ENGS2613 Intro Electrical Science Week 13 Dr. George Scheets Read: 9.3 – 9.5 Problems: 9.1, 2, 7, 12, 17, 28 Quiz 7 this Friday: Discharging RL or RC circuit

Leonhard Euler 1707 – 1783 Swiss Mathematician Considered one of History's Greatest Mathematicians Published & proved ejθ = cos(θ) + jsin(θ) in 1748 source: Wikipedia

Phasor Representation of a Time Domain Cosine Useful for analyzing steady state response Time Domain Waveform: Vp cos(ωt + θ) Got a sine wave? Convert to a cosine via sin(ωt + θº) = cos(ωt + θº - 90º) Phasor Representation = Vp ∟θ = Vp ejθ = Vp Contains amplitude and cosine phase info Does not directly contain frequency info Phasors are vectors Can be plotted on real [cosθ)] & imaginary [jsin(θ)] axis via Vp [cos(θ) + jsin(θ)]. Vector tail at origin? Tip is at coordinates (Vpcos(θ), Vpsin(θ)). θ = angle in radians, θº = angle in degrees

Phasor Addition Example

100 Hz Cosine thru Resistor Current thru a Resistor Have same phase shift Voltage across a Resistor

200 Hz Cosine thru Inductor Current thru an Inductor Voltage leads Current by 90º =1/4 wavelength = (1/4)(1/200) = 1/800 second Voltage across an Inductor

100 Hz Cosine thru Capacitor Current thru an Capacitor Current leads Voltage by 90º =1/4 wavelength = (1/4)(1/100) = 1/400 second Voltage across a Capacitor

Time Domain Format Resistor: v = i R Inductor: v = L di/dt Capacitor: i = C dv/dt Captures entire circuit behavior Transient Response Steady State Response Worked Some Examples in Class Capacitor discharged a voltage Inductor discharged a current Exponential Decay

Current vs Voltage Phasor & Time Domain Form Resistor: V = I R v = i R Inductor: V = jωL I v = L di/dt Capacitor: I = jωC V i = C dv/dt

Phasor Format Impedances Resistor: V = I R Inductor: V = jωL I = I jωL Capacitor: I = jωC V → V = I (1/jωC) Captures Steady State Behavior Useful for evaluating sine wave changes Voltage Transfer Function H(jω) ≡ Vout/Vin On circuits with R's, L's, and/or C's Sinusoid amplitudes and phases likely change The frequency does not change

Notes on Terminology Impedance Z Reactance Term that goes in Ohm's Law: V = I Z Reactance Imaginary portion of the impedance Element Impedance Reactance Resistor R - Inductor jωL ωL Capacitor 1/(jωC) -1/(ωC)

Notes on Terminology Element Impedance Z (V = I Z) Resistor R Inductor jωL Capacitor 1/(jωC) Impedances Z in series? Zeq = Z1 + Z2 + … + Zn Impedances Z in parallel? 1/Zeq = 1/Z1 + 1/Z2 + … + 1/Zn

Voltage Transfer Function (RL Highpass Filter)

Transfer Function (RLC Bandpass Filter) Wider pass bands can be obtained with more complicated filters.

A 5 Hz Somewhat Square Wave + 25 Hz 35 Hz 1.5 -1.5 1.0 vin(t) = cos2π5t - (1/3)cos2π15t + (1/5)cos2π25t - (1/7)cos2π35t) cos2π5t → Vin5 = 1∟0º → Vout5 = Vin5*H(5 Hz) = 1∟0º (0.007498∟89.57º) → 0.007498 cos(2π5t + 89.57º) -(1/3)cos2π15t → Vin15 = 1/3∟180º → Vout15 = Vin15*H(15 Hz) = 1/3∟180º (0.02292∟88.69º) → 0.007639 cos(2π15t + 268.7º)

A 5 Hz Somewhat Square Wave + 25 Hz 35 Hz 1.5 -1.5 1.0 vin(t) = cos2π5t - (1/3)cos2π15t + (1/5)cos2π25t - (1/7)cos2π35t) input to this RLC Bandpass Filter has vout(t) = 0.007498 cos(2π5t + 89.57º) + 0.007639 cos(2π15t + 268.7º) + 0.007814 cos(2π25t + 87.72º) + 0.008439 cos(2π15t + 266.6º)

Output