Brief historical perspective on how mathematical principles dating from 2300 to 205 years ago are used to design modern wireless electronic devices such as cell phones. Created to show an algebra 1 math class that thanks to many brilliant mathematicians, simple algebra can be used to calculate and predict electronic circuit behavior.

1. From Ancient Greece to cell phones Mathematics Used Everyday in Modern Electronics by David and Justin Sorrells

2. Euclidean SpaceEuclid of Alexander, Greece, 300BCE Symbol En Every point in 3 dimensional Euclidean Space (E3)can be located or mapped to a unique x, y, and z coordinate value The x, y, and z axes in Euclidean space are Orthogonal (Perpendicular) Copyright David F and Justin W Sorrells, 2011

3. Cartesian CoordinatesRenè Descartes, France, 1600 CE 2 Dimensional Euclidean Space E2 AKA a Plane The Cartesian x and y axes are Orthogonal Every point in a 2 dimensional Cartesian Coordinate Plane can be mapped to a unique x and y coordinate value Y axis (1,3) 3 X axis 1 Copyright David F and Justin W Sorrells, 2011

4. The Unit Circle Gupta Period, India, 550 CEPythagoras, Greece, 490 CE Unit Circle Radius = 1 Symbol S1 x2 + y2 = h2 = r2 = 1 Unique x and y coordinates can be expressed as Polar coordinates (r,θ) Y axis (0,1) r y (-1,0) θ X axis x (1,0) (-1,-1) Copyright David F and Justin W Sorrells, 2011

5. Cartesian/Polar Coordinates to Trigonometric IdentitiesHipparchus, Greece, 2 CE Unit Circle Radius = 1 sin(θ) Identities: x = r * cos(θ) y = r * sin(θ) Since x2 + y2 = r2 --and-- r = 1 --then– sin2(θ)+cos2(θ) = 1 r y θ cos(θ) x Copyright David F and Justin W Sorrells, 2011

6. Complex PlaneHeron of Alexandria, Greece, 10-70 CERafael Bombelli, Italy, 1572 CE Cartesian Coordinates can be expressed as a real axis and an imaginary axis instead of x axis and y axis Named the Complex Plane because of the complex number (1+i3) notation. i = j = -1 ; (1+i3) = (1+j3) In electronics, i is the variable for current so j was chosen to represent complex notation. Imaginary axis (1,i3) or 1+i3 i3 Real axis 1 Copyright David F and Justin W Sorrells, 2011

7. Complex Polar Plane with Unit CircleJean-Robert Argand, France, 1806 CE The notation cos(θ) + jsin(θ) defines the position of V which is known as a Vector Simply by knowing the angle θ on the complex plane, we can describe any Vector by calculating cos(θ) for the x-coordinate and jsin(θ) for the y-coordinate jsin(θ) Unit Circle Radius = 1 V jsin(θ) θ cos(θ) cos(θ) Copyright David F and Justin W Sorrells, 2011

8. Euler Makes another LeapLeonhard Euler, Switzerland, 1783 ejθ = cos(θ) + jsin(θ) With Euler’s formula, we can express any Vector in the complex plane simply by writing ejθ. jsin(θ) Unit Circle Radius = 1 V jsin(θ) θ cos(θ) cos(θ) Copyright David F and Justin W Sorrells, 2011

9. Laplace Ties it all TogetherPierre-Simon Laplace, France, 1800 Laplace Transform Laplace uses Euler’s ejθ relationship and extends it to e-st with s defined as j*2*π*f, which can be expanded to: e-st = -(cos(2*π*f*t) + jsin(i*2*π*f*t)) Now we can define the response of f(t) in terms of frequency instead of θ (angle) Who uses this information? Copyright David F and Justin W Sorrells, 2011

10. Electrical Engineers Electrical Engineers use mathematics that date back 205 to 2300 years to mathematically describe all basic passive electronic components circuit responses using simple algebra in the frequency domain. Time domain Equations Components Laplace Transform Impedance Laplace, and all those before him makes it so that we don’t have to solve differential time domain equations to calculate how resistors, capacitors, and inductors behave at any given frequency. Copyright David F and Justin W Sorrells, 2011

11. Easy as Pi Imaginary Axis The inductive impedance is plotted on the +j or positive imaginary axis The capacitive impedance is plotted on the –j or negative imaginary axis The resistance is plotted on the real axis f = frequency L = inductance C = capacitance R = resistance j2πfL R Real Axis -j 2πfC Copyright David F and Justin W Sorrells, 2011

13. Voltage = Current * Resistance

14. V = i * R

15. Ohm’s Law for Alternating Current (AC):

16. Voltage = Current * Impedance

17. V = i * Z

18. Impedance is a complex parameter defined as Re+jXCopyright David F and Justin W Sorrells, 2011

20. Step 1: Plot the Complex Impedance (Z) Z = 50 + j62.83 – j159.16 Z = 50 – j96.33 Zmag = 502 – j96.332 = 108.53 θ = -tan-1(96.33/50) = -62.57deg Xl = j62.83 = R = 50 -62.57deg Zmag=108.53 Xl = -j159.16 Copyright David F and Justin W Sorrells, 2011

21. Step 2: Calculate the Complex Current i = VSinInput / Z i = 1 / (108.53 -62.57) i = 9.214x10-3 62.57 imag=9.214x10-3 +62.57deg -62.57deg Zmag=108.53 Copyright David F and Justin W Sorrells, 2011

22. Step 3: Calculate the Voltage across the Capacitor From Ohm’s Law: V = i * Z ; and in this case Z is the Impedance of the Capacitor (Zc) Zc = -jXc = -j159.16 or in Polar Coordinates Zc = 159.16 -90 Vc = (9.214x10-3 62.57) * 159.16 -90 Vc = 1.466 -27.43 Copyright David F and Justin W Sorrells, 2011

23. Convert back to Complex Coordinates for Completeness Vc = 1.466 -27.43 Re (aka x) = r * cos(θ) Re = 1.466 * cos(-27.43) Re = 1.301 Im (aka y) = r * sin (θ) Im = 1.466 * sin(-27.43) Im = -.675 Vc = 1.301 - j.675 VSinInput -27.43deg Vc_mag=1.466 Copyright David F and Justin W Sorrells, 2011

24. Let’s Check our Work We calculated Vc as 1.466V -27.43 Correct! Copyright David F and Justin W Sorrells, 2011

25. Result Today we manipulated and solved a 2nd order differential calculus equation using simple algebra and Cartesian coordinates thanks to many brilliant mathematicians dating back to Ancient Greece Copyright David F and Justin W Sorrells, 2011

26. Conclusion Engineers use the mathematical techniques in this presentation to calculate complex voltages, currents, and impedances to design and optimize radio frequency (RF) circuitry. Their goal is to continually improve the distance, coverage, and reliability of one of our most modern devices – Cell Phones Copyright David F and Justin W Sorrells, 2011