Argand Diagrams

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Argand Diagrams

The Argand diagram is an ingenious method of representing complex numbers as a graphical plot, the imaginary number represented by (i) in mathematics was considered a rotation of 90 degrees.

This ingenious method was devised by self-taught amateur mathematician Jean Robert Argand (July 18th 1768 – August 13th 1822)

Note: The Argand diagram shown in Fig. 1 uses (j) to represent imaginary numbers on the y axis of the plot and real numbers are represented on the x axis of the plot.

In electrical and electronics engineering very often imaginary numbers crop up particularly when dealing with phase diagrams in electrical engineering and oscillations that occur in electronics circuits.

In physics the description of electron spin will result in the use of imaginary numbers.

In control systems transient responses can very often be oscillatory and the solutions to problems in these areas involve solving quadratic equations that sometimes gives rise tothe square root of minus one (√-1) as the answer.

Although (√-1) is seemingly an impossible answer, by denoting (√-1) as j,then it can be manipulated algebraically using all the rules of algebra.

Example: simplify the following quadratic equation expansion:-

z = 2+2j+5+4j+4-3

Answer: add all real numbers together2+5+4-3 = 8

Add all the imaginary numbers +2j+4j = +6j

Giving a simplified answer of 8+6j

This can be plotted on an Argand diagram as shown below:

Multiplying Complex Numbers

The Argand diagram gives a pictorial description of what could otherwise be considered meaningless numbers.

For instance if the above complex number 8+6j was multiplied by j what would be the outcome algebraically?

(8+6j) j = 8j+6j²

Remembering that j =√-1 then j² must be -1

Thus 8j+6j² = -6+8j

Plot this on an Argand diagram gives:

As can be seen from the plot of 8+j6 when it is multiplied by j it rotates the whole expression by 90°. It follows by similar reasoning that:-

j² = -1 a rotation of 180°

j³ = -j a rotation of 270°

j⁴ = 1 a rotation of 360

In Electrical engineering it is sometimes necessary to know the magnitude and phase of sinusoidal voltages and currents and these can be converted from the Argand diagram (rectangular to polar form):-

Fig. 4

Using Pythagoras Theorem to find the magnitude of the complex number then from Fig. 1:-

r² = a²+b²

The magnitude ǀ r ǀ = √ (a² + b²)

The phase angle is tanˉ¹ (b/a)

Polar to Rectangular Conversion

Using trigonometry from Fig. 4

The length of side a is:-

Cosine Ɵ = adjacent/hypotenuse

= a / r

Rearranging to make a the subject:

a =rCosƟ

Sin Ɵ = opposite/hypotenuse

= jb/r

Rearranging to make b the subject:

b = rjSinƟ

Hence:

a +jb = r(cosƟ+jsinƟ)

r/Ɵ = r(cosƟ+jsinƟ)

Special Case of (1+j)

This is an interesting case where real and imaginary numbers are both unity values. This is plotted on an Argand diagram below:

Finding the magnitude of r is by Pythagoras theorem is:-

ǀ r ǀ = √ (a² + b²)

= √ (1² + 1²)

= √ 2

= 1.414

This particular result will be very familiar for anyone who has studied electrical engineering where peak values of sinusoidal currents and voltages are concerned.

The phase angle is tanˉ¹ (b/a)

= tanˉ¹ (1)

= 45°

The Argand diagram pictorially conveys the concepts of complex numbers when dealing with angles, sinusoids and rotational bodies met by students in science and engineering mathematics.

A picture paints a thousand words!

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