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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)
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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)
Copyright David F and Justin W Sorrells, 2011
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)
Copyright David F and Justin W Sorrells, 2011
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
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
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
Copyright David F and Justin W Sorrells, 2011
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
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
Copyright David F and Justin W Sorrells, 2011
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!
Copyright David F and Justin W Sorrells, 2011
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
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