ECEN3714 Network Analysis Lecture #6 26 January 2015 Dr. George Scheets n Read 13.5 n Problems: 13.8, 10, 12 n Friday, Quiz #2 n Exam #1, Lecture 11, Friday 6 February Open Book & Notes, Closed Instructor

ECEN3714 Network Analysis Lecture #7 28 January 2015 Dr. George Scheets n Read 13.7 n Problems: Old Quiz #2 n Friday, Quiz #2 n Exam #1, Lecture 11, Friday 6 February Open Book & Notes, Closed Instructor n Quiz #1 Results Hi = 10, Low = 6.0, Average = 8.31 Standard Deviation = 1.41

Laplace Transform X(s) = x(t) e -st dt 0-0- ∞ n Equation above can be used to find answer n Table 13-1 & 13-2 usually easier

3 Circuit Equations to Remember n V = iR(Resistor) n V = L (di/dt)(Inductor) u Resists changes in current u L ≈ 0 for DC (0 Hertz) u L ≈ ∞ for very high frequencies n I = C (dv/dt)(Capacitor) u Embraces changes in the current u C ≈ 0 for very high frequencies u C ≈ ∞ for DC (0 Hertz)

RC Filter R Ω y(t) C μf x(t) n Voltage Transfer Function H(s) = Y(s)/X(s) = R/[R + 1/(Cs)] n Frequency Response: Let s = σ + jω with σ = 0 H(jω) = Y(jω)/X(jω) = R/[R + 1/(Cjω)] n High Pass or Low Pass Filter? u Capacitors embrace change

RC Filter H(jω) = R/[R + 1/(Cjω)]

V(s) = 10s/D(s) D(s) = s s + 25 = (s + 5) 2 Re(s) D(s) -5 n Denominator has two real roots at s = -5 n Using Partial Fraction Expansion, can write as V(s) = -50/(s+5) /(s+5) u These two terms have transform pairs in the tables

V(s) = -50/(s+5) /(s+5) ↕ v(t) = -50te -5t + 10e -5t t v(t) 10

V(s) has 2 nd order pole at -5 Zero at 0 Re(s) V(s) = 10s/(s+5) 2 -5 Re(s) = σ Im(s) = jω x x -5 |V(s)| The figure above is a plot along the real axis.

3D Plot of V(s) = 10s/(s+5) 2 ω σ ω = 0 axis = real axis

v(t) = -50te -5t + 10e -5t t v(t) 10 Re(s) = σ Im(s) = jω x x -5 |V(s)| Frequency Content along σ = 0 axis

V(s) = 10s/D(s) D(s) = s 2 + 2s + 25 Re(s) D(s) n Roots exist even if equation doesn't hit zero u They'll be complex

Need to find Roots for as 2 + bs + c n Use Quadratic Equation n Roots at [-b + √ (b 2 - 4ac) ] / 2a

Leonhard Euler n Born 1707 n Died 1783 n Swiss Mathematician & Physicist n Introduced modern notation for trig functions & complex numbers (phasors) n Considered to be pre-eminent mathematician of 18th Century source: Wikipedia