Review of Complex numbers Rectangular Form: Imag π§ = π₯ + ππ¦ π§ r=|z| Exponential Form: π§ = ππ ππ = π§ π ππ y f x Real Real & Imaginary Parts of Rectangular Form The real and imaginary parts of a complex number in rectangular form are real numbers: Imag π§ π π π§ = π₯ πΌπ π§ = π¦ Therefore, rectangular form can be equivalently written as: π₯ + ππ¦ = π π π§ + ππΌπ(π§) y=Im(z) x=Re(z) Real Geometry Relating the Forms The real and imaginary components of exponential form can be found using trigonometry: Imag π§ r=|z| f x Imag π₯ = π cos π πππ π¦ → sin π = = βπ¦π π π¦ = π sin π Real π§ r=|z| y f πππ π₯ cos π = = → βπ¦π π Real Geometry Relating the Forms: Real & Imaginary Parts The real and imaginary parts of a complex number can be expressed as follows: Imag π¦ = πΌπ π§ = π sin π π§ r=|z| π π π§ = π₯ = πcosπ = π§ cos π πΌπ π§ = π¦ = πsinπ = π§ sin π π π₯ = π π π§ = π cos π Real Geometry Relating the Forms: Quadrants In exponential form, the positive angle, π, is always defined from the positive real axis. If the complex number is not in the first quadrant, then the “triangle” has lengths which are negative numbers. Imag π§ r=|z| y πππ |π₯| cos π = = βπ¦π π π₯ = −π cos π = π cos π π x f Imag π₯<0 π¦>0 π₯>0 π¦>0 Real Real π₯<0 π¦<0 π₯>0 π¦<0 Geometry Relating the Forms: π in terms of π₯ and π¦ Imag Use Pythagorean Theorem π§ π2 = π§ r=|z| y x 2 = π₯2 + π¦2 to find π in terms of π₯ and π¦: Real π= π§ = π₯2 + π¦2 Geometry Relating the Forms: π in terms of π₯ and π¦ Use trigonometry hyp opp πππ tan π = πππ π to find π in terms of π₯ and π¦: adj Imag π§ π sin π πΌπ(π§) π¦ tan π = = = π cos π π π(π§) π₯ r=|z| y f x Real π = tan −1 π¦ π₯ Summary of Algebraic Relationships between Forms Imag π§ π₯ = π cos π r=|z| y π¦ = π sin π f x Real π= π§ = π = tan−1 π₯2 + π¦2 π¦ π₯ Euler’s Formula π ππ = cos π + π sin π Consistency argument π§ = π₯ + ππ¦ π§ = ππ ππ = |π§|π ππ If these represent the same thing, then the assumed Euler relationship says: Rectangular Form: π§ = π₯ + ππ¦ = π cos π + ππ sin π = π§ cos π + π π§ sin π = π (cos π + π sin π) = π§ (cos π + π sin π) Exponential Form: = ππ ππ = |π§|π ππ Euler’s Formula π ππ = exp(ππ) = cos π + π sin π Can be used with functions: π ππ0 π‘ = exp(ππ0 π‘) = cos π0 π‘ + π sin π0 π‘ 11 Addition & Subtraction of Complex Numbers Addition and subtraction of complex numbers is easy in rectangular form π§2 = π + ππ π§1 = π + ππ π§ = π§1 + π§2 = π + ππ + π + ππ = (π + π) + π(π + π) Addition and subtraction are analogous to vector addition and subtraction π§2 = π π₯ + ππ¦ π§1 = π π₯ + ππ¦ π§ = π§1 + π§2 = π + π π₯ + (π + π)π¦ y Imag π§1 b a Real c d π§2 π§1 b a c π§ π§2 d π§ π§1 12 x Multiplication of Complex Numbers Multiplication of complex numbers is easy in exponential form π§1 = π1 π ππ π§ = π§1 π§2 = Multiplication by a complex number, π§ππ ππ , can be thought of as scaling by π and rotation by π π§2 = π2 π ππ π1 π ππ π2 π ππ = π1 π2 π π(π+π) = |π§1 ||π§2 |π π(π+π) Imag π§ππ ππ Angle rotated counterclockwise by π Magnitude scaled by π π§ π Real 13 Division of Complex Numbers Division of complex numbers is easy in exponential form π§1 = π1 π ππ π§2 = π2 π ππ π§1 π1 π ππ π§= = π§2 π2 π ππ π1 π(π−π) = π π2 |π§1 | π(π−π) = π |π§2 | = |π§|π π(π−π) Division of complex numbers is sometimes easy in rectangular form π + ππ π§= π + ππ π + ππ π − ππ = π + ππ π − ππ Multiply by 1 using the complex conjugate of the denominator ππ + ππ + π(ππ − ππ) = π 2 + π2 ππ + ππ (ππ − ππ) = 2 +π 2 2 π +π π − π2 = π π(π§) + ππΌπ(π§) 14 Complex Conjugate Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c.: change i → -i Imag π§ = π₯ + ππ¦ π§ ∗ = π₯ − ππ¦ π§ ππ π§ = ππ π§ ∗ = ππ −ππ r=|z| y f Real x The complex conjugate is a reflection about the real axis π§∗ Common Operations with the Complex Conjugate Addition of the complex number and its complex conjugate results in a real number π§ + π§ ∗ = π₯ + ππ¦ + π₯ − ππ¦ = 2π₯ The product of a complex number and its complex conjugate is REAL. π§π§ ∗ = ππ ππ ππ −ππ Imag π§ r=|z| y f x π§+ = π 2 π π(π−π) = π2 = |π§|2 π§∗ π§∗ = 2x Real Phasors A phasor, or phase vector, is a representation of a sinusoidal wave whose amplitude , phase , and frequency are timeinvariant. The phasor spins around the complex plane as a function of time. Phasors of the same frequency can be added. This is an animation But it’s a known fact Phasor Diagram ο΅To simplify the analysis of complex numbers, a graphical constructor called a phasor diagram can be used. ο΅A phasor is a vector whose length is proportional to the maximum value of the variable it represents. ο΅The vector rotates counterclockwise at an angular speed equal to the angular frequency associated with the variable. ο΅The projection of the phasor onto the vertical axis represents the instantaneous value of the quantity it represents. A Phasor is Like a Graph The phase space in which the phasor is drawn is similar to polar coordinate graph paper. ο΅ The radial coordinate represents the amplitude. ο΅ The angular coordinate is the phase angle. ο΅ The vertical axis coordinate of the tip of the phasor represents the instantaneous value. ο΅ The horizontal coordinate does not represent anything. Example of usage: Alternating currents can also be represented by phasors.
