Introduction to Complex Numbers AC Circuits I

Outline Complex numbers basics Rectangular form – Addition – Subtraction – Multiplication – Division Rectangular to Polar form conversion Polar form – Multplication – Division Euler’s formula Polar form to Rectangular form conversion 2AC Circuits I

Solutions to Quadratic Equations What are the roots of the following quadratic equation ? Roots are complex!!!! What does this mean? 3AC Circuits I

The meaning of i i represents Mathematicians use i to represent Since i already represents current, from now on we shall use j instead of i i.e. 4AC Circuits I

The complex plane Imaginary axis Real axis 5AC Circuits I

Complex, real and imaginary A complex number is typically comprised of a real and imaginary portions i.e. X 1 is a complex number -1 is the real number is the imaginary number 6AC Circuits I

Examples Simplify the following: – 7AC Circuits I

Rectangular Representation of Complex Numbers Complex numbers can be represented in both rectangular and polar formats Lets start with the Rectangular formats Addition - – E.g. 8AC Circuits I

Rectangular Representation of Complex Numbers Subtraction – – E.G. 9AC Circuits I

Rectangular Representation of Complex Numbers Multiplication – E.g. 10AC Circuits I

Rectangular Representation of Complex Numbers Division – In dividing a+jb by c+jd, we rationalized the denominator using the fact that: The complex numbers c+jd and c−jd are called complex conjugates If z = c+jd then z * =c-jd 11AC Circuits I

Rectangular Representation of Complex Numbers Division Example 12AC Circuits I

Polar Representation of Complex Numbers Consider the point represented by 4+j6 in the complex plane θ r x = r cos θ y =r sin θ Can be represented as a coordinate i.e. (4+j6) or via r (radius) and angle θ i.e. Or 13AC Circuits I

Converting from Rectangular to Polar For a complex number x+jy: Note that θ is defined -  to  θ r x = r cos θ y =r sin θ 14AC Circuits I

Converting from Rectangular to Polar Example – Convert 4+j6 to polar notation Plot the polar points 6+j8, 4-j3, -2-j6 on the complex plane and then convert these complex values to their polar form 15AC Circuits I

Polar Representation of Complex Numbers In polar number representation, we can only multiply and divide. Multiplication – E.g – easy!! Try: 16AC Circuits I

Polar Representation of Complex Numbers Division – Easy! E.g. 17AC Circuits I

Euler’s Formula can be written as: A complex number in terms of this formula can be written as: This allows us to convert a complex number in polar form back to rectangular form!!!! Euler’s Formula 18AC Circuits I

Converting from Polar to Rectangular Convert the following complex number from polar form to rectangular form: Try converting these o, o, 19AC Circuits I

Calculators Many scientific and graphic calculators have built in complex number functionality. Use it to aid you in your calculations. But make sure you know how to manipulate complex numbers without a calculator as well!!! 20AC Circuits I