Circuit Connections
Student Extras
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PhET Simulation: Circuit Construction Kit (DC Only)
Build a virtual circuit right here on your computer using this Java applet from PhET.
Ohm Zone
Use this virtual playground of switches, cells, wires, resistors and more to explore circuit principles. Best of all, you won't get shocked.
Teacher's Guide
Looking for a lab that coordinates with this page? Try the Combination Circuits Lab from The Laboratory.
Curriculum CornerPractice makes perfect with this computational activity from The Curriculum Corner.
Curriculum CornerPractice makes perfect with this computational activity from The Curriculum Corner.
PhET Simulation: Circuit Construction Kit (DC Only)
This applet from PhET allows students to easily create circuits and analyze numerical relationships between electrical quantities.
All About Circuits: Experiments: Voltmeter Usage
Learn about the use of voltmeters and multimeters.
Ohm Zone
The Ohm Zone provides an electronic breadboard on which circuits can be constructed; or simply choose a pre-made circuit to analyze.
Combination Circuits
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 that 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.
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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 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.
This method is consistent with the formula
1 / Req = 1
/ R1 + 1 / R2 + 1 / R3 +
...
where R1, R2, and R3 are the resistance values of the individual resistors that 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 R2 and R3 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
Rtot =
R1 + R2 + R3 +
...
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 that has an equivalent resistance. Two 8
resistors in series is equivalent to a single 4
resistor. Thus, the two branch resistors (R2 and
R3) can be replaced by a single resistor with a
resistance of 4 .
This 4
resistor is in series with R1 and R4.
Thus, the total resistance is
+ R4 = 5
+ 4
+ 6
Rtot = 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.
Vtot
/ Rtot = (60 V) / (15 )
Itot = 4 Amp
The 4 Amp current calculation represents the current at the battery location. Yet, resistors R1 and R4 are in series and the current in series-connected resistors is everywhere the same. Thus,
For parallel branches, the sum of the current in each individual branch is equal to the current outside the branches. Thus, I2 + I3 must equal 4 Amp. There are an infinite number of possible values of I2 and I3 that satisfy this equation. Since the resistance values are equal, the current values in these two resistors are also equal. Therefore, the current in resistors 2 and 3 are both equal to 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.
V1 = 20 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:
1 / Req = 1 / (4
) + 1 / (12
)
1 / Req = 0.333 -1
Req = 1 / (0.333 -1)
Req = 3.00
Based on this calculation, it can be said that the two
branch resistors (R2 and R3) can be
replaced by a single resistor with a resistance of 3
.
This 3
resistor is in series with R1 and R4.
Thus, the total resistance is
+ R4 = 5
+ 3
+ 8
Rtot = 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.
Vtot
/ Rtot = (24 V) / (16 )
Itot = 1.5 Amp
The 1.5 Amp current calculation represents the current at the battery location. Yet, resistors R1 and R4 are in series and the current in series-connected resistors is everywhere the same. Thus,
For parallel branches, the sum of the current in each individual branch is equal to the current outside the branches. Thus, I2 + I3 must equal 1.5 Amp. There are an infinite possibilities of I2 and I3 values that 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 that 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 (R1 and R4) 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.
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 (R1 and R4). 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 R1 and R4. Thus,
V2
= V3
= 4.5 VKnowing the voltage drop across the parallel-connected
resistors (R1 and R4) allows one to
use the Ohm's law equation (V
= I • R) to determine the current in the two
branches.
I2 =I2 = 1.125 AI3 =
I3 = 0.375 A
The analysis is now complete and the results are summarized in the diagram below.
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
Itot, I1, R3, V2,
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 I2, it
is important to substitute the V2
and the R2 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.