complex number use in analysing alternating signals - short

8/2/2019 Complex Number Use in Analysing Alternating Signals - Short

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An introduction to ComplexNumbers in ElectricalEngineering

Phil Illingworth



Learning Objectives

• To understand how to represent a quantityin both Complex Cartesian and Polar forms(and convert between the two forms)

• To be able to manipulate (add, subtract,and multiply) these complex numbers

Complex numbers are widely used in the analysis of

electrical networks supplied by alternating voltages, such as:In deriving balance equations with AC bridges, in analyzing AC circuitsusing Kirchhoff’s laws, mesh and nodal analysis, the superpositiontheorem, with Thevenin’s and Norton’s theorems, and with delta-star andstar-delta transforms.



Why use Complex Numbers?

• One dimension or two?

• From Scalar to Vector

• Complex numbers extend the idea of a 1

dimensional number line, to a 2 dimensional“complex plane”

x

y

Cartesian

Polar



The Complex Plane• The complex plane (known as an Argand diagram)

allows us to represent magnitude and direction

“real”

axis

“Imaginary”

axis (j)

Magnitude:

Direction:

j is known as the j-operator, and

indicates a phase shift of 90o

Note: j is used in engineering, but i is often used in pure maths

The phase shift is measured withregard to the reference signal

resistance, reactance,

voltage or current

phase shift



Cartesian and Polar Representation inthe Complex PlaneFor a complex quantity with an “in-phase” element (a), and

an “out of phase” element (b), in this case +90o, we canrepresent this in both Cartesian (rectangular) and polar forms

a

The j-operator shows a 90o phase shift

b

a + jb

“real”

axis

“Imaginary”

axis (j)

Cartesian

r∠ø

“real”

axis

“Imaginary”

axis (j)

Polar

r

ø

Z Z



Converting between Cartesian and Polar

How do we write a and b

in terms of r and θ?

Z = a + jb Z = r∠ø

“real”axis

“Imaginary”

axis (j)

r

ø

Z

r is called the modulus (magnitude of Z), written mod Z or |Z|

How do we write r (or |Z|)in terms of a and b?

θ is called the argument of Z(arg Z). In terms of a and b?b=r sinθ

a

b

a=r cosθ

r or |Z| = (a 2 + b 2 )

arg Z or θ = tan-1 (b/a)

Z = r cosθ + j r sinθ



Manipulating Complex Numbers

• j has a numerical value which is negative when

squared.

Adding and Subtracting Cartesian Complex Numbers

We simply collect the “real” elements together and

the “imaginary” elements together, for example:

a. (3 + j2) + (2 – j4) =

b. (3 + j2) - (2 – j4) =

Multiplying Cartesian Complex Numbers

We simply multiply brackets togetherremembering that j2 = -1, for example :

c. (3 + j2) (2 – j4) =

(5 - j2)

(1 + j6)

j is defined as -1 and therefore j2

= -1

(14 – j8)



Using the Complex Numbers in context

“real”axis

“Imaginary”

axis ( j)Analysing an AC circuit

(series)

R-L Series Circuit

VR

VL

The supply voltage V given by:

V = VR + j VL

V

As current I is common to both, impedance Z = R + j XL

And so, an impedance expressed as

(3 + j4) means that the resistance is3 and the inductive reactance is 4



Assessment

1. What is another name for the complex plane?

2. What does the “real” axis represent in a complex

plane?

3. What does the “imaginary” axis represent in acomplex plane?

4. Find the modulus and argument of (3 + j 4)

5. If two complex numbers are equal, then their realparts are equal and their imaginary parts are equal.Hence solve:

a. (2 + j )(-2 + j ) = x + j y



Objectives Achieved?

• To understand how to represent a quantityin both Complex Cartesian and Polar forms(and convert between the two forms)

• To be able to manipulate (add, subtract,and multiply) these complex numbers



Assessment

1. What is another name for the complex plane?2. What does the “real” axis represent in a complex

plane?

3. What does the “imaginary” axis represent in acomplex plane?

4. Find the modulus and argument of (3 + j 4)

5. If two complex numbers are equal, then their realparts are equal and their imaginary parts are equal.Hence solve:

a. (2 + j )(-2 + j ) = x + j y

Name:



Using the Complex Numbers in context

“real”axis

“Imaginary”

axis (j)Analysing Impedance in

an AC circuit (series)

Pure resistance

VR

IR

The impedance Z, is given by

R

I

V Z

o

R

o

R

0

0

complex number use in analysing alternating signals - short

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