Worked Through Examples
1. In the series-parallel circuit shown, calculate the total equivalent resistance and all unknown voltages and currents using Ohm’s law and circuit reduction techniques.
First, you can find RT by circuit reduction techniques. Since R2
and R3 are of equal value and are connected in parallel, the
equivalent resistance, R2,3 can be found with the formula:
Req = RS
N
RS equals 18 kilohms and N equals 2, so:
R2,3 = Req = RS
N =
18 k
2
R2,3 = 9 k
After the first circuit reduction, the circuit now consists of R1 in
You can find the total resistance of the circuit by simply using the series circuit law which says that the total resistance of a series circuit equals the sum of the individual resistances. In formula form: RT = R1 + R2 + R3 + . . . or in this case: RT = R1 + R2.3 RT = 15 k + 9 k RT = 24 k
Once you know the total resistance, you can find the total
current by using Ohm’s law in the form IT = ET/RT. Substituting
the appropriate values in the formula gives: IT = ET
RT
72 V
24 k
IT = 3 mA
This total current can be used to find the voltage across R1.
Remember, since R1 is in series with the rest of the circuit, the
total current must flow through R1. If you use Ohm’s law in the
ER1 = IT X R1
ER1 = 3 mA X 15 k
ER1 = 45 V
Remember that in a series circuit the total voltage equals the sum of the individual voltage drops. You know the total voltage and the voltage across R1; the remainder of the voltage must be
dropped across R2,3. In formula form:
ER2,3 = ET -ER1
ER2,3 = 72 V - 45 V
ER2,3 = 27 V
You can find the current through R2 or R3 by using Ohm’s law
in the form I = E/R. Remember, R2 and R3 are in parallel, so
they have the same 27 volts dropped across them. IR2 = ER2
R2
27 V
18 k
Since R2 and R3 have the same resistance value and the same
voltage across them, they have the same current flow through them. You could have found the current through R2 and R3 by
simply realizing that they must divide the total current of 3 milliamps equally between them.
IR2 = IR3 =
IT
2
3 mA
2
If R2 and R3 did not have the same resistance value, you could
have found the current through R3 by subtraction. You know
the total current and you know the current through R2, so the
remainder of the current must flow through R3.
IR3 = IT - IR2
IR3 = 3 mA - 1.5 mA
IR3 = 1.5 mA
and the circuit is completely solved.
2. In the series-parallel circuit shown, calculate the total equivalent resistance and all unknown voltages and currents using Ohm’s law and circuit reduction techniques.
In order to keep track of all the knowns and unknowns, make a chart as shown on the next page and fill in the known values. Then you can fill in the unknown values as you calculate them.
Notice that since R1 and R2 are in parallel, the voltage across
them is the same.
You can use Ohm’s law in the form I = E/R to calculate IR1 and
IR2.
IR1 = ER1
R1 IR2 =
ER2 R2
IR1 = 36 V
10 k IR2 = ER2
R2
IR1 = 3.6 mA IR2 = 2.4 mA
You know that the total current in a parallel circuit equals the sum of the individual branch currents. In this circuit, the total current flows through the combination of R1 and R2; you can
add IR1 and IR2 to get IT.
IT = IR1 + IR2
IT = 3.6 mA + 2.4 mA
IT = 6.0 mA
You can now fill in these calculated values on the chart as shown.
Looking at the chart or the circuit, you can see that you know two things about R5, you know its resistance, and you know the
current flow through it. You can use Ohm’s law in the form E = I X R to find ER5.
ER5 = IR5 X R5
ER5 = 2 mA X 27 k
ER5 = 54 V
Because R3, R4, and R5 are in parallel, they have 54 volts
dropped across them. If they all have the same voltage across them and they all have the same resistance value, then the
current must be the same through all of them. Since IR5 equals 2
milliamps, the IR3 and IR4 also equal 2 milliamps each.
You could check your work at this point by adding IR3, IR4 and
IR5 to see that they do add up to the total current of 6 milliamps.
Because R6 is in series with the rest of the circuit, the total
current must flow through it. Thus IR6 equals 6 milliamps and
you can now use this information to find ER6.
ER6 = IR6 X R6
ER6 = 6 mA X 2 k
ER6 = 12 V
As shown, you know the voltage across and current flow through each portion of the circuit.
The voltage across R1 and R2 is the same; ER1,2 equals 36 volts.
The voltage is also the same across R3, R4, and R5; ER3,4,5 equals
54 volts. You also know the voltage across R6; ER6 equals 12
volts. From series circuit laws, these voltages can be added to find the total voltage applied to the circuit.
ET = ER1,2 + ER3,4,5 + ER6
ET = 36 V + 54 V + 12 V
The only unknown quantity remaining to be calculated is the total resistance. This can be found in either of two ways. One way is to use Ohm’s law in the form:
RT = ET
IT
When you substitute the appropriate values in the formula, you obtain:
RT = 102 V
6 mA
RT = 17 k
Circuit reduction techniques can also be used to find RT. First,
consider R1 in parallel with R2. Using the product-over-the-sum
formula: R1,2 = R1 X R2 R1 R2 R1,2 = 10 k X 15 k 10 k 15 k R1,2 = (1 X 10 4 ) X (1 .5 X 10 4) (1 X 10 4) (1 .5 X 10 4) R1,2 = 1 .5 X 10 8 2 .5 10 4 R1,2 = 0.6 X 10+4 = 6 k
Because R3, R4 and R5 all have the same resistance value, they
Req = RS
N
Req = 27 k 3
Req = 9 k
These three resistance are now in series and can be added to find RT.
RT = R1,2 + R3,4,5 + R6
RT = 6 k + 9 k + 2 k
RT = 17 k
and this agrees with the previous calculation.
The chart can be filled in as shown, and the circuit is completely solved.
Practice Problems
The key objective of this lesson has been achieved if you can analyze any series parallel circuit in a variety of situations such as:
1. Given a series-parallel wired network of resistors, calculate their equivalent resistance, Req.
2. Given a series-parallel circuit with all of the resistor values and the applied voltage labeled, calculate any or all of the voltages across and currents through each resistor, as well as the total circuit current and equivalent resistance.
3. Given a series-parallel circuit schematic with several known values labeled, calculate any unknown values required.
The practice problems that follow are designed to give you as much practice as you may need in these areas. It is suggested that you work enough of these to enable you to approach and analyze any series-parallel circuit without referring back to the lesson.
Depending upon the approach you use in solving these problems and how you round off intermediate results, your answers may vary slightly from those given here. However, any differences you encounter should only occur in the third significant digit of your answer. If the first two significant digits of your answers do not agree with those given here, recheck your calculations.
Problems
1. Find Req for the following circuits.