Introduction

In earlier work you will have met the idea that the current flowing through each of several resistors connected in series with a power supply is the same, even if their individual resistances are different.

By applying Ohm's Law to the type of circuit shown above, we can show that different resistances connected in series will have different p.d.s across them and that the sum of the individual p.d.s gives the voltage of the supply connected to the circuit. In this unit we will see how two resistors in series with a power supply can be treated as a

In earlier work you will have met the idea that the current flowing through each of several resistors connected in series with a power supply is the same, even if their individual resistances are different.

By applying Ohm's Law to the type of circuit shown above, we can show that different resistances connected in series will have different p.d.s across them and that the sum of the individual p.d.s gives the voltage of the supply connected to the circuit. In this unit we will see how two resistors in series with a power supply can be treated as a

**voltage divider**and can be used as the basis for environmental sensors. (Note that some texts refer to such circuits as**potential dividers**.)Voltage divider formula

Alter the value of the variable resistor in Fig.1 to confirm the laws governing the

**current**

**series circuit**Click on the figure below to interact with the model.

When the resistors in Fig.1 have the same value, the supply

**voltage**When the resistors in Fig.1 are in the ratio two to one, the supply voltage is divided in the ratio two-thirds to one-third.

When the resistor values are in the ratio two to one, the p.d.s across them are two-thirds and one-third of the supply voltage. The larger resistor has the larger share of the total supply voltage.

In each of these examples you should have noticed that the p.d. across the variable resistor is the same fraction of the total p.d. as its

**resistance**The circuit of Fig.2 is similar to that of Fig.1 except that some labels have been added.

The fraction of the total p.d. across the variable resistor V_{1} is the same as the fraction that the variable resistor is of the total resistance. |

We also know that the total resistance of two resistors in series is equal to the sum of their individual values. |

We can combine the two equations. |

And rearranging them. |

This

**voltage divider**These circuits are often used in sensors. voltage divider equation can also be expressed as:

We can use this voltage divider equation to calculate the p.d. across the 10 kΩ resistor

*V*

_{10k}, in Fig.3.

We use the voltage divider formula. |

We then substitute the values from above. |

We finally simplify to find the value of V_{10k}. |

We can also use the voltage dividers equation to show that the p.d. across the 5 kΩ resistor is 3 V. This acts as a cross-check for the validity of the voltage divider equation.

We use the voltage divider formula again. |

We then substitute the values from above. |

We finally simplify to find the value of V_{5k}. |

This is as we expect. |

Set the variable resistor in Fig.4 below to 10 kΩ and note that the 9 V p.d. from the supply is shared equally between the two 10 kΩ resistors.

Click on the figure below to interact with the model.

When the resistors of Fig.4 are of equal value, the p.d. of the supply is shared equally between the two. Alter the value of the variable resistor in Fig.4 and observe how the supply voltage is shared between the two resistors.

A convenient way of remembering how the voltage divider behaves is to think that the larger resistance 'grabs' the bigger share of the supply voltage.

Understanding the formula

The voltage at the midpoint of the voltage divider in Fig.5 has been labelled *V*

_{mp}.

Click on the figure below to interact with the model.

When the variable resistor in Fig.5 is set to 15 kΩ the midpoint of the voltage divider is 3.60 V above the 0 V terminal of the power supply and 5.40 V below the 9 V terminal. The midpoint is said to be at a voltage 3.60 V (

*V*

_{mp}= 3.60 V). These voltages are relative to the 0 V terminal of the power supply, so the 0 V terminal of the power supply is used as a reference point.

The voltage at the midpoint of the voltage divider in Fig.5 reduces when the resistance of the variable resistor increases.

In Fig.6 the positions of the variable and fixed resistors have been interchanged. The voltage at the midpoint of the voltage divider has again been labelled

*V*

_{mp}.

Click on the figure below to interact with the model.

If the fixed and variable resistors are arranged as shown in Fig.6, the voltage at the midpoint of the voltage divider increases when the resistance of the variable resistor increases.

The circuits of Fig.5 and 6 are very similar and can be easily confused. When considering how the voltage at the midpoint of a voltage divider changes as one of the resistance values alters, you must:

- consider how the p.d. across that resistor changes;
- consider how the p.d. across the resistor connected to the 0 V terminal of the supply changes;
- determine the effect on the voltage at the midpoint.

The behaviour of each of these voltage dividers is summarized below.

Sensors

Your senses (such as touch, sight, and sound) are the inputs to your brain. In much the same way, sensors can be inputs
in electronic systems. Sensors respond to changes in external conditions (such as heat or light). They do this because they
are semiconductors and the resistance of a semiconductor changes when certain conditions change.
Click on the figure below to interact with the model.

Fig.7 shows two common devices based on semiconductors. When these are used in circuits, moving the sliders simulates altering the surrounding conditions The resistance of the

**thermistor**

**LDR**When the temperature of the thermistor is as low as possible, the resistance of the thermistor is at its maximum value. As the temperature increases, the resistance of the thermistor decreases and when the temperature is at its maximum, the resistance of the thermistor is at its minimum value.

When the light shining on the LDR is at its brightest, the resistance of the LDR is at its minimum value. Decreasing the amount of light reaching the LDR increases the resistance of the LDR.

Temperature-sensing circuit

The thermistor can be used in the type of voltage divider circuit shown in Fig.8 to make a temperature-sensing circuit. Click on the figure below to interact with the model.

The reading on the voltmeter in Fig.8 increases as the temperature increases and falls as the temperature decreases.

Placing a voltmeter across the fixed resistance in a circuit such as Fig.8 produces a voltage readout which changes with temperature. This relationship is exploited in commercial devices which use additional electronics to provide a direct readout of the temperature.

Light-sensing circuits

An LDR can be used in a voltage divider such as shown in Fig.9 to make a light-sensing circuit. Use the slider to alter the
quantity of light entering the LDR and observe the changes in the voltmeter reading.Click on the figure below to interact with the model.

In commercial applications additional circuits convert the voltage from the voltage divider circuit into a light-level reading so that the readout on the lightmeter's display indicates the light level entering the LDR.

A potential problem

At first sight the circuit in Fig.10 might be considered suitable for a system that switches on a lamp automatically in the
dark. When darkness falls, the resistance of the LDR increases and so we would expect the p.d. across the LDR to increase.
This should make the voltage at the midpoint of the voltage divider high enough to light the lamp. Click on the figure below to interact with the model.

The operation of a voltage divider relies on the fact the current through both resistors in a series circuit is the same. In Fig.10 the lamp is connected to the midpoint of the voltage divider and so not all the current passing through the top resistor passes through the lower resistor. In fact, the resistance of the lamp is much lower than the LDR so almost all the current in the circuit passes through the lamp regardless of conditions surrounding the LDR. However, this is still not sufficient to light the lamp due to the current limiting effect of the 10 kΩ resistor.

In this situation electronic engineers would say that the lamp is 'crippling' the operation of the voltage divider.

Fig.11 shows how electronic engineers can overcome this

**loading effect**. Decrease the level of light entering the LDR and note what happens to the lamp now.

Click on the figure below to interact with the model.

The additional component (a

**transistor***directly*from the supply. The current in both resistors in the voltage divider is almost the same, so the voltage divider switches on the current to the lamp.

You will learn more about these types of circuits in specialist electronics courses.

Summary

When resistors are connected in series with a power supply, the fraction of the total p.d. across any specific resistor is the same as the fraction that its resistance is of the circuit's total resistance.

The voltage across one resistor in a voltage divider can be calculated using the equation:

Voltage dividers incorporating LDRs or thermistors can be used to produce circuits giving output voltages which depend on the environment in which the sensor is placed.

When resistors are connected in series with a power supply, the fraction of the total p.d. across any specific resistor is the same as the fraction that its resistance is of the circuit's total resistance.

The voltage across one resistor in a voltage divider can be calculated using the equation:

Voltage dividers incorporating LDRs or thermistors can be used to produce circuits giving output voltages which depend on the environment in which the sensor is placed.

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