Previously in Lesson 4, it was
mentioned that there are two different ways to connect two
or more electrical devices together in a circuit. They can
be connected by means of series connections or by means of
parallel connections. When all the devices in a circuit are
connected by series connections, then the circuit is
referred to as a series circuit.
When all the devices in a circuit are connected by
parallel
connections, then the circuit is referred to as a parallel
circuit. A third type of circuit involves the dual use
of series and parallel connections in a circuit; such
circuits are referred to as compound circuits or combination
circuits. The circuit depicted at the right is an example of
the use of both series and parallel connections within the
same circuit. In this case, light bulbs A and B are
connected by parallel connections and light bulbs C and D
are connected by series connections. This is an example of a
combination circuit.
When analyzing combination circuits, it is
critically important to have a solid understanding of the
concepts which pertain to both series
circuits and parallel circuits.
Since both types of connections are used in combination
circuits, the concepts associated with both types of
circuits apply to the respective parts of the circuit. The
main concepts associated with series and parallel circuits
are organized in the table below.
Series
Circuits
The current is the same in every resistor;
this current is equal to that in the
battery.
The sum of the voltage drops across the
individual resistors is equal to the voltage
rating of the battery.
The overall resistance of the collection of
resistors is equal to the sum of the individual
resistance values,
R_{tot} = R_{1} +
R_{2} + R_{3} + ...
Parallel
Circuits
The voltage drop is the same across each
parallel branch.
The sum of the current in each individual
branch is equal to the current outside the
branches.
The equivalent or overall resistance of the
collection of resistors is given by the equation
1/R_{eq} = 1/R_{1} +
1/R_{2} + 1/R_{3}
...
Each of the above concepts has a mathematical expression.
Combining the mathematical expressions of the above concepts
with the Ohm's law equation (V
= I • R) allows one to conduct a complete analysis of a
combination circuit.
Analysis of
Combination Circuits
The basic strategy for the analysis of combination
circuits involves using the meaning of equivalent resistance
for parallel branches to transform the combination circuit
into a series circuit. Once transformed into a series
circuit, the analysis can be conducted in the usual manner.
Previously in Lesson 4, the
method for determining the equivalent resistance of parallel
branches was discussed. If the resistance of the branches
are equal, then the total or equivalent resistance of those
branches is equal to the resistance of one branch divided by
the number of branches.
where R_{1}, R_{2}, and R_{3} are
the resistance values of the individual resistors which are
connected in parallel. If the two or more resistors found in
the parallel branches do not have equal resistance, then the
above formula must be used. An example of this method was
presented in a previous
section of Lesson 4.
By applying one's understanding of the
equivalent resistance of parallel branches to a combination
circuit, the combination circuit can be transformed into a
series circuit. Then an understanding of the equivalent
resistance of a series circuit can be used to determine the
total resistance of the circuit. Consider the following
diagrams below. Diagram A represents a combination circuit
with resistors R_{2} and R_{3} placed in
parallel branches. Two 4-
resistors in parallel is equivalent to a resistance of 2
.
Thus, the two branches can be replaced by a single resistor
with a resistance of 2 .
This is shown in Diagram B. Now that all resistors are in
series, the formula for the total resistance of series
resistors can be used to determine the total resistance of
this circuit: The formula for series resistance is
R_{tot} =
R_{1} + R_{2} + R_{3} +
...
So in Diagram B, the total resistance of the circuit is
10 .
Once the total resistance of the circuit is determined,
the analysis continues using Ohm's law and voltage and
resistance values to determine current values at various
locations. The entire method is illustrated below with two
examples.
Example
1:
The first example is the easiest case - the resistors
placed in parallel have the same resistance. The goal of the
analysis is to determine the current in and the voltage drop
across each resistor.
As discussed above, the first step is to simplify the
circuit by replacing the two parallel resistors with a
single resistor which has an equivalent resistance. Two 8
resistors in series is equivalent to a single 4
resistor. Thus, the two branch resistors (R_{2} and
R_{3}) can be replaced by a single resistor with a
resistance of 4 .
This 4
resistor is in series with R_{1} and R_{4}.
Thus, the total resistance is
R_{tot} = R_{1} + 4
+ R_{4} = 5
+ 4
+ 6
R_{tot} = 15
Now the Ohm's law equation (V
= I • R) can be used to determine the total current in
the circuit. In doing so, the total resistance and the total
voltage (or battery voltage) will have to be used.
I_{tot} = V_{tot}
/ R_{tot} = (60 V) / (15 )
I_{tot} = 4
Amp
The 4 Amp current calculation represents the current at
the battery location. Yet, resistors R_{1} and
R_{4} are in series and the current in
series-connected resistors is everywhere the same. Thus,
I_{tot} =
I_{1} = I_{4} = 4
Amp
For parallel branches, the sum of the current in each
individual branch is equal to the current outside the
branches. Thus, I_{2} + I_{3} must equal 4
Amp. There is an infinite possibilities of I_{2} and
I_{3} values which satisfy this equation. Since the
resistance values are equal, the current values in these two
resistors is also equal. Therefore, the current in resistors
2 and 3 are both equal to 2 Amp.
I_{2} =
I_{3} = 2 Amp
Now that the current at each individual resistor location
is known, the Ohm's law equation (V
= I • R) can be used to determine the voltage drop
across each resistor. These calculations are shown
below.
V_{1}
= I_{1} • R_{1} = (4 Amp) • (5
)
V_{1}
= 20 V
V_{2}
= I_{2} • R_{2} = (2 Amp) • (8
)
V_{2}
= 16 V
V_{3}
= I_{3} • R_{3} = (2 Amp) • (8
)
V_{3}
= 16 V
V_{4}
= I_{4} • R_{4} = (4 Amp) • (6
)
V_{4}
= 24 V
The analysis is now complete and the results are
summarized in the diagram below.
Example
2:
The second example is the more difficult case - the
resistors placed in parallel have a different resistance
value. The goal of the analysis is the same - to determine
the current in and the voltage drop across each
resistor.
As discussed above, the first step is to simplify the
circuit by replacing the two parallel resistors with a
single resistor with an equivalent resistance. The
equivalent resistance of a 4
and 12
resistor placed in parallel can be determined using the
usual formula for equivalent resistance of parallel
branches:
Based on this calculation, it can be said that the two
branch resistors (R_{2} and R_{3}) can be
replaced by a single resistor with a resistance of 3
.
This 3
resistor is in series with R_{1} and R_{4}.
Thus, the total resistance is
R_{tot} = R_{1} + 3
+ R_{4} = 5
+ 3
+ 8
R_{tot} = 16
Now the Ohm's law equation (V
= I • R) can be used to determine the total current in
the circuit. In doing so, the total resistance and the total
voltage (or battery voltage) will have to be used.
I_{tot} = V_{tot}
/ R_{tot} = (24 V) / (16 )
I_{tot} = 1.5
Amp
The 1.5 Amp current calculation represents the current at
the battery location. Yet, resistors R_{1} and
R_{4} are in series and the current in
series-connected resistors is everywhere the same. Thus,
I_{tot} =
I_{1} = I_{4} = 1.5
Amp
For parallel branches, the sum of the current in each
individual branch is equal to the current outside the
branches. Thus, I_{2} + I_{3} must equal 1.5
Amp. There are an infinite possibilities of I_{2}
and I_{3} values which satisfy this equation. In the
previous example, the two resistors in parallel had the
identical resistance; thus the current was distributed
equally among the two branches. In this example, the unequal
current in the two resistors complicates the analysis. The
branch with the least resistance will have the greatest
current. Determining the amount of current will demand that
we use the Ohm's law equation. But to use it, the voltage
drop across the branches must first be known. So the
direction which the solution takes in this example will be
slightly different than that of the simpler case illustrated
in the previous example.
To determine the voltage drop across the parallel
branches, the voltage drop across the two series-connected
resistors (R_{1} and R_{4}) must first be
determined. The Ohm's law equation (V
= I • R) can be used to determine the voltage drop
across each resistor. These calculations are shown
below.
V_{1}
= I_{1} • R_{1} = (1.5 Amp) •
(5 )
V_{1}
= 7.5 V
V_{4}
= I_{4} • R_{4} = (1.5 Amp) •
(8 )
V_{4}
= 12 V
This circuit is powered by a 24-volt source. Thus, the
cumulative voltage drop of a charge traversing a loop about
the circuit is 24 volts. There will be a 19.5 V drop (7.5 V
+ 12 V) resulting from passage through the two
series-connected resistors (R_{1} and
R_{4}). The voltage drop across the branches must be
4.5 volts to make up the difference between the 24 volt
total and the 19.5 volt drop across R_{1} and
R_{4}. Thus,
V_{2}
= V_{3}
= 4.5 V
Knowing the voltage drop across the parallel-connected
resistors (R_{1} and R_{4}) allows one to
use the Ohm's law equation (V
= I • R) to determine the current in the two
branches.
I_{2} = V_{2}
/ R_{2} = (4.5 V) / (4 )
I_{2} =
1.125 A
I_{3} = V_{3}
/ R_{3} = (4.5 V) / (12 )
I_{3} =
0.375 A
The analysis is now complete and the results are
summarized in the diagram below.
Developing
a Strategy
The two examples above illustrate an effective
concept-centered strategy for analyzing combination
circuits. The approach demanded a firm grasp of the series
and parallel concepts discussed earlier.
Such analyses are often conducted in order to solve a
physics problem for a specified unknown. In such situations,
the unknown typically varies from problem to problem. In one
problem, the resistor values may be given and the current in
all the branches are the unknown. In another problem, the
current in the battery and a few resistor values may be
stated and the unknown quantity becomes the resistance of
one of the resistors. Different problem situations will
obviously require slight alterations in the approaches.
Nonetheless, every problem-solving approach will utilize the
same principles utilized in approaching the two example
problems above.
The following suggestions for approaching combination
circuit problems are offered to the beginning student:
If a schematic diagram is not provided, take the time
to construct one. Use schematic
symbols such as those shown in the example
above.
When approaching a problem involving a combination
circuit, take the time to organize yourself, writing down
known values and equating them with a symbol such as
I_{tot}, I_{1}, R_{3}, V_{2},
etc. The organization scheme used in the two examples
above is an effective starting point.
Know and use the appropriate formulae for the
equivalent resistance of series-connected and
parallel-connected resistors. Use of the wrong formulae
will guarantee failure.
Transform a combination circuit into a strictly
series circuit by replacing (in your mind) the parallel
section with a single resistor having a resistance value
equal to the equivalent resistance of the parallel
section.
Use the Ohm's law equation (V
= I • R) often and appropriately. Most answers will
be determined using this equation. When using it, it is
important to substitute the appropriate values into the
equation. For instance, if calculating I_{2}, it
is important to substitute the V_{2}
and the R_{2} values into the equation.
For further practice analyzing combination circuits,
consider analyzing the problems in the Check Your
Understanding section below.
Check Your
Understanding
1.
A combination circuit is shown in the diagram at the right.
Use the diagram to answer the following questions.
a. The current at location A is _____ (greater than,
equal to, less than) the current at location B.
b. The current at location B is _____ (greater than,
equal to, less than) the current at location E.
c. The current at location G is _____ (greater than,
equal to, less than) the current at location F.
d. The current at location E is _____ (greater than,
equal to, less than) the current at location G.
e. The current at location B is _____ (greater than,
equal to, less than) the current at location F.
f. The current at location A is _____ (greater than,
equal to, less than) the current at location L.
f. The current at location H is _____ (greater than,
equal to, less than) the current at location I.
2.
Consider the combination circuit in the diagram at the
right. Use the diagram to answer the following questions.
(Assume that the voltage drops in the wires themselves in
negligibly small.)
a. The electric potential difference (voltage drop)
between points B and C is _____ (greater than, equal to,
less than) the electric potential difference (voltage drop)
between points J and K.
b. The electric potential difference (voltage drop)
between points B and K is _____ (greater than, equal to,
less than) the electric potential difference (voltage drop)
between points D and I.
c. The electric potential difference (voltage drop)
between points E and F is _____ (greater than, equal to,
less than) the electric potential difference (voltage drop)
between points G and H.
d. The electric potential difference (voltage drop)
between points E and F is _____ (greater than, equal to,
less than) the electric potential difference (voltage drop)
between points D and I.
e. The electric potential difference (voltage drop)
between points J and K is _____ (greater than, equal to,
less than) the electric potential difference (voltage drop)
between points D and I.
f. The electric potential difference between points L and
A is _____ (greater than, equal to, less than) the electric
potential difference (voltage drop) between points B and
K.
3. Use the concept of equivalent resistance to determine
the unknown resistance of the identified resistor that would
make the circuits equivalent.
4. Analyze the following circuit and determine the values
of the total resistance, total current, and the current at
and voltage drops across each individual resistor.
5. Referring to the diagram in question #4, determine the
...
a. ... power rating of resistor 4.
b. ... rate at which energy is consumed by resistor
3.