E
R
IR
-
-
I
I
b
3
Series Circuits
We can add another resistor, R2, as shown below.
Now the two resistors are connected in series.
Two circuit elements are in series if the
following two conditions are met

• They have only one terminal in common
• The terminal which is common to the two elements
  is not common to a third.

I

R1
IR1

-
E
-

R2
IR2
-
4
Series Circuits
If these two conditions are met, the current
leaving the first element in the series (R1 in
the circuit below) has nowhere to go except to
enter the next element in the series.
I

R1
IR1
Exactly the same current must flow through both
elements.

-
E
-
I

R2
IR2
-
5
Series Circuits
If we violate the first condition, the two
elements would have two terminals in common. The
current leaving the battery is I, but when it
reaches the common terminals some of it goes to
the right while the rest goes to the left.
The same current may not flow through both
elements.
I

E
-
R1
R2
6
Series Circuits
If we violate the second condition, by adding a
third element which also shares the common
terminal, the current leaving R1 has two paths it
can take.
Again, the same current may not flow through both
R1 and R2. R1 and R2 are not in series. R1 and
R2 are not in series.
I
R1

E
I
-
R3
R2
7
Series Circuits
Terminology A Branch of a circuit is any part
of the circuit that has one or more elements in
series. In our circuit, R1, R2 and the battery
are branches.
By this definition, the series combination of R1
and R2 also is a branch. Note that the definition
of a series connection implies that the same
current flows through any two branches in a
series connection. This is there is no path for
the current flowing through the first branch to
take, except through the second.
I

R1
IR1

-
E
-

R2
IR2
-
8
Series Circuits
In this circuit, Ohms law says the voltage drops
across the two resistors are IR1 and IR2. KVL
says
I

R1
IR1

-
E
-

R2
IR2
-
9
Series Circuits
Consider the circuit below. Let E 12 V, R
470 W, and let VLED 2 V. Which way should the
current arrow be drawn?
R

E

VLED
-
10
Series Circuits
The current arrow comes out of the batterys
positive terminal. This means the polarity of VR
should be as shown, so the voltage across the
resistor is a voltage drop. Current flows in the
proper direction for a forward biased diode, so
VLED 2 V (as long as E is at least 2V, which it
is). Applying KCL around the loop,
Ohms law tells us
R
-

I
VR
And we already know VLED is 2V, so

E

VLED
-
-
11
Series Circuits
Returning to our previous example of two
resistors in series, we had found that KVL says
I
We could define a resistance RT such that

R1
IR1

-
E
-
then

R2
IR2
-
12
Series Circuits
So the two resistors R1 and R2 in series are
equivalent to a single resistor, RT R1 R2.
I

I
R1
IR1



-
E
E
RT
IRT
-
-
-

R2
IR2
-
13
Series Circuits
If an Ohmmeter is connected to either the series
combination or its equivalent, it will measure a
resistance of RT Ohms.
R1
RT
W
W
R2
14
Series Circuits
Of course, this can be extended to n resistors in
series
I

R1
IR1
-

R2
IR2

E
-
-

Rn
IRn
-
15
Current Meter Loading
Consider yet again two resistors in series. The
current flowing in this loop is given by Ohms
law
I

R1
IR1

-
E
-

R2
IR2
-
16
Current Meter Loading
Ideally, if we connect an ammeter into the
circuit to measure I, nothing is affected because
the resistance of the ammeter is zero.

R1
IR1

-
Of course, in the real world nothing is ideal
(except maybe superconductors). The meter leads
have a small but nonzero resistance, and the
meter has nonzero internal resistance. The total
of the lead resistance and the internal
resistance can be modeled as a resistor Rmeter in
series with the meter.
E
-

R2
IR2
-
I
17
Current Meter Loading
Its as if Rmeter were entirely lumped into one
of the meter leads.
Rmeter

R1
IR1

-
E
-

R2
IR2
-
I
18
Current Meter Loading
Lets compare the measured current, Imeasured, to
the true current (the current that would be
measured with an ideal meter) I
Rmeter

R1
IR1

-
E
-

R2
IR2
-
I
19
Current Meter Loading
The percent error due to nonzero meter resistance
is
Rmeter

R1
IR1

-
E
-

R2
IR2
-
I
As long as Rmeter is small compared to RT, the
error is proportionately small.
20
Current Meter Loading
Conclusion As long as the internal resistance
of the ammeter (and its leads) is much smaller
than the resistance in the circuit,
Rmeter
the ammeter does not significantly load (disturb
the operation of) the circuit, and the measured
current is very nearly equal to the current which
would flow in the circuit if the meter were not
there.

R1
IR1

-
E
-

R2
IR2
-
I
21
Voltage Divider Rule
Returning to our example of n resistors in
series, we see (again) that there is a Voltage
drop across each of the n resistor, and the sum
of the n Voltage drops is equal to the source
Voltage.
I

R1
VR1 IR1
-

R2
VR2 IR2

E
-
-
Weve already seen that

VRn IRn
Rn
-
22
Voltage Divider Rule
Substituting,
I

R1
VR1 IR1
-

R2
VR2 IR2

E
-
-
In other words, the source voltage is divided
among the n resistors. The fraction of the
source voltage which appears across each
individual resistor is equal to the fraction of
the total resistance represented by that resistor.

VRn IRn
Rn
-
23
Voltage Divider Rule
Example
I

R1 3 W
VR1
-

R2 4 W
VR2

E 12 V
-
-

R3 5 W
VRn
-
24
BJT Voltage-Divider Bias
This BJT circuit has two power supplies. The 5 V
supply provides base bias The base voltage is
-

2 V
LED
-
Practical BJT amplifiers usually require only one
supply, so another method of biasing the base
must be used.
IB

20V

RB
-

0.7 V
5V
RE
25
BJT Voltage-Divider Bias
Heres another way The base voltage is set by
the voltage divider formed by R1 and R2. One may
notice that R1 and R2 arent really in series,
because the base is connected to the node which
connects R1 and R2. Applying KCL at that node,
IR1
-
R1

2 V
LED
-

IB
20V
But if we select the values of R1 and R2 so that
IR1 gtgt IB, then

-
0.7 V
R2
RE
IR1
and we can treat R1 and R2 as if theyre in
series, and they do in effect form a Voltage
divider.
26
BJT Voltage-Divider Bias
Heres the same circuit with actual resistance
values, to be analyzed. First, we see that,
assuming IB ltlt IR1,
IR1
-
R1 10K

2 V
LED
-

IB
20V

-
R2 3.3K
As usual, the Voltage drop across the
base-emitter junction is 0.7 V, so
0.7 V
RE 1K
IR1
27
BJT Voltage-Divider Bias
Continuing,
IR1
-
KCL says
R1 10K

2 V
LED
-

IB
20V
If b gtgt 1, then clearly

-
R2 3.3K
0.7 V
RE 1K
IR1
IE
28
BJT Voltage-Divider Bias
Weve fount that IE 4.3 mA., so
IR1
-
Assuming that b 50 (a conservative figure for
b),
R1 10K

2 V
LED
-

IB
20V

-
R2 3.3K
0.7 V
RE 1K
IR1
so our treatment of R1 and R2 as if they were
connected in series is valid.
IE
29
Nonideal Voltage Source
All the voltage sources weve considered so far
are ideal voltage sources. If a load resistor
(Rload in the figure below) is connected across a
Voltage source E, load current Iload flows in
the circuit, and load Voltage Vload appears
across Rload. If the Voltage source is an ideal
Voltage source, Vload E regardless of the value
of Rload. If Rload is infinite, the load is an
open circuit, and Vload E. If Rload approaches
0 Ohms (a short circuit), Vload is still equal to
E. Ohms law applies, so
Iload
For an ideal source, Vload E for any value of
Iload, from zero to approaching infinity.
Obviously, an ideal Voltage source cant exist in
the real world.


Rload
E
Vload
-
30
Nonideal Voltage Source
A nonideal voltage source can be modeled by an
ideal source in series with a nonzero resistance
Rsupply. Applying the Voltage divider rule,
Nonideal Voltage Source
Iload
If Rload is an open circuit (infinite
resistance), Vload E. But if Rload is less
than infinite, the load current is nonzero, and
there is a nonzero Voltage drop across Rsupply.
If Rload is a short circuit, Vload drops to zero
Volts.
Rsupply


Rload
E
Vload
-
31
Nonideal Voltage Source
The no-load (or open-circuit) Voltage of a
nonideal Voltage source is simply the Voltage of
the internal ideal source
If the load is an open circuit, Iload 0 and the
Voltage drop across Rsupply is zero. KVL says
the full amount of E must appear across the
open-circuited terminals. The open-circuit is the
lightest possible load, because no current is
drawn from the source.
Nonideal Voltage Source
Iload

Rsupply

E
Voc
-
32
Nonideal Voltage Source
We can increase the load by placing a large but
finite load resistance across the terminals of
the nonideal source. Then we can increase the
load further by reducing the load resistance,
which increases the load current. If we keep
reducing the load resistance, eventually the load
current will increase to the maximum current the
source is capable
supplying without damage. At this point the
source is at full load. The terminal Voltage
under full load is VFL The value of Rload which
results in full-load conditions is the minimum
load resistance, Rminimum, which may be connected
across the sources terminals. If the maximum
load current is Imaximum,
Nonideal Voltage Source
Iload
Rsupply


Rload
E
Vload
-
33
Voltage Regulation
For an ideal source, the terminal voltage does
not vary as the load resistance changes. The
terminal Voltage is constant under any load
conditions no-load, full load, or anything in
between
The nonideal sources terminal Voltage drops as
the load increases. We can get an idea of the
goodness of a Voltage source by comparing its
no-load Voltage to its full-load Voltage. One
way to do this is called percent Voltage
regulation
Nonideal Voltage Source
Iload
Rsupply


Rload
E
Vload
-
An ideal Voltage source has 0 regulation.
34
Voltage Regulation
For a nonideal source, if E and Rsupply are
known, we can calculate VNL and VFL. Therefore,
we can calculate the percent regulation
Nonideal Voltage Source
Iload
Rsupply


Rload
E
Vload
-
35
Voltage Regulation
For a nonideal source, if E and Rsupply are
known, we can calculate VNL and VFL. Therefore,
we can calculate the percent regulation
Nonideal Voltage Source
Iload
Rsupply


Rload
E
Vload
-
36
Thevenins Theorem
Thevenins Theorem says that any linear bilateral
network may be modeled as an ideal Voltage source
in series with a resistance. This means that if
we have a circuit that can be divided in two
parts, Network A which is a linear, bilateral
network (with two terminals), and Network B,
which is connected to Network A via that Network
As two terminals, we could replace the Network A
with an equivalent network
(the Thevenin equivalent of Network A) which
consists of an ideal Voltage source in series
with a resistance. If we do this, everything
outside of Network A behaves exactly as if
Network A were connected instead of its Thevenin
equivalent.
Iload
Network B
Network A
Rsupply
Linear, Bilateral Network

E
37
Thevenins Theorem
The figure below shows Network A replaced by its
Thevenin equivalent. There would be no way to
tell by measuring any Voltage, resistance or
current to the right of the terminals (that is,
outside of Network A or its Thevenin equivalent)
which is connected to Network B Network A or
its Thevenin equivalent. All we need to do is
obtain values for the
Thevenin Voltage, ETH and the Thevenin resistance
RTH.
Thevenin equivalent of Network A
Iload
Network B
RTH

ETH
38
Thevenins Theorem
Whats meant by linear, bilateral network? A
linear network is any network which contains only
linear elements, and a linear element is one
which has an I-V characteristic curve which is a
straight line. Of the circuit elements weve
considered so far, resistors and voltage sources
are linear. On the left, we have the I-V curve
of a resistor, which is clearly a straight line.
On the right is the I-V curve of an ideal Voltage
source, which is a vertical straight line.
I
I
I
I
V
V
E
The slope of the I-V characteristic of an ideal
Voltage source is infinity, which means its
resistance is zero. The source Voltage is E, no
matter what current is drawn.
39
Thevenins Theorem
This is the I-V characteristic of a nonideal
voltage source. Its still a straight line, and
the slope is the reciprocal of the source
resistance. It doesnt pass through the origin,
it intercepts the V axis at the no-load Voltage.
Nonideal Voltage Source
I
I
Iload

Rsupply
VNL

VNL
V
V
-
40
Thevenins Theorem
How do we obtain values for ETH and RTH? Notice
that ETH is equal to Network As open circuit
Voltage, the Voltage we would measure or
calculate across the network terminals with
nothing connected to them (except an ideal
Voltmeter, if were measuring instead of
calculating.
Thevenin equivalent of Network A
Iload

RTH

ETH
Voc
-
41
Thevenins Theorem
We could measure (or calculate) RTH directly, if
we could kill any of the ideal Voltage sources
inside Network A by setting their Voltages to
zero. Setting the Voltage of an ideal source,
which has zero source resistance, to zero is
equivalent to replacing it with a short circuit.
If we could do this, we could simply measure RTH
by connecting an
Ohmmeter to the network terminals. If were
analyzing the network on paper, we can calculate
RTH by replacing any ideal Voltage sources in the
network with short circuits, then calculating the
resistance we would measure across the terminals.
All Voltage sources short-circuited.
Iload

RTH
Ohmmeter
-
42
Thevenins Theorem
If were trying to measure the Thevenin
resistance of an actual, physical network (such
as a function generator or power supply) in the
lab, we usually cant kill the internal ideal
Voltage sources by turning them off, so we cant
measure RTH directly. We must resort to indirect
means.
Notice that if we connect a short circuit to the
network terminals, and measure the current
flowing through the short circuit (the
short-circuit current, Isc), we can use Ohms law
to calculate RTH.
Thevenin equivalent of Network A
Iload

RTH

ETH
Isc
Its usually not a good idea to shor-circuit a
network in the lab, but this method can be used
when analyzing a network on paper.
-
43
Thevenins Theorem
Heres a better way to obtain RTH in the lab
Connect a load resistor Rload across the network
terminals. Select a resistance value small
enough to draw measureable current, but large
enough to avoid overloading the network. Measure
Vload and Iload. Using Ohms Law and
KVL,
Thevenin equivalent of Network A
Iload
RTH

RLoad

VLoad
ETH
-
44
Thevenins Theorem
Heres another way Connect a known load
resistor Rload across the network terminals.
Again, select a resistance value small enough to
draw measureable current, but large enough to
avoid overloading the network. Measure Vload.
Using the Voltage divider rule,
Thevenin equivalent of Network A
Iload
RTH

RLoad

VLoad
ETH
-
45
Thevenins Theorem
Heres an example of using Thevenins theorem.
Remember this BJT circuit? We replaced the 5 V
supply and RB with a Voltage divider.
-

2 V
LED
-
IB

20V

RB
-

0.7 V
5V
RE
46
Thevenins Theorem
Heres the circuit with a Voltage divider
providing base bias. Lets find the Thevenin
equivalent of the Voltage divider consisting of
the 20 Volt source, R1 and R2.
IR1
-
R1 10K

2 V
LED
-

IB
20V

-
R2 3.3K
0.7 V
RE 1K
IR1
IE
47
Thevenins Theorem
Heres the network were going to replace with
its Thevenin equivalent. VB is the voltage
across the network terminals. The open-circuit
Voltage (which is equal to the Thevenin Voltage)
can be obtained using the Voltage divider rule
IR1
R1 10K

IB
20V

R2 3.3K
VB
-
IR1
48
Thevenins Theorem
We can calculate the short-circuit current. If
the terminals are short-circuited, all the
current will flow through the short and none
through R2, so
IR1
R1 10K

20V

R2 3.3K
Isc
-
0
49
Voltage Sources in Series
Voltage sources may be connected in series, as
shown below. The Voltage across the series
connection is the algebraic sum of the source
Voltage in other words, voltage sources in
series add. The two sources on the left are
connected so their Voltages add, this connection
is called series aiding. On the left, E2 has
been reversed so E2 is subtracted from E1. This
is called series opposing.
A common example of a series aiding connection is
the two 1.5 Volt cells in a battery. The
Voltages add, so the total voltage applied to the
bulb is 3 Volts.




E1
E1
-
-
E1E2
E1-E2

-
E2
E2
-

-
-
50
Battery Charger
A battery is a nonideal Voltage source. A
battery charger is also a nonideal Voltage
source. A battery may be charged by connecting
it in series with a battery charger, as shown
below. KVL and Ohms law tell us that
So the charging current, the current flowing from
the charger into the battery, is
Rcharger
Rbattery
I


If I were 100 mA, a 1500 mAH battery would fully
charge in 15 hours.
Echarger
Ebattery
-
-
51
Better Diode Model
The diodes weve talked about turn on when a
forward bias Voltage of 0.7 Volts is applied. A
truly ideal diode would turn on when a forward
bias of gt 0 Volts is applied. This is the first
approximation, or roughest approximation model.
Its I-V characteristic is shown below
V
If the applied bias Voltage is less than 0 Volts,
the diode is an open circuit. If the applied
bias is greater than or equal to 0 Volts, it is a
short circuit, so the first approximation model
for a diode with 0 Volts or more applied is shown
below
-

I
I
First Approximation Model of a Forward-Biased
Diode
V
0 V
52
Better Diode Model
A second approximation model is somewhat more
accurate than the first approximation, but its
still an approximation and therefore not perfect.
We can create a second-approximation model which
exhibits the 0.7 Volt drop (when forward biased),
shifting the I-V characteristic
to the right by 0.7 V. This model consists of
the first approximation model of a
forward-biased diode. The second-approximation
model consists of the first-approximation model
(a short circuit) in series with a 0.7 Volt
Voltage source.
V
-

I
I
0.7 Vr
-

Second Approximation Model of a Forward-Biased
Diode
V
0.7 V
53
Better Diode Model
A third approximation model is still more
accurate. A real diode has nonzero resistance
when forward biased, so the I-V characteristic
would not have infinite slop in the forward bias
region. We can create a third-approximation
model which exhibits this behavior by connecting
A resistance in series with the second
approximation model. The resistance represents
the resistance of the doped silicon from which
the diode is made.
V
-

I
I
-

RB
0.7 Vr
Third Approximation Model of a Forward-Biased
Diode
V
0.7 V
54
Zener Diode Voltage Regulator
A zener diode is a diode, usually silicon, which
is designed to be operated in the reverse-bias
region. Its reverse breakdown Voltage is
specified with a high degree of accuracy, and is
called its Zener Voltage (VZ). While the forward
bias Voltage of any silicon diode is 0.7 Volts,
Zener diodes are available with a wide variety of
Zener Voltages.
V
-

I
I
VZ
V

RB
0.7 V
Second Approximation Model of a Reverse-Biased
Zener Diode
55
Zener Diode Voltage Regulator
A zener diode is a diode, usually silicon, which
is designed to be operated in the reverse-bias
region. Its reverse breakdown Voltage is
specified with a high degree of accuracy, and is
called its Zener Voltage (VZ). While the forward
bias Voltage of any silicon diode is 0.7 Volts,
Zener diodes are available with a wide variety of
Zener Voltages.
Heres a second-approximation of a reverse biased
Zener diode, operating at its Zener voltage. Its
series resistance is zero.
V
-

I
I
VZ
V

0.7 V
Second Approximation Model of a Reverse-Biased
Zener Diode
56
Zener Diode Voltage Regulator
The Zener diode model can be improved by adding a
series resistance, since the actual resistance of
a reverse-biased Zener diode is not zero.
Heres a third-approximation of a reverse biased
Zener diode, operating at its Zener voltage. Its
series resistance is zero.
V
-

I
I
VZ
V
RZ
0.7 V
Second Approximation Model of a Reverse-Biased
Zener Diode
57
Zener Diode Voltage Regulator
Lets see how a Zener diode can be used to
improve the regulation of a nonideal Voltage
supply. Suppose we have a 5-V supply, with a
series resistance of 1 Ohm. The full load
current is 100 mA. KVL and Ohms law tell us that
Nonideal Voltage Source
At no-load, Iload is zero, so
Iload
1 W

At full-load, Iload 100 mA, so
Rsupply

Vload
5 V
-
This may sound like good regulation, but its
really very poor.
58
Zener Diode Voltage Regulator
Heres how we do it with a Zener diode regulator.
First, increase the nonideal supply Voltage to
10 V, then connect a series resistance of 49
Ohms, to make the total series resistance 50
Ohms. Now connect a 5-Volt Zener diode across
the load terminals.
Nonideal Voltage Source
Iload
49 W
1 W

Rsupply
Rsupply

5 V
Vload
5 V
-
59
Zener Diode Voltage Regulator
Now lets replace the Zener with its
third-approximation model. Assume that RZ .01
(This resistance is normally very low) Ohm. At
No-load, Iload 0, and KVL tells us
Nonideal Voltage Source
Iload
49 W
1 W

Rseries
Rsupply
IZ

RZ 1 W
Vload
10 V
5 V
-
60
Zener Diode Voltage Regulator
So the percent regulation is
A very large improvement.
Nonideal Voltage Source
Iload
49 W
1 W

Rseries
Rsupply
IZ

RZ 1 W
Vload
10 V
5 V
-