ECE 3301 General Electrical Engineering Presentation 32 Complex Numbers

Real Numbers Real numbers consist of:

Real Numbers Real Numbers may be represented on the Real Number Line.

Imaginary Numbers Imaginary Numbers are the square-roots of negative numbers: The Imaginary Unit is defined as:

Imaginary Numbers The custom for electrical engineers is to use j instead of i (as the mathematicians do) to represent the imaginary unit, because i is used to represent current.

Imaginary Numbers Using the definition for the imaginary unit:

Imaginary Numbers The reciprocal of the imaginary unit number is also of interest.

Imaginary Numbers Any Imaginary number can be expressed using the Imaginary Unit j.

Imaginary Numbers All imaginary numbers may be represented on the Imaginary Number Line.

Complex Numbers Complex Numbers have a Real Part and an Imaginary Part: z = a + jb where a is the Real Part and b is the Imaginary Part.

Complex Numbers Two complex numbers: z1 = a1 + jb1 and z2 = a2 + jb2 are equal if and only if a1 = a2 (the Real Parts are equal) and b1 = b2 (the Imaginary Parts are equal)

The Complex Plane

The Real Part

The Imaginary Part

Rectangular Form Any Complex Number may be expressed in Rectangular Form:

Polar Form The same Complex Number may be expressed in Polar Form:

Polar Form In Polar Form, r is the distance from the origin:

Polar Form In Polar Form,  (in degrees) is the angle formed with respect to the positive Real Axis:

Polar Form The relationship between Rectangular and Polar coordinates can be derived using simple trigonometry :

Polar Form Given a Complex Number in Polar Form, the Rectangular Form is found by:

Polar Form Given a Complex Number in Polar Form, the Rectangular Form is found by:

Polar Form Recall Euler’s Formula:

Complex Numbers Any Complex Number is defined in one of two forms: Real Part and Imaginary Part (Rectangular Form) Magnitude and Phase Angle (Polar Form)

Complex Numbers Electrical Engineers use a short-hand notation often called the Steinmetz Form:

Complex Numbers For any Complex Number there are four equivalent ways of expressing the number:

Complex Numbers Each Complex Number has a Real Part: Each Complex Number has an Imaginary Part:

Complex Numbers Each Complex Number has a Magnitude: Each Complex Number has a Phase:

Conversion From Rectangular to Polar Form Because the inverse-tangent operation of most calculators returns values in the range – 90    + 90 care must be taken to place the Complex Number in the appropriate quadrant of the Complex Plane.

Conversion From Rectangular to Polar Form

Complex Conjugates For each Complex Number there is a Complex Conjugate that is found by reversing the sign of the Imaginary Part of the Complex number. z = a + jb z* = a – jb

Complex Conjugates In Trigonometric Form, the conjugate pair is: z = r cos( ) + j r sin( ) z* = r cos( ) – j r sin( ) z* = r cos( ) + j r sin(– ) recall sin(– ) = – sin( )

Complex Conjugates In Exponential Form, the conjugate pair is: z = r e j z* = r e – j

Complex Conjugates In Polar Form the conjugate pair is: z = r 

Complex Conjugates When a number is multiplied by its complex conjugate: z z* = (a + jb)(a – jb) z z* = aa – a jb + jba – jb jb z z* = a 2 + b 2 = | z | 2

Complex Conjugates When a number is multiplied by its complex conjugate: z z* = r e j r e – j z z* = r 2 e j( –  ) = r 2 e 0 z z* = r 2 = | z | 2

Addition of Complex Numbers Given the Complex Numbers z1 = a1 +jb1 and z2 = a2 +jb2 the sum of the two Complex Numbers is z1 + z2 = (a1 + a2) + j(b1 + b2).

Addition of Complex Numbers

Multiplication of Complex Numbers Given the Complex Numbers z1 = a1 +jb1 and z2 = a2 +jb2 The product of the two numbers is z1 z2 = (a1 +jb1) (a2 +jb2)

Multiplication of Complex Numbers z1 z2 = (a1 + jb1) (a2 + jb2) z1 z2 = a1 a2 + a1 jb2 + jb1a2 +jb1 jb2 z1 z2 = (a1 a2 – b1b2) + j (a1b2 + b1a2)

Multiplication of Complex Numbers This procedure is much easier to carry out in Exponential Form z1 = r1 e j 1 and z2 = r2 e j 2 z1 z2 =(r1 e j 1)(r2 e j 2) z1 z2 = r1 r2 e j( 1 +  2)

Multiplication of Complex Numbers Or in Polar Form: z1 = r1 1 and z2 = r2 2 z1 z2 = (r1 1)(r2 2) z1 z2 = r1 r2 (1 + 2)

Division of Complex Numbers Given the Complex Numbers z1 = a1 +jb1 and z2 = a2 +jb2 The quotient of the two numbers is

Division of Complex Numbers Multiplying both top and bottom by the complex Conjugate of the denominator:

Division of Complex Numbers Carrying out the multiplications:

Division of Complex Numbers Using the definition j2:

Division of Complex Numbers Separating into real and imaginary parts:

Division of Complex Numbers This procedure is much easier to carry out in Exponential Form z1 = r1 e j 1 and z2 = r2 e j 2

Division of Complex Numbers Using the Law of Exponents:

Division of Complex Numbers In Polar Form:

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