Justin Math Presentation Rev1.2

FROM ANCIENT GREECE TO CELL

PHONES

Mathematics Used Everyday in Modern Electronics

byDavid and Justin Sorrells

Copyright David F and Justin W Sorrells, 2011

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)


Cartesian CoordinatesRenè Descartes, France, 1600 CE

X axis

Y axis 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

(1,3)3

1


Unit Circle Radius = 1

X axis

Y axis

The Unit CircleGupta Period, India, 550 CEPythagoras, Greece, 490 CE

(1,0)

(0,1)

(-1,0)

(-1,-1)

Symbol S1

x2 + y2 = h2 = r2 = 1

Unique x and y coordinates can be expressed as Polar coordinates (r,θ) x

yr

θ


Cartesian/Polar Coordinates to Trigonometric IdentitiesHipparchus, Greece, 2 CE

Unit Circle Radius = 1

cos(θ)

sin(θ)

x

yr

θ

Identities: x = r * cos(θ) y = r * sin(θ) Since x2 + y2 = r2

--and-- r = 1--then– sin2(θ)+cos2(θ) = 1


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.

Real axis

Imaginary axis

(1,i3) or 1+i3i3

1


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

cos(θ)

jsin(θ) Unit Circle Radius = 1

V

θcos(θ)

jsin(θ)


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. cos(θ)

jsin(θ) Unit Circle Radius = 1

V

θcos(θ)

jsin(θ)


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?


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.


Easy as Pi 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

Real Axis

ImaginaryAxis

R

j2πfL

-j2πfC


Ohm’s Law (one more simple equation)Georg Ohm, Germany, 1827 CE

Ohm’s Law for Direct Current (DC): Voltage = Current * Resistance V = i * R

Ohm’s Law for Alternating Current (AC): Voltage = Current * Impedance V = i * Z

Impedance is a complex parameter defined as Re+jX


A Real (and Imaginary) Example

VSinC

VSinInput

VtSineSRC1

Phase=0Damping=0Delay=0 nsecFreq=1 GHzAmplitude=1 VVdc=0 V

CC1C=1.0 pF

LL1

R=L=10 nH

RR1R=50 Ohm

Consider the following circuit:From Ohm’s law we know:VsinInput = i * Z

VsinInput = R*i + jXl*i - jXc*i

Z = R + jXl – jXc

f = 1 Ghz (1*109)

R = 50 ohms

Xl = j2* π*f*10nH (10*10-9) = j62.83 ohms

Xc = -j2* π*f*1pF (1*10-12) = -j159.16 ohms

Let’s calculate the voltage across the capacitor


Step 1: Plot the Complex Impedance (Z)

Z = 50 + j62.83 – j159.16Z = 50 – j96.33

Zmag=108.53

-62.57deg

Xl = j62.83

Xl = -j159.16

R = 50 =

Zmag = 502 – j96.332 = 108.53

θ = -tan-1(96.33/50) = -62.57deg


Step 2: Calculate the Complex Current i = VSinInput / Z

i = 1 / (108.53 -62.57)

i = 9.214x10-3 62.57Zmag=108.53

-62.57deg

imag=9.214x10-3+62.57deg


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


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

Vc_mag=1.466

-27.43deg

VSinInput


Let’s Check our Work

VSinC

VSinInput

TranTran1

MaxTimeStep=1psStopTime=10 nsec

TRANSIENT

VtSineSRC1

Phase=0Damping=0Delay=0 nsecFreq=1 GHzAmplitude=1 VVdc=0 V

CC1C=1.0 pF

LL1

R=L=10 nH

RR1R=50 Ohm

2 4 6 80 10

-1

0

1

-2

2

time, nsec

VSinI

nput,

VVS

inC, V

Readout

m1

time,

sec

m1time=VSinC=1.466

5.322nsec

We calculated Vc as 1.466V -27.43

Correct!


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


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

Documents

Justin Math Presentation Rev1.2