Figure 20.21 shows the variation with time of a signal voltage, V, over a 5-hour period.

Question 1

Figure 20.21 shows the variation with time of a signal voltage, V, over a 5-hour period.

a State the name of the type of signal shown in the diagram.

b The signal is turned into a digital signal with 4-bit binary numbers. The value is (0000) when V is 3V and each subsequent binary unit represents an extra 0.2V. Copy and complete the table to give the value of V as a decimal number and as a binary number for times 0, 2 and 4 hours.

Reference: Cambridge International AS and A Level Physics Coursebook with CD-ROM by David Sang, Graham Jones, Gurinder Chadha, Richard Woodside - Cambridge University Press, 2014 Pg 326 Chapter 20: Communication systems





Solution:

(a)

Analogue signal

(b)

A 4-bit binary number can only consist of ‘0’s and ‘1’s. [these are called bits] The 4-bit binary numbers have the bits 0 0 0 0 (in this order) corresponding to 2³ 2² 2¹ 2⁰ respectively [which corresponds to 8 4 2 1 respectively]. 

{The values given in square brackets next are the numbers of binary units}
That is, for the right-most bit, a ‘0’ corresponds to 0 x (2⁰) [= 0 binary unit] while a ‘1’ corresponds to 1 x (2⁰) [= 1 binary unit]. For the left-most bit, a ‘0’ would therefore correspond to 0 x (2³) [= 0 binary unit] while a ‘1’ would correspond to 1 x (2³) [= 8 binary units]

The total number of binary units represented by the 4-bit binary number is equal to the sum of binary units of each bit.

Examples:

4-bit binary number: 0000

Number of binary bits: 0 + 0 + 0 + 0 = 0

4-bit binary number: 0001

Number of binary bits: 0 + 0 + 0 + 1 = 1

4-bit binary number: 0011

Number of binary bits: 0 + 0 + 2 + 1 = 3

4-bit binary number: 1011

Number of binary bits: 8 + 0 + 2 + 1 = 11

Now, it is said in the question that the value is (0000) when V is 3V and that each binary unit represents an extra 0.2V. 

So, for example, consider V = 3.4V [this value is given in decimal]. We need a number of binary units that would represent the extra here (which is 3.4 – 3.0 = 0.4). Since 0.2V is represented by 1 binary unit, the number of binary units representing 0.4V is 0.4 / 0.2 = 2 binary units. 

Now, in the 4-bit binary number, 2 binary units is represented by 0010. So, the same process is done to complete the table.

The decimal values can be read directly from the graph. The binary number is obtained as explained above.

From the graph, 5 small squares represent 1V. So, 1 small square = 0.2V.

At time = 0, V = 5.2V. The extra is 5.2 – 3.0 = 2.2V. This is represented by 2.2 / 0.2 = 11 binary units. In the 4-bit binary number, 11 is represented by 1011 {that is, 1(8) + 0(4) + 1(2) + 1(1) = 11}

At time = 2, V = 4.0V. The extra is 4.0 – 3.0 = 1.0V. This is represented by 1.0 / 0.2 = 5 binary units. In the 4-bit binary number, 5 is represented by 0101 {that is, 0(8) + 1(4) + 0(2) + 1(1) = 5}

At time = 4, V = 3.8V (it is a little less than 3.8V but we cannot read to value to that accuracy. Data is lost at such points). The extra is 3.8 – 3.0 = 0.8V. This is represented by 0.8 / 0.2 = 4 binary units. In the 4-bit binary number, 4 is represented by 0100 {that is, 0(8) + 1(4) + 0(2) + 0(1) = 4}

Time / hours    V / V (decimal)           V / V (binary)

0                      5.2V                            1011

2                      4.0V                            0101

4                      3.8V                            0100