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UE EE 210 - EE210 – Circuits Complex Numbers and Your Calculator

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EE210 – CircuitsComplex Numbers and Your CalculatorTony RichardsonNote: I did not have access to a TI85/86 when this was written. If you have one of these calculators please test the accuracy of the statements here.Let's explore evaluating the following complex number expression on a variety of calculators:240 ∡75˚160∡−30 ˚ 60− j 8067 j 8420 ∡32 ˚Texas Instruments – TI83/84In the MODE menu set the default angle unit to Degree and the default complex format to re^θi (exponential) mode. These calculators allow you to directly enter the imaginary unit i. (This is NOT the same as the alphabetic i key that is also available). They do not allow you to enter complex numbers in polar form. You must use exponential mode instead. Angles in exponential mode can only be entered in radians. So to enter a number that is expressed in polar form (where the angle is in degrees) into the calculator you must convert the angle to radians. An easy way to do this is to multiply the angle by π/180. So to enter, for example, the polar form number (240∡75) into the calculator you must enter 240 e^(i 75 π/180). With the calculator in DEGREE mode this will then display 240 e^(i 75) corresponding to the polar form number (240∡75). These calculators will display complex numbers in exponential form with the angle in degrees, but will not allow you to enter the angle in degrees.You can use the following trick to allow you to enter angles directly in degrees. Store the expression i π/180 as variable I (that is a capital alphabetic i), you can then enter (240∡75) as 240 e^(I*75) and the calculator will then display 240 e^(i 75).With this trick, you can enter the expression above as:((240 e^(I*75)+160 e^(I*-30)) (60 – i80))/((67 + i84) 20 e^(I*32))The calculator then displays:11.709888 e^(i*-99.444742)The ►Rect operator can be used to convert complex numbers to rectangular form, applying this operator to the previous result gives:-1.9215496 – 11.551152 iTexas Instruments – TI85/TI86In the MODE menu set the default Angle unit to DEGREE and the default Complex Format to POLAR. You must enclose complex numbers expressed in polar form in parentheses. A number in rectangular form is entered as (R, I) where R and I are the real and imaginary parts of the number. To enter a complex number representing i, enter (0,1) or (1∡90).11/12/2007 1 of 3When entered in the calculator the expression above looks like this:(((240∡75)+(160∡-30)) (60,–80))/((67,84) (20∡32))The calculator then displays:(11.709888∡-99.444742)The ►Rect operator can be used to convert complex numbers to rectangular form, applying this operator to the previous result gives:-1.9215496 – 11.551152 iTexas Instruments – TI89/TI92/Voyage 200In the MODE menu set the default Angle unit to DEGREE and the default Complex Format to POLAR. You must enclose complex numbers expressed in polar form in parentheses. These calculators allow you to directly enter the imaginary unit i. (This is NOT the same as the alphabetic i key that is also available).When entered in the calculator the expression above looks like this:(((240∡75)+(160∡-30)) (60 – i80))/((67 + i84) (20∡32))The calculator then displays:(11.7098879325∡-99.44474228)The ►Rect operator can be used to convert complex numbers to rectangular form, applying this operator to the previous result gives:-1.92154959538 – 11.5511524336 iMatlab/Scilab/OctaveBy default, these software packages only allow complex numbers to be entered in rectangular form. In Scilab use %i to represent the imaginary unit i. In both Matlab and Octave you can just use i. You can, of course, enter complex numbers in exponential form, but it is convenient to define functions p2z() and z2p(). p2z() allows you to easily enter numbers in polar form and converts numbers to rectangular form. z2p() displays complex numbers in polar form. (It does not convert the number, but just returns the original complex number.) These functions are explained in a separate handout and are available from the course web site.To enter the expression above in Scilab just type:((p2z(240,75)+p2z(160,-30))*(60-%i*80))/((67+%i*84)*p2z(20,32))In both Matlab and Octave the expression would look like:((p2z(240,75)+p2z(160,-30))*(60-80i))/((67+84i)*p2z(20,32))11/12/2007 2 of 3All packages return a result similar to:ans = - 1.9215 - 11.5512iTo display the result in polar form enter:z2p(ans);and the software will display (the semicolon at the end of the line suppresses the display of the answer):11.710 -99.445so in polar form the number is:(11.710∡-99.445)DeriveThe imaginary unit i can be entered by either typing #i or by clicking on the icon that looks something like î. Derive does not support direct entry of complex numbers in polar form. You can use a trick similar to that used for the TI83 calculator to simplify entry of complex numbers in polar form. Define I as i π/180 by entering:I := î·π/180You can then enter our test expression as:((240 ê^(I*75)+160 ê^(I*-30)) (60-î80))/((67+î84) 20 ê^(I*32))Clicking on the Approximate icon yields:#2: -1.921549595 - 11.55115243·î(The #2 is the tag assigned to the result.) We can display the components of the corresponding polar form number by entering:[abs(#2), phase(#2)*180/π]and clicking on the Approximate icon. Derive responds with:#3: [11.70988793, -99.44474227]11/12/2007 3 of


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UE EE 210 - EE210 – Circuits Complex Numbers and Your Calculator

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