Series and Parallel Resistors
Introduction


Electrical circuits nearly always contain more than one component. For example, you can plug a radio, a toaster, and a microwave into different sockets in your kitchen but all will be connected to the same electricity supply.

Calculating the effective resistance of a combination of components allows circuit designers to prevent appliances from overheating or drawing too much current from the power supply.



For safety and other reasons, it is useful to know how the current in a circuit will vary depending on which appliances you use.

Effective resistance
We sometimes call the network of wires and components connected to a power supply the external circuit or load circuit (since it is the power-consuming part of the system). The total
resistance
The opposition to the flow of current provided by a circuit is called resistance. Resistance is measured in units called Ohms.
resistance
of the
external circuit
All the components connected across the terminals of a battery or power supply are collectively known as the external circuit.
external circuit
is sometimes called the load
resistance
The opposition to the flow of current provided by a circuit is called resistance. Resistance is measured in units called Ohms.
resistance
.

Close the switches in Fig.1 to connect the batteries to the resistors. Set the variable resistor in circuit (b) to its maximum value by moving the slider as far up as it will go.


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 Figure 1.  A complicated and a simple circuit.



The two circuits in the above diagram look very different indeed, but there are some similarities. Both circuits are powered from 9 V batteries and both contain a switch and an ammeter.

Complete the following sentences (to the nearest whole number).

  • When the switch in Fig.1(a) is closed, the current shown on the ammeter is mA. With the variable resistor set to its maximum value and the switch closed, the current in circuit Fig.1(b) is mA.
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Complete the following sentences.

  • The current flowing from the battery in Fig.1(a) is than the current flowing from the battery in Fig.1(b). Since the battery voltages are the same for both circuits, we can conclude that the resistance in Fig.1(a) is than the resistance in Fig.1(b).
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Now reduce the value of the variable resistor until the
current
The rate of flow of charge past any specific point in a circuit. The base unit of current is the Ampere.
current
in Fig.1(b) is equal to the current in Fig.1(a).

What is the value of the variable resistor when the currents in both circuits have the same value?
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When the current flowing from each of the 9 V batteries in Fig.1 is the same, we can say that both circuits have the same effective resistance.

Fig.1(a) looks complicated, but the 10 resistors connected to the 9 V battery are providing the same resistance as a single 75 Ω resistor! Fig.1(a) therefore has an effective resistance of 75 Ω. The effective resistance is also sometimes called the total resistance and is given the symbol RT.

Resistors in series
Close the switch in each of the two circuits In Fig.2 below. We can see that the resistance in both circuits is the same since equal currents flow from identical batteries. Therefore the effective resistance of both circuits in Fig.2 is the same.

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 Figure 2.  Circuits with equal current.





From our earlier consideration of series circuits in the unit Current and Voltage, we can add labels to the circuit of Fig.2 as follows:


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 Figure 3.  The current, I, in both circuits is the same.

Since the p.d.s across resistors in series add up we can say that:
From Ohm's Law,
So
     Therefore



The effective resistance RT of resistors R1 and R2 connected in series is determined by the equation:

Resistors in parallel
Fig.4 below shows a circuit in which a 100 Ω resistor is connected in series with a 9 V battery.


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 Figure 4.  Resistors in parallel.

With one resistor connected, what is the current in the circuit?
  •  mA   (to the nearest whole number)
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Connect the second 100 Ω resistor into the circuit and watch what happens to the current flowing from the battery.

Notice the effect of connecting the second resistor in parallel with the first and complete the following sentences.

  • When the second 100 Ω resistor is connected in parallel with the first, the current taken from the battery . This observation, coupled with the fact that the p.d. of the battery , means that the overall resistance of the circuit .
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At first it seems very strange that adding an extra resistor to create the
parallel circuit
Components connected in parallel offer alternative paths for charges from a supply.
parallel circuit
in Fig.4 makes it easier for current to flow. However the set-up in Fig.5 might help clarify the situation.

Figure 5.   Resistors in parallel.
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Complete the following sentences.

  • The total charge flowing through the combination of two wires in is than through the single wire and battery connected in . Consequently more is supplied to the two wires from the same p.d., so the effective is .
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We can calculate the effective resistance of two resistors connected in parallel. In the circuits in Fig.6 below, the p.d.s across the resistors are all the same.


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 Figure 6.  Circuits with resistors.



The circuits in Fig.6 have the same resistance since the same total current flows from identical batteries.
The total current into any junction IT equals the total current flowing out of that junction, so

From Ohm's Law,

So

     Simplifying this equation gives:




The effective resistance RT of resistors R1 and R2 connected in parallel is determined by the equation:



This equation can be extended for any number of resistors in parallel.


 
For two resistors in parallel the equation can be rearranged into a more convenient form, giving the value of RT if R1 and R2 are known:
    



The effective resistance RT of two resistors R1 and R2 in parallel can be determined from:

Resistors in practice
In circuit diagrams, resistors are represented as a rectangle and the value of the resistance is normally written as a number beside the symbol.
 
Figure 7.   Symbol and picture of a resistor.
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In practice, resistors are normally very small and any text written on the resistor would be difficult to read. So the value of a resistor is usually indicated by a series of coloured bands around the body of the resistor. Each colour and each band has a value based on a standard colour code.

Summary


Connecting different components across a power supply alters the current flowing into an external circuit.

The effective resistance of components joined in series or parallel can be derived using the laws governing the behaviour of current and voltage in circuits.


For resistances in series For resistances in parallel



Once the effective resistance of the components connected to a power supply has been determined the total current taken from the power supply can be calculated.

Exercises
1.
Ohm's law
Ohm's Law states that at constant temperature the current in a conductor is directly proportional to the d.p. across the conductor. The constant of proportionality is called the resistance of the sample.
Ohm's law
links a circuit (I) with the resistance (R) and p.d. (V) by the mathematical law:
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2. In which of the following circuits could the perfect ammeter and voltmeter readings be used to calculate the value of resistor, R?
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3. In the circuit shown in Fig.8 the battery has no
internal resistance
All batteries or power supplies have internal resistance. This resistance has the effect of reducing the output p.d. as the current supplied increases.
internal resistance
. The current in the circuit is 20 mA. Calculate the p.d. across the 75 Ω resistor.
  •  V   (to 1 d.p.)
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Figure 8.  
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4. What is the p.d. across resistor X, as shown in Fig.8?
  •  V   (to 1 d.p.)
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5. Calculate the resistance of resistor X in Fig.8.
  •   Ω   (to the nearest whole number)
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6. What single value of resistor could be used to replace the two resistors in the circuit, as shown in Fig.8, so that the current from the 6 V battery is still 20 mA?
  •  Ω   (to the nearest whole number)
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Figure 9.  
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7. Resistor X and the 75 Ω resistor are now connected in parallel and connected to the 6 V battery as shown in Fig.9.

  • What is the current at point A1 in the circuit?
     A   (to 3 d.p.)

    What is the current at point A2?
     A   (to 3 d.p.)

    What is the current at point A3?
     A   (to 3 d.p.)
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8. A pupil studying potential dividers sets up the circuit shown in Fig.10. When the switch is not pressed the voltmeter shows a reading of 3.5 V. What is the
voltage
The voltage across a component is the electrical energy transferred by 1 coulomb of charge passing through the component.
voltage
across the 100 Ω resistor?
  •  V   (to 1 d.p.)
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9. in Fig.10, what is the current in the 100 Ω resistor when the switch is open?
  •  mA   (to the nearest whole number)
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10. What is the value of the resistor R in Fig.10?
  •  Ω   (to the nearest whole number)
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Figure 10.  
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11. The pupil now closes the switch in Fig.10. What happens to the reading on the voltmeter?
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12. Calculate the reading shown in the voltmeter in Fig.10 when the switch is closed.
  •  V   (to 2 d.p.)
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Figure 11.  
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13. A single in-line (SIL) resistor package, which is commonly used in printed circuit boards, contains six 10 kΩ resistors. When this resistor package is soldered into the part of the printed circuit board shown in Fig.11, what is the resistance between the points marked X and Y?
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Figure 12.  
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14. Resistors X and Y are connected in parallel with the supply as shown in Fig.12. The p.d. of the output of the supply is set at 3.5 V.

  • What is the current in resistor X?
     mA   (to the nearest whole number)

    What is the current in resistor Y?
     mA   (to the nearest whole number)

    What is the current in the ammeter?
     mA   (to the nearest whole number)
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Figure 13.  
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15. The student now connects resistor X and lamp Z in series, as shown in Fig.13. The reading recorded on the ammeter is 30 mA.

  • What p.d. across resistor X will cause a current of 30 mA in the resistor?
     V   (to 1 d.p.)

    What p.d. across lamp Z will cause a current of 30 mA in lamp Z?
     V   (to the nearest whole number)

    Calculate the p.d. of the output of the supply needed to produce a current of 30 mA in the ammeter.
     V   (to 1 d.p.)
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Figure 14.  
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Figure 15.  
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16. A student investigating electronic components sets up the test circuit shown in Fig.14. She is given three components labelled A, B, and C. A and B are resistors while C is a light bulb.

When testing the components she adjusts the
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit of power is the watt.
power
supply until the ammeter reads certain values and she records pairs of ammeter and voltmeter readings.

She repeats this procedure with each component and then plots the results on a single set of graph axis. Her results are shown in Fig.15.

  • What is the current in the lamp when the voltage across it is 3 V?
     mA   (to the nearest whole number)

    What is the current in resistor A when the voltage across it is 3 V?
     mA   (to the nearest whole number)

    Which resistor has the larger resistance, A or B?
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17. The student is now asked to use her graphs to answer questions about the circuit of Fig.16 where resistors A and B are connected in parallel with the power supply.

  • When the output voltage from the power supply is 3 V, what is the reading on the ammeter?
     mA   (to the nearest whole number)

    Using your answer to the previous part, calculate the effective resistance of A and B in parallel.
     Ω   (to 1 d.p.)

    What is the reading on the ammeter when resistor B is replaced by the lamp and the supply voltage left at 3 V?
     mA   (to the nearest whole number)
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Figure 16.  
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18. Resistor A is removed from the circuit and the resistor B and the lamp are instead connected in series with the ammeter and the power supply.

What is the voltage of the power supply when the reading of the ammeter is 50 mA?
  •  V   (to the nearest whole number)
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Figure 17.  
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19. In the circuit shown in Fig.17, P and Q are 3 Ω and 1 Ω resistors respectively. They are connected to a 6 V battery of negligible internal resistance.

Which of the following changes would make the current in the circuit equal 2.4 A?
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Conditions LDR resistance
Daylight 500 Ω
Darkness 90 kΩ

Figure 18.  

20. A light dependent resistor (
LDR
The resistance of a light dependent resistor reduces as the light intensity increases. This feature makes LDRs ideal for use in light sensing circuits.
LDR
) is connected in series with a 10 kΩ resistor and a 1.5 V battery of negligible internal resistance. If the LDR's resistance in daylight and darkness is as shown in Fig.18, approximately what values would you expect for the voltmeter readings …

  • in daylight?
     V (to 2 d.p.)

    in darkness?
     V (to 2 d.p.)
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Figure 19.  
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21. In circuit in Fig.19, X, Y, and Z are identical 10 kΩ resistors and V1, V2, and V3 are identical ideal voltmeters. A pupil is asked to connect another resistor in parallel with X but before doing so he makes the following predictions. Decide whether his predictions are true or false.
  • The reading on V1 will not change.
    The reading on V2 will increase.
    The calculated value of V1 − (V2 + V3) will be greater.
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