ECE 222 Electric Circuit Analysis II Chapter 1 Mathematical Formulae & Identities

1. ECE 222Electric Circuit Analysis II Chapter 1Mathematical Formulae & Identities Herbert G. Mayer, PSU Status 5/1/2016 For use at CCUT Spring 2016

2. Syllabus • Identities • Differentiation of f( x ) • Differentiation of f( ex ) • Integrals of f( ex ) • Integrals of f( 1/x) • Radian • Euler’s Identity & Formula • Bibliography

3. Identities √1 = ± 1 √1 = 11/2 11/2 = ± 1 √-1 = ± j-- E. Engineers use ‘j’, not ‘i’ -11/2 = ± i -- Mathematicians use ‘i’ 1¼ = ± 1, and ± j -- EEs use ‘i’ for current! -j = 1 / j -- proof: 1/j = j/( j * j ) = j/-1 = -j 1! = 1 -- 1 factorial 0! = 1 -- 0 factorial e = Σ 1 / n! -- n = 0 to ∞ = 2.718281828… e x = Σ xn / n! -- n = 0 to ∞ e j π + 1 = 0 -- Euler’s identity, see later

4. Identities e j π = -1 -- Euler’s identity, rewritten! e = ( 1 + 1/n )n-- for n -> ∞ E = 2.718281828 . . . -- Removal of complex denominator: x = ( a + j b ) / ( c – j d ) x = ( a + j b ) ( c + j d ) / ( ( c – j d ) ( c + j d ) ) x = ( a + j b ) / ( c – j d ) / ( c2 + d2 ) x = ( ac + bd + j ( bc – ad ) / ( c2 + d2 ) x = ( ac + bd ) / ( c2 + d2 ) + j ( bc – ad ) / ( c2 + d2 ) -- now only real denominator!

5. Trigonometric Identities sin( α + β ) = sin( α ) cos( β ) + cos( α ) sin( β ) sin( α - β ) = sin( α ) cos( β ) - cos( α ) sin( β ) cos( α + β ) = cos( α ) cos( β ) - sin( α ) sin( β ) cos( α - β ) = cos( α ) cos( β ) + sin( α ) sin( β ) sin( 2 α ) = 2 sin( α ) cos( α ) cos( 2 α ) = cos2( α ) - sin2 ( α ) sin( α )’ = cos( α ) -- first derivative ’ w.r.t. α cos( α )’ = -sin( α ) -- first derivative ’ w.r.t. α tan( α + β ) = ( tan( α ) + tab( β ) ) / ( 1 - tan( α ) tab( β ) )

6. Identities ( x + y )2 = x2 + 2 x y + y2 ( x + y )4 = x4 + 4 x3 y + 6 x2 y2 + 4 x y3 + y4 . . . Pascal’s Triangle

7. Identities: Simplification Rules for f(x), with ^ for Exponentiation, and & x for ln(x)

8. Differentiation of f(x) = f( u, v )

9. Differentiation of f( ex ) f1(x) = ex f1(x)’ = ex = e^x -- ^ different notation f2(x) = 5 x e2x -- use product rule f2(x)’ = 5 e2x + 5 x 2 e2x f2(x)’ = 5 e2x + 10 x e2x = 5 e2x ( 1 + 2 x ) f3(x) = e-x/2 sin( a x ) -- use product rule f3(x)’ = (-1/2) e-x/2 sin( a x ) + a e-x/2 cos( a x )

10. Integrals of f( ex ) ex dx = ex -- 1. ecx dx = ecx/c -- 2. x ecx dx = x/c ecx - 1/c2 ecx -- 3. prove! x ecx dx = ( x/c - 1/c2 ) ecx-- 3’. x2 ecx dx = ( x2/c - 2x/c2 + 2/c3 ) ecx -- 4. x3 ex dx = ( x3 - 3x2 + 6x - 6 ) ex -- 5. xn ecx dx = xn ecx / c - n/c xn-1 ecx dx -- 6. x3 ecx dx = exercise in class -- 7.

11. Integral of f( x ecx ) Informal proof for 3: Starting with g(x) = x ecx dx, assume g(x), find derivative g(x)’, and verify that in fact the result is as assumed: g(x) = x ecx dx g(x) = x/c ecx - 1/c2 ecx-- guess you know! g(x)’ = c x/c ecx - 1/c ecx + 1/c ecx g(x)’ = x ecx -- q.e.d.

12. Integral of f( x3 ecx ) Exercise xn ecx dx = xn ecx / c - n/c xn-1 ecx dx For n = 3 it follows: x3 ecx dx = x3 ecx / c - 3/c x2 ecx dx But we know that: x2 ecx dx = ( x2/c - 2x/c2 + 2/c3 ) ecx x3 ecx dx = x3 ecx /c - 3/c ( x2/c - 2x/c2 + 2/c3 ) ecx x3 ecx dx = ecx ( x3/c - 3 x2/c2 + 6 x/c3 - 6/c4 )

13. Integrals of f( 1/x) 1 / xdx = ln | x | -- singularity at x=0 1 / ( x + a )2 dx = -1 / ( x + a ) Prove via quotient rule demonstration: d( -1 / ( x + a ) ) / dx= ( 0 * ( x + a ) - ( -1*1 ) ) / ( x + a )2 d( -1 / ( x + a ) ) / dx= 1 / ( x + a )2-- q.e.d. x / ( a2 + x2 )dx = ½ ln | a2 + x2 | x3 / ( a2 + x2 )dx = ½ ( x2 - a2 ln | a2 + x2 | )

14. Radian Radian another way of specifying parts of circle Common way: circle divided into 360 degrees Def: A radian is that angle of a circle's subtended part that encloses a circumference portion of length exactly equal to the radius Convert degrees to radians: A total circle's radian is defined to be: 2 π Or a half-circle’s radian is π A one degree angle is equal to π / 180o radians Convert radians to degrees: One radian spans an angle of 360o / 2 π = 180o / π Or 2 π radians are 360o degrees

15. Euler’s Identity & Formula Euler’s Formula is one of the most beautiful, simple and complex formulae; derive Euler’s Identity! See [2] Euler uses i for the imaginary axis of complex numbers; we’ll use j in ECE 222, to avoid confusion with current i e j π + 1 = 0 e base for natural logs, AKA Euler’s Number, e = 2.718281828... j imaginary unit, such that j2 = -1 π another transcendental number, π = 3.14159265... And for any real number x, Euler’s Formula is: e j x = cos( x ) + j sin( x )

16. Euler’s Identity Using Euler’s Formula: e j x = cos( x ) + j sin( x ) More generally, noting the sign: e ± j x = cos( x ) ± j sin( x ) Then for x = -90o e -j 90 = cos( -90o ) + j sin( -90o ) e -j 90 = 0 + j sin( -90o ) e -j 90 = - j

17. Bibliography • Differentiation rules: http://www.codeproject.com/KB/recipes/Differentiation.aspx • Euler’s Identity: https://en.wikipedia.org/wiki/Euler%27s_identity • Table of integrals: http://integral-table.com/downloads/single-page-integral-table.pdf • Radian: https://www.khanacademy.org/math/geometry/cc-geometry-circles/intro-to-radians/v/introduction-to-radians