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Published byElise Mowery Modified over 2 years ago

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Complex numbers i or j

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Complex numbers An Imaginary Number, when squared, gives a negative result. imaginary 2 = negative

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Complex numbers i = -1 i is used in maths But j is used in electronics and engineering (because i is already used as a symbol for current)

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Complex numbers i = -1 i 2 = -1 i 3 = --1 i 4 = 1 i 5 = -1

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Complex numbers Example What is i 6 ? i 6 = i 4 × i 2 = 1 × -1 = -1

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Adding complex numbers (4 +j3) + (3 + j5) 4 +j j5 7 + j8

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Adding complex numbers (3 +j6) + (2 – j3) 3 +j6 + 2 – j3 5 + j3

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Subtracting complex numbers ( 6 +j8) - (2 + j3) 6 +j8 - 2 – j3 4 + j5 Note the change of sign

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Multiplying complex numbers Example 1, 6(3 +j4) = 18 + j24 Example 2 j8 + 3(3 – j2) = j8 + 9 – j6 j2 + 9

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Multiplying complex numbers (3 + j2)(4 + j) Use F.O.I.L. (3x4) + (3xj) + (j2 x4) + (j2 x j) 12 + j3 + j8 + j 2 2 j 2 = j11 – j11

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Multiplying complex numbers (5 - j2)(2 + j2) Use F.O.I.L. (5 x 2) + (5 x j2) - (j2 x2) - (-j2 x j2) 10 + j10 – j4 - j 2 4 j 2 = – j j10

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Multiplying complex numbers (4 - j2)(3 - j) Use F.O.I.L. (4 x 3) - (4 x j) - (j2 x3) + (j2 x j) 12 – j4 – j6 + j 2 2 j 2 = j10 – j10

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Multiplying a conjugate pair (4 - j2)(4 + j2) Use F.O.I.L. (4 x 4) + (4 x j2) - (j2 x 4) - (j2 x j2) 16 + j8 – j8 - j 2 4 j 2 =

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Dividing complex numbers (2 +6j)/2j = 2/2j + 6j/2j = 1/j +3 J

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Dividing complex numbers (6 + j3)/ (3+j2) Multiply by the conjugate of the denominator (6 + j3) x (3 – j2) (3 + j2) (3 - j2) 18 – j12 +j9 –j – j6 + j6 –j 2 4

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Dividing complex numbers 18 - j – j 2 4= 24 – j – j3 13

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Argand Diagrams Imaginary axis y Real axis x Z = x +yj

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Argand Diagrams r r = (x 2 + y 2 )

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Argand Diagrams Φ tanΦ = y/x

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Argand Diagrams Imaginary axis y Real axis x Z = x +yj r Φ yj = r sinΦ x = r cosΦ

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Example Argand diagrams are used to calculate impedance in RLC circuits

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Example The impedance of a circuit is given by the complex number 3 +j4 Construct the Argand diagram for 3 +j4

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Example Imaginary axis y Real axis x Z = 3 +j4 j4 3 r

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Example From the Argand diagram derive the expression for the impedance in polar form

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Example Imaginary axis y Real axis x Z = 3 +j4 j4 3 r r = ( ) = (9 + 16) (25) = 5

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Example Imaginary axis y Real axis x Z = 3 +j4 j4 3 r tanΦ = 4/3 Φ = 53.13

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Example Imaginary axis y Real axis x Z = 3 +j4 j4 3 r Answer Z =

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Multiplying and dividing polar form 620° x 430° Multiply the length (modulus) and add the argument (angle) = 2450° 910° / 340° = 9/3 (10°-40°) divide the length (modulus) and subtract the argument (angle) = 3-30°

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Argand diagrams as phasor diagrams The voltage of a circuit is given as V = 3 + j3 and the current drawn is given as I = 8 + j2 Find the phase difference between V and I Find the power (VI.cosФ)

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Argand diagrams as phasor diagrams Voltage = ( ) = 18 = 4.24 Volts Current = ( ) = 68 = 8.25 amps

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Argand diagrams as phasor diagrams Voltage phase angle tanΦ = 3/3 =1, Φ = 45 o Current phase angle tanΦ = 2/8 =0.25, Φ = 14.0 o

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Argand diagrams as phasor diagrams Phase difference between V and I = 45 o o = 31 o power = VIcosΦ 4.24 x 8.25 cos31 o 4.24 x 8.25 x.86 = 30 watts

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