Argand Diagram
An Argand diagram is a graphical representation of complex numbers in the complex plane. It consists of a horizontal axis representing the real part of a complex number and a vertical axis representing the imaginary part. This diagram is useful for visualizing complex numbers and understanding their properties, such as magnitude and phase.
3 Key excerpts on "Argand Diagram"
eBook - ePubEngineering Science
For Foundation Degree and Higher National
- Mike Tooley, Lloyd Dingle(Authors)
- 2013(Publication Date)
- Routledge(Publisher)
...Thus for example, a 10 Ω reactance would be referred to as j10, not 10j. The Argand Diagram The Argand Diagram provides a useful method of visualising complex quantities and allowing us to solve problems graphically. In common with any ordinary ‘ x – y ’ graph, the Argand Diagram has two sets of axes as right angles, as shown in Figure 26.2. The horizontal axis is known as the real axis whilst the vertical axis is known as the imaginary axis (don’t panic – the imaginary axis isn’t really imaginary; we simply use the term to indicate that we are using this axis to plot values that are multiples of the j operator). Figure 26.2 Argand Diagram showing complex impedances In Figure 26.2 we have plotted an impedance which has a real (resistive) part, A, and an imaginary (reactive) part, B. We can refer to this impedance as (A + j B). The brackets help us to remember that the impedance is made up from two components; one imposing no phase shift whilst the other changes phase by 90°. Examples 26.1 and 26.2 show you how this works. 26.2 Series impedance As you have just seen, the j operator and the Argand Diagram provide us with a useful way of representing impedances. Any complex impedance can be represented by the relationship: where Z represents impedance, R represents resistance and X represents reactance, all three quantities being measured in ohms. The ± j term simply allows us to indicate whether the reactance is attributable to inductance, in which case the j term is positive (i.e. + j) or to capacitance (in which case the j term is negative (i.e...
- Tony Roskilly, Rikard Mikalsen(Authors)
- 2015(Publication Date)
- Butterworth-Heinemann(Publisher)
...Appendix B Mathematics Background This appendix presents some of the mathematics background required for the material covered in the main text. It is intended for revision and for use as a reference, and the presentation is therefore simplified. For a more comprehensive description of these topics, please refer to an engineering mathematics textbook. B.1 Complex Numbers A complex number, z, has real and imaginary parts and can be represented in the complex plane (known as an Argand Diagram) as shown below. We can represent the complex number z on the Argand Diagram in two ways. First, in rectangular coordinates z = x + j y where j is the imaginary operator. 1 Here, x is the real part of z (x = Re(z)) and y is the imaginary part of z (y = Im(z)). We can also specify this point using polar coordinates, in terms of magnitude r and angle θ to the real axis: z = r ∠ θ. Here, x = r cos θ and y = r sin θ, and the polar form of z becomes z = r (cos θ + j sin θ). r is the absolute value, or modulus of z, and is equal. to | z | = r = x 2 + y 2. θ is the argument of z, and is equal to θ = arg (z) = arc tan y x. Using the trigonometric identity e j A = cos A + j sin A we have: e j θ = x r + j y r ⇒ r e j θ = x + j y = z. So we. have z = x + j y = r e j θ where r = x 2 + y 2 and θ = arc tan y x. B.1.1 Arithmetic with complex numbers Using rectangular coordinates, the following rules apply for arithmetic operations with two complex numbers z 1 = x 1 + j y 1 and z 2 = x 2 + j y 2 : For addition, we. have z 1 + z 2 = (x 1 + j y 1) + (x 2 + j y 2) = (x 1 + x 2) + j (y 1 + y 2). Similarly, for. subtraction: z 1 − z 2 = (x 1 + j y 1) − (x 2 + j y 2) = (x 1 − x 2) + j (y 1 − y 2). Multiplication follows normal rules (remember that j 2 =. −1): z 1 ⋅ z 2 = (x 1 + j y 1) ⋅ (x 2 + j y 2) = x 1 x 2 + j x 1 y 2 + j y 1 x 2 + j 2 y 1 y 2 = (x 1 x 2 − y 1 y 2) + j (x 1 y 2 + x 2 y 1). Division of complex numbers can be derived from the multiplication...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...If a point A represents z = a + bi on the complex plane, then the distance from the origin O to the point A is the modulus of z Note that compex conjugates have the same modulus: E XAMPLE 11.4 (a) |3 + 4 i | = 5 (b) |3 − 4 i | = 5 (c) (d) Using complex numbers we can solve any quadratic equation. Recall that the solution for ax 2 + bx + c = 0 is given by where We can use the formula over the complex numbers even when the discriminant D is negative. E XAMPLE 11.5 Solve Solution : E XAMPLE 11.6 Prove that. Solution : Let z 1 = a + bi and z 2 = c + di. Then E XAMPLE 11.7 Show that. Solution : Let z 1 = a + bi and z 2 = c + di. Then 11.2 Polar and trigonometric form of complex numbers So far we have discussed the so-called Cartesian representation of the complex numbers. In essence, we have assigned to each complex number a point on the Cartesian (or complex) plane. However, complex numbers have also a representation in polar coordinates. A complex number z = a + bi is assigned coordinates (r, θ) in the polar system, where and θ is such that as shown in Figure 11.3. The pair (r, θ) specifies a unique point on the complex plane. However, a given point on the complex plane does not have a unique polar representation. In fact, a point z = (r, θ) has infinitely many polar representations of the form z = (r, θ + 2 πn), with n ∈ ℤ. A number in the form (θ + 2 πn) is called an argument of z. The argument of z lying in the range (−π, π] is referred to as the principal argument of z or Arg(z). For instance, the principal argument of 2 i is and the principal argument of (2 i + 2) is Figure 11.3 A complex number z can also be written in a trigonometric form: The trigonometric form of complex numbers is used extensively in time-series analysis. E XAMPLE 11.8 Find the trigonometric form of z = 2 + 2 i. Solution : It follows that Hence Complex numbers can also be written in an exponential form...
