Resistors in Series
Introduction
Two types of current are readily available, direct
current (dc) and sinusoidal alternating current (ac)
We will first consider direct current (dc)
Insert Fig 5.1
Introducing the basic current flow of an electric
circuit.
Defining the direction of conventional flow for
single-source dc circuits.
Defining the polarity resulting from a conventional current I
through a resistive element.
Resistors in series.
Schematic representation
for a dc series circuit
Example of five resistors in series.
Series Resistors
The total resistance of a series configuration
is the sum of the resistance levels.
RT  R1  R2  R3  R4  ...  RN
The more resistors we add in series, the
greater the resistance (no matter what their
value).
Resistance “seen” at the terminals of a series circuit.
Total Series Resistance
• The total resistance of a series circuit is equal to the
sum of the resistances of each individual series
resistor
Series connection of resistors.
Using an ohmmeter to measure the total resistance of a
series circuit.
Two series combinations of the same elements with the same
total resistance.
Series connection of resistors
Series combination of resistors
Series Resistors
• When series resistors have the same value,
RT  NR
• Where N = the number of resistors in the string.
• The total series resistance is found by multiplying
the value of the same resistor times the number of
resistors
Series connection of four resistors of the same value
Four 2.2Kohms resistors
Resistors in Series
• A series circuit provides only one path for current
between two points so that the current is the
same through each series resistor
Current in a Series Circuit
• The current is the same through all points in a
series circuit
• The current through each resistor in a series
circuit is the same as the current through all
the other resistors that are in series with it
• Current entering any point in a series circuit is
the same as the current leaving that point
Current entering any point in a series circuit is the same as the
current leaving that point.
Current is the same at all points in a series circuit.
Series Circuits
• Total resistance (RT) is all the
source “sees.”
• Once RT is known, the current
drawn from the source can be
determined using Ohm’s law:
E
Is 
RT
• Since E is fixed, the magnitude of
the source current will be totally
dependent on the magnitude of RT
Ohm’s Law in Series Circuits
• Current through one of the series resistors
is the same as the current through each of
the other resistors and is the total current
• If you know the total voltage and the total
resistance, you can determine the total
current by using:
•
IT = VT/RT
Measuring the current throughout the series circuit
Find RT then find IS
Find Current in circuit ?
Find the voltage of the Source ?
Find the voltage of the Source
Ohm’s Law in Series Circuits
• Current through one of the series resistors is
the same as the current through each of the
other resistors and is the total current
• If you know the voltage drop across one of the
series resistors, you can determine the current
by using:
I = VR/R
Notation
Single-subscript notation
The single-subscript notation Va specifies the
voltage at point a with respect to ground (zero
volts). If the voltage is less than zero volts, a
negative sign must be associated with the
magnitude of Va .
Notation
Double-subscript notation
• Because voltage is an “across” variable and exists
between two points, the double-subscript notation
defines differences in potential.
• The double-subscript notation Vab specifies point a as
the higher potential. If this is not the case, a negative
sign must be associated with the magnitude of Vab .
• The voltage Vab is the voltage at point (a) with respect to
point (b).
Inserting the polarities across a resistor as
determined by the direction of the current
Ohm’s Law in Series Circuits
• If you know the total current, you can find the
voltage drop across any of the series resistors by
using:
VR = ITR
• The polarity of a voltage drop across a resistor is
positive at the end of the resistor that is closest to
the positive terminal of the voltage source
• The resistor current is in a direction from the positive
end of the resistor to the negative end
Voltage in a Series dc circuit to be analyzed
Using voltmeters to measure the voltages across the resistors
Series circuit to be investigated
Series circuit to be analyzed
The source voltage appears across
the open series resistor
Voltage Sources in Series
• When two or more voltage sources are in series,
the total voltage is equal to the the algebraic sum
(including polarities of the sources) of the
individual source voltages
Reducing series dc voltage sources to a
single source.
Series connection of dc supplies: (a) four 1.5 V batteries in series to establish a terminal voltage of 6 V; (b) incorrect connections for
two series dc supplies; (c) correct connection of two series supplies to establish 60 V at the output terminals.
Kirchhoff’s Voltage Law
The applied voltage of a series circuit equals the sum
of the voltage drops across the series elements:
V
rises
 Vdrops
The sum of the rises around a closed loop must equal the
sum of the drops.
The application of Kirchhoff’s voltage law need not
follow a path that includes current-carrying
elements.
When applying Kirchhoff’s voltage law, be sure to
concentrate on the polarities of the voltage rise or drop
rather than on the type of element.
Do not treat a voltage drop across a resistive element
differently from a voltage drop across a source.
Kirchhoff’s Voltage Law
• Kirchhoff’s voltage law (KVL) states that the
algebraic sum of the potential rises and drops
around a closed loop (or path) is zero.
Kirchhoff’s Voltage Law
• The algebraic sum of all
the voltage drops
around a single closed
loop in a circuit is equal
to the total source
voltage in that loop
VS = V1 + V2 + V3 + … + Vn
Applying Kirchhoff’s voltage law to a series dc circuit.
Another Way to state
Kirchhoff’s Voltage Law
• The algebraic sum of all voltages (both
sources and drops) around a closed path is
zero
VS - V1 - V2 - V3 - … - Vn = 0
Illustration of a verification of Kirchhoff’s voltage law.
Sum of n voltage drops equals the source voltage.
Voltage Dividers
• Since each resistor has the same current, the
voltage drops are proportional to the resistance
values
The voltage divider as a bias circuit for a transistor amplifier.
Voltage-Divider Formula
• The voltage drop Vx across any resistor or
combination of resistors in a series circuit is
equal to the ratio of that resistance value Rx to
the total resistance RT, multiplied by the
source voltage VS
Vx = (Rx/RT)VS
Potentiometer as an Adjustable
Voltage Divider
• The potentiometer shown below is equivalent to a tworesistor voltage divider that can be manually adjusted
• The two resistors are between terminals 1 & 3 and 2 & 3
Adjusting the voltage divider.
A voltage divider used for volume control.
A potentiometer voltage divider used as an automotive fuel-level
sensor.
Example of a two-resistor voltage divider.
A five-resistor voltage divider.
Circuit Ground
• Voltage is relative
• The voltage at one point in a circuit is always
measured relative to another point
• This reference point in a circuit is usually the
ground point
Measuring Voltages with Respect to
Ground
• When voltages are measured with respect
to ground in a circuit, one meter lead is
connected to the circuit ground, and the
other to the point at which the voltage is to
be measured
Simple illustration of circuit ground.
Measuring a voltage with respect to negative ground.
.
Measuring Voltages with Respect to Ground
Measuring voltages at several points in a
circuit with respect to ground
Series circuit (without a short) with correct voltages marked.
Measuring Voltage Across an
Ungrounded Resistor
• Voltage can normally (as long as the meter
is isolated from the power line ground) be
measured across a resistor even though
neither side of the resistor is connected to
circuit ground
• The reading will be the voltage drop across
the resistor
Measuring voltage directly across a resistor.
Open Circuit
• The most common failure in a series circuit is
an open
• When an open occurs in a series circuit, all of
the source voltage appears across the open
The source voltage appears across the
open series resistor.
Ohm’s Law in Series Circuits
• An open in a series circuit prevents current;
and, there is zero voltage drop across each
series resistor
• The total voltage appears across the points
between which there is an open
An open in a circuit prevents current.
Troubleshooting a series circuit for an open using half-splitting.
Short Circuit
• When there is a short, a portion of the series
resistance is bypassed, thus reducing the total
resistance
• A short in a series circuit results in more
current than normal through the circuit
• The voltage across a shorted series
component (or circuit) is 0 volts
A Short in a Series Circuit
Example of shorts on a PC board.
Power in a Series Circuit
• The total amount of power in a series resistive
circuit is equal to the sum of the powers in
each resistor in series
PT = P1 + P2 + P3 + . . . + Pn
Power Distribution in a Series Circuit
The power applied by the dc supply must equal that
dissipated by the resistive elements.
PE  PR1  PR2  ...  PRN
Power in a Resistor
• The amount of power in a resistor is important
because the power rating of the resistor must
be high enough to handle the expected power
in the circuit
Series circuit to be investigated for Power
Identifying Parallel Circuits
• If there is more than one current path
(branch) between two separate points, and if
the voltage between those two points also
appears across each of the branches, then
there is a parallel circuit between those two
points
Resistors in Parallel
• Each current path is called a branch
• A parallel circuit is one that has more than one
branch
Voltage in Parallel Circuits
• The voltage across any
given branch of a
parallel circuit is equal
to the voltage across
each of the other
branches in parallel
Voltage across parallel branches is the same.
Kirchhoff’s current law: The current into a node equals the
current out of that node.
Generalized Circuit Node Illustrating
KCL
Total Parallel Resistance
• When resistors are connected in parallel, the
total resistance of the circuit decreases
• The total resistance of a parallel circuit is
always less than the value of the smallest
resistor
Parallel Resistors
For parallel elements, the total conductance is
the sum of the individual conductance values.
GT  G1  G2  G3  ...  GN
As the number of resistors in parallel increases, the
input current level will increase for the same applied
voltage.
Parallel Resistors
For resistors in parallel, the total resistance is
determined from
Note that the equation is for the reciprocal of
RT rather than for RT.
Once the right side of the equation has been
determined, it is necessary to divide the result into
1 to determine the total resistance
Parallel Resistors
The total resistance of any number of parallel
resistors can be determined using
1
RT 
1
1
1
1


 ... 
R1 R2 R3
RN
The total resistance of parallel resistors is always
less than the value of the smallest resistor.
Connecting resistors in parallel reduces total resistance and
increases total current.
Circuit with n resistors in parallel.
Notation for Parallel Resistors
• To indicate 5 resistors, all in parallel, we would
write:
R1||R2||R3||R4||R5
Illustration of a verification of Kirchhoff’s current law.
Total current divides between the two branches.
Generalized Circuit Node Illustrating KCL
Kirchhoff’s Current Law (KCL)
• The sum of the currents into a node (total
current in) is equal to the sum of the currents
out of that node (total current out)
IIN(1) + IIN(2) + . . . + IIN(n) = IOUT(1) + IOUT(2) +
. . . +IOUT(m)
Kirchhoff’s Current Law
• Kirchhoff’s current Law (KCL) can be stated
another way:
The algebraic sum of all the currents entering
and leaving a junction is equal to zero
FIGURE 5-18
Two-node configuration
Four-node configuration
.
Total Parallel Resistance
• When resistors are connected in parallel, the
total resistance of the circuit decreases
• The total resistance of a parallel circuit is
always less than the value of the smallest
resistor
Formula for Total Parallel Resistance
1/RT = 1/R1 + 1/R2 + 1/R3 + . . . + 1/Rn
Parallel Resistors
For equal resistors in parallel:
Where N = the number of parallel resistors.
Two Resistors in Parallel
• The total resistance for two resistors in
parallel is equal to the product of the two
resistors divided by the sum of the two
resistors
RT = R1R2/(R1 + R2)
Current Divider Rule
The current divider rule (CDR) is used to find the
current through a resistor in a parallel circuit.
General points:
For two parallel elements of equal value, the current
will divide equally.
For parallel elements with different values, the smaller
the resistance, the greater the share of input current.
For parallel elements of different values, the current
will split with a ratio equal to the inverse of their
resistor values.
Current Divider Rule
RT
Ix 
IT
Rx
General Current-Divider Formula
• The current (Ix) through any branch equals the
total parallel resistance (RT) divided by the
resistance (Rx) of that branch, and then
multiplied by the total current (IT) into the
junction of the parallel branches
Ix = (RT/Rx)IT
Notation for Parallel Resistors
• To indicate 5 resistors, all in parallel, we would
write:
R1||R2||R3||R4||R5
Application of a Parallel Circuit
• One advantage of a parallel circuit over a series
circuit is that when one branch opens, the other
branches are not affected
Current Dividers
• A parallel circuit acts as a current divider
because the current entering the junction of
parallel branches “divides” up into several
individual branch currents
Current Dividers
• The total current divides among parallel resistors into
currents with values inversely proportional to the
resistance values
FIGURE 5-53
FIGURE 5-51 A 10 mA meter.
FIGURE 5-52 A milliammeter with three ranges.
General Current-Divider Formula
• The current (Ix) through any branch equals the
total parallel resistance (RT) divided by the
resistance (Rx) of that branch, and then
multiplied by the total current (IT) into the
junction of the parallel branches
Ix = (RT/Rx)IT
Open Branches
• When a parallel resistor opens, IT is always less
than its normal value
• Once IT and the voltage across the branches
are known, a few calculations will determine
the open resistor when all the resistors are of
different values
Open Branches
• When an open circuit occurs in a parallel branch,
the total resistance increases, the total current
decreases, and the same current continues
through each of the remaining parallel paths
When a lamp filament opens, total current decreases by the amount
of current in the lamp that opened. The other branch currents
remain unchanged.
All parallel branches (open or not) have the same voltage.
Power in Parallel Circuits
• Total power in a parallel circuit is found by
adding up the powers of all the individual
resistors, the same as for series circuits
PT = P1 + P2 + P3 + . . . + Pn
Power flow in a dc parallel network.
When a lamp filament opens, total current decreases by the
amount of current in the lamp that opened. The other branch
currents remain unchanged.
All parallel branches (open or not) have the
same voltage
Voltmeter Loading Effects
Voltmeters are always placed across an element
to measure the potential difference.
The resistance of parallel resistors will always be less
than the resistance of the smallest resistor.
A DMM has internal resistance which may alter the
resistance of the network under test.
The loading of a network by the insertion of a meter
is not to be taken lightly, especially if accuracy is a
primary consideration.
Voltmeter Loading Effects
A good practice is to always check the meter resistance
against the resistive elements of the network before
making a measurement.
Most DMMs have internal resistance levels in excess of
10 MW on all voltage scales.
The internal resistance of a VOM depends on the scale
chosen.
Internal resistance is determined by multiplying the
maximum voltage of the scale setting by the
ohm/volt (W / V) rating of the meter, normally
found at the bottom of the face of the meter.
Voltmeter loading.
Applications
Car system
The electrical system on a car is essentially a parallel
system.
Parallel computer bus connections
The bus connectors are connected in parallel with
common connections to the power supply, address
and data buses, control signals, and ground.
Expanded view of an automobile’s electrical system.
Application of a Parallel Circuit
• One advantage of a parallel circuit over a series
circuit is that when one branch opens, the other
branches are not affected
Applications
House wiring
Except in some very special circumstances the basic
wiring of a house is done in a parallel
configuration.
Each parallel branch, however, can have a
combination of parallel and series elements.
Each branch receives a full 120 V or 208 V, with the
current determined by the applied load.
Application of a Parallel Circuit
• All lights and appliances in a home are wired in
parallel
• The switches are located in series with the lights