Argand Diagram and Polar Form

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Argand Diagram and Polar Form
Mathematics for AVT Dr Ian Drumm

• Aims To Introduce alternative forms of complex
  number representation
• Learning Outcomes polar form, arithmetic with
  polar form, exponential form, real world
  application

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Argand Diagram
Can think of complex numbers as vectors
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Adding Vectors

• Consider z and w

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Argument of Complex Number

• Consider Argand diagram

Magnitude
Argument
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The Polar Form of a Complex Number
Given
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Multiplication and Division Using Polar
Coordinates

• Given
• Multiplication, moduli multiply and arguments add
• Division, moduli divide and arguments subtract

Used addition formulae
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Complex Numbers and Rotations
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Exponential Form

• Well show in later lecture
• This very important equation is called Eulers
  formula and should be remembered
• This gives us yet another useful way to represent
  a complex number

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Simplifying complex numbers

• An import skill is to express equations involving
  complex numbers in a meaningful form such as
• For example given

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Complex Number Applications

• As we will see throughout the degree programme
  complex numbers are an important tool for
  describing oscillating systems such as acoustic
  media, loudspeakers, microphones, electronic
  circuits, etc
• We would use complex numbers to represent time
  varying forces, pressures, displacements,
  velocities, impedances, currents, voltages, etc

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Complex Pressure

• For example the pressure of a sound wave at a
  given point can be described using a complex
  number
• When we add pressures from a number of sound
  waves we add complex numbers to get a new complex
  number
• Hence can find the resulting magnitude and phase
  to relate the mathematics to real world
  observations

where
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Simple forced mass spring damper system

• Even the simplest vibrating systems require
  complex number descriptions
• We can show displacement is
• And impedance is
• From such results we can make important
  inferences (tacoma narrows)