ENGS2613 Intro Electrical Science Week 12 Dr. George Scheets Read: 7.2, 9.1, & 9.2 Problems: 7.1, 14, 22, 29, & 30

Capacitors Capacitor Open Circuit if DC in ↕ Short Circuit if Very Hi Freq in i = C dv/dt v(t) - v(0-) = 1 i dt C i i + + - - ∫ t Absorbing Power (Voltage is Increasing) Releasing Power (Voltage is Decreasing) + - i → Note Reference Current Direction

Inductors Inductor Short Circuit if DC in ↕ Open Circuit if Very Hi Freq in v = L di/dt i(t) - i(0-) = 1 v dt L i i + - ∫ t - + Absorbing Power (Current is Increasing) Releasing Power (Current is Decreasing) + - i → Note Reference Current Direction

Electrical Material Chapters 1 – 5 R's & Circuit Behavior at DC (Constant I & V) Chapter 6 – 7 L's & C's (Time Varying i & v) i = i(t) and v = v(t) Chapter 9 Sinusoids Any time domain waveform = sum of sinusoids Resistors Treat all sinusoids the same L's & C's Don't

Cycles/Second Heinrich Hertz 1857 - 1894 German Physicist Around 1887 Proved existance of EM waves University of Karlsruhe Units of Hertz = Sinusoid Cycles/Second Source: Wikipedia

Frequency a(t)sin[2πft + θ(t)] or a(t)cos[2πft + θ(t)] Frequency = value of f, units of Hertz (cycles/sec) U.S. Electricity mostly uses 60 Hz sinusoids Wall socket ≈ 120 Vrms ≈ 170 Vpeak = 340 Vp-p Smart Phone radios are up around 1 GHz Any signal can be constructed by adding together the appropriate sinusoids

5 Hertz Square Wave... 1 volt peak, 2 volts peak-to-peak, 0 V average 1.5 -1.5 1.0 1 volt peak, 2 volts peak-to-peak, 0 V average

Generating a Square Wave... 1.5 -1.5 1.0 1 vp 5 Hz 1.5 -1.5 1.0 1/3 vp 15 Hz

Generating a Square Wave... 1.5 5 Hz + 15 Hz -1.5 1.0 1.5 -1.5 1.0 1/5 vp 25 Hz

Generating a Square Wave... 5 Hz + 15 Hz + 25 Hz 1.5 -1.5 1.0 1.5 -1.5 1.0 1/7 vp 35 Hz

Generating a Square Wave... 5 Hz + 15 Hz + 25 Hz 35 Hz 1.5 -1.5 1.0 cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)

Generating a Square Wave... 1.5 -1.5 1.0 5 cycle per second square wave generated using first 50 cosines, Absolute Bandwidth = 495 Hertz.

Generating a Square Wave... 1.5 -1.5 1.0 5 cycle per second square wave generated using first 100 cosines, Absolute Bandwidth = 995 Hertz.

Time Varying Waveforms High Frequencies Rapidly Changing Waveforms Low Frequency content Slowly Changing Wavefom Could be a single sinusoid Could be a arbitrary waveform shape = ∑ (bunch of sinusoids) Note: DC = 0 Hertz sinusoid 5 V DC = 5cos(2π0t) volts

Sinusoidal Behavior Can construct any time domain signal Add together appropriate sine waves & cosine waves Appropriate frequencies & amplitudes How do sinusoids react with Resistors? Capacitors? Inductors?

Resistor: v = iR Suppose time varying current thru resistor i = 2cos(2πft) amps If R = 10 Ω, then voltage across resistor v = 20cos(2πft) volts Note: Voltage & Current are time aligned Sinusoid frequency doesn't affect v/i ratio If one freq has v/i ratio = 10, they all do

100 Hz Cosine thru Resistor

200 Hz Cosine thru Resistor

Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t 5 Hz + 15 Hz + 25 Hz 35 Hz 1.5 amps -1.5 1.0 i(t) = cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t) Run this current into a resistor network? Voltage shape will be… identical.

Inductor: v = L di/dt Suppose time varying current thru inductor i = 2cos(2πft) amps If L = 0.1 H, then voltage across inductor v = -0.4πf sin(2πft) volts Note: Voltage & Current are 90º out of phase Sinusoid frequency does affect v/i ratio Voltage drop at 200 Hz is 2x drop at 100 Hz Inductor impedance at 200 Hz = 2x that at 100 Hz Higher frequencies attenuated more than lower freqs

100 Hz Cosine thru Inductor Voltage leads Current by 90º =1/4 wavelength = (1/4)(1/100) = 1/400 second

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

Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t 5 Hz + 15 Hz + 25 Hz 35 Hz 1.5 amps -1.5 1.0 i(t) = cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t) Run this thru an inductor & shape will… … change. Amplitude attenuation & time shift differs with freq.

Square-ish Wave Current 1.5 thru 0.1 H Inductor v = L di/dt amps -1.5 -1.5 1.0 Voltage across 0.1 H Inductor

∫ Capacitor: v = (1/C) i dt t Capacitor: v = (1/C) i dt Suppose time varying current thru inductor i = 2cos(2πft) amps If C = 0.1 F, then voltage across inductor v = 10sin(2πft)/πf volts Note: Voltage & Current are 90º out of phase Sinusoid frequency does affect v/i ratio Voltage drop at 200 Hz is 1/2 drop at 100 Hz Capacitor impedance at 200 Hz = 1/2 that at 100 Hz Higher frequencies attenuated less than lower freqs

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

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

Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t 5 Hz + 15 Hz + 25 Hz 35 Hz 1.5 amps -1.5 1.0 i(t) = cos2*pi*5t - (1/3)cos2*pi*15t + (1/5)cos2*pi*25t - (1/7)cos2*pi*35t) Run this thru an capacitor & shape will… … change. Attenuation & time shift differs with freq.

Square-ish Wave Current 1.5 thru 0.1 F Capacitor v = amps (1/C) ∫ i dt -1.5 1.0 Voltage across 0.1 F Capacitor

Passive Devices Resistors Inductors Capacitors Treat all electrical frequencies identically v & i show no phase shift Inductors High freqs show larger voltage drop than low freqs Attenuate high frequencies more than low freqs v leads i by 90° (or v lags i by 270°) Capacitors Low freqs show larger voltage drop than hi freqs Attenuate low frequencies more than high freqs i leads v by 90° (or i lags v by 270°) NOTE: Output frequency f = input frequency

Inductors & Capacitors Are frequency selective Provide opportunity to change signal's shape Unwanted noise has different freqs than desired signal? Can filter out noise RF signal received distorted? Can undo much of this distortion (if characteristics known) These are examples of frequency selective filtering How do we analyze this? Phase shift Frequency selectivity

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