1. 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

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

3. 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

4. Phasor Addition Example

5. 100 Hz Cosine thru Resistor
Current thru a Resistor
Have same phase shift
Voltage across a Resistor

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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) /(ωC)

12. 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

13. Voltage Transfer Function (RL Highpass Filter)

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

15. 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 ( ∟89.57º) → cos(2π5t º)
-(1/3)cos2π15t → Vin15 = 1/3∟180º → Vout15 = Vin15*H(15 Hz)
= 1/3∟180º ( ∟88.69º) → cos(2π15t º)

16. 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) = cos(2π5t º) cos(2π15t º) cos(2π25t º) cos(2π15t º)

17. Output