I) What are complex numbers?

 

A.  Numbers that can be expressed as A + jB

where  and A and B are real numbers. 

                                or

          B.  numbers that can be expressed as  where C and  are real numbers. 

 

II)  How are definitions A and B related?

By Euler's theorem:

                     

                                     

                                      =A+jB

 

Proof :

              

                                   

 

III) What good are they?

           

When you have to solve differential equations relating harmonic (wave-like) functions they make

life much easier.

 

In particular, if

            

 

Thus,         

 

This means that differentiation and integration can

be (usually) replaced by multiplication and division!

 

IV)  Anything else?  Of course! 

 

If    then         

 (look familiar?).

Thus a wave, for example, can be written as

 

 

,   etc.

 

We will explore this part more in quantum mechanics.

 

V)  Do complex numbers have a geometric representation?

Well, that’s actually how they arose...

let  

Because of Euler's formula we can represent the

real and imaginary parts of Z as the components of a vector in the complex plane.

Text Box: Imaginary Axis  

 

 

 

 

 

 

 

 

 

… may be broken into its constituent parts

 
 

 


 

 

 

z 

 

           

 

 

 

 

 

VI) How do I find the Magnitude of Complex Numbers…

Answer: Its easy, just multiply the number by its complex conjugate (and then take the square root). 

 

But…what is a complex conjugate and how do I do it?

Okay…if  ,

Then: 

 

So to get the complex conjugate you change all the j’s to –j’s..that's  it.  

So…what’s the point?

 

Well, if you multiply Z*Z,  you get:

.   This means that you can find the magnitude of a complex number by taking the square root of Z*Z.

 

Example:  How can you use complex numbers to derive double slit  intensity distribution? 

 

A:  Let E1 = Eo, and let E2 = Eo

Then Enet  = E1 + E2 = Eo+ Eo

 

 

 

Since the Intensity  I =

Then we can find I by using E*E  to get |Enet |2

So…


 

 

 

 

 


Problems Using Complex Numbers

 

Work through the following examples below.  Turn your work in on paper, showing all your steps (even if the work seems obvious). 

 

Exercise #1   Show that the formula for adding two traveling waves of equal amplitude and frequency, but traveling in opposite direction is:

                                y = y1 + y2 = 2Aejkxcos (wt)

 

  Where        y1 = Aej(kx -wt)  and y2= Aej(kx +wt )

 

show that if we choose the imaginary part of y for our “real” solution, then the answer is the familiar standing wave equation as described in the text (HR & W Chapters 17 and 18).   

 

Exercise #2  Show that the complex number

                        Z = 1/jw

                        can be written as

                         Z = (1/w)e-jp/2

 

Exercise #3   

(a)   A circuit  has only an inductor of inductance  L , and a  Sinusoidal  AC power supply with amplitude E. 

Apply Kirchoff’s laws to determine a complex expression for the current in the circuit.    Hint: Let the voltage source be represented by   e = Eejwt  and recall that the voltage drop across the inductor is –L dI/dt.  Integrate (using the rules for integrating complex #’s) to find I. 

 

  (b) Show that the value of I you determined in part (a) can be written as

                        I = Ioejwt-jf   where f= p/2, and Io = E/wL.  

 

Exercise #4  Repeat exercise #2 and #3 with the inductor replaced by a capacitor...... only  this  time, show that f= -p/2, and Io = E wC.  

Hint:  this time the voltage drop on the capacitor is -q/C, where q is equal to the time integral of I(t). 

 

 

Exercise #5.  Just like the total resistance in a circuit, the total impedance in a circuit  can be used to determine the current flowing in or out of the power supply. 

 

 


                                       I = e/Z  = Ioej(wt + f)     (Equation #1) 

     I

     

e                     Z

 

 

 

Now Consider for example a circuit with complex impedance

                                         Z = 100ejp/6

 

If the power supply can be described the equation

                                        e = 10ej(30pt)

Determine the value of Io and f.