3/31/2020USF Physics 1011 Physics 101 AC Circuits

3/31/2020USF Physics 1012 Agenda Administrative matters –EVB is still ill –Homework due today? AC Circuits –AC in R, L, C Phase Shifts Filters –Series LCR Circuit Phasors –Parallel LCR Circuit –Resonance

3/31/2020USF Physics 1013 AC Circuits Note that I could just as well use cos with a different  ’ =  + 90° or  2 rad. sin lags cos by 90° We will assume a sinusoidal voltage source which supplies a current

3/31/2020USF Physics 1014 R only: I and V are in phase Energy is transformed onto heat

3/31/2020USF Physics 1015 L Only: In an inductor the current lags the voltage in phase by 90°. Alternatively, the voltage leads the current by 90°. On the average, no energy is transformed into heat

3/31/2020USF Physics 1016 Flow of charge impeded by back EMF as energy is stored in L In analogy to Ohm’s Law The inductive reactance Units:  Notes: V 0 and I 0 are peak values. Can also write V and I do not peak at the same time so V ≠ I X L at a particular time. For a resistor V = IR  t. X L = 0 for DC (  = 0) (End of previous)

3/31/2020USF Physics 1017 Example: Inductive reactance of a coil: R = 1.00  L  What is current for (a) 120 VDC, (b) 120 V RMS at 60 Hz Cannot simply add R and X L. There are phase considerations. (Later)

3/31/2020USF Physics 1018 C only: In an capacitor the current leads the voltage in phase by 90°. Alternatively, the voltage lags the current by 90°. Again = 0. On average, no energy → heat

3/31/2020USF Physics 1019 Flow of charge impeded by back EMF as energy is stored in C In analogy to Ohm’s Law The capacitive reactance Units:  Notes: V 0 and I 0 are peak values. Can also write V and I are not in phase so V ≠ I X L at a particular time. X C =  for DC (  = 0)

3/31/2020USF Physics Example: Peak and RMS currents in C = 1.0  F, V RMS = 120 V for (a) f = 60 Hz and (b) 600 Khz

3/31/2020USF Physics Filters: High pass Low pass Replace Cs with Ls

3/31/2020USF Physics Series LCR Circuit: DD At any time t, loop rule  Continuity  currents same in all elements at any time t Consequence: everywhere in the series circuit. Because of their phase differences, the voltages add in a more complicated fashion. 2 Approaches: Complex variables Graphical analysis, phasors

3/31/2020USF Physics Phasors: Represent voltages as vectors in a plane t = 0 Length of each arrow = peak V  gives phase w.r.t. I Let this diagram rotate, angular velocity 

3/31/2020USF Physics The vector sum of these voltages is the voltage across the whole circuit. Source V is out of phase with I by  Define impedance, Z Pythagoras 

3/31/2020USF Physics Phase angle Power dissipated cos  is called the power factor R alone:  = 0, cos  = 1 L or C alone:  = ± 90 °, cos  = 0, no power dissipated

3/31/2020USF Physics Example: In series, R = 25.0 , L = 30 mH and C = 12.0  F. Driven by 90.0 V RMS at 500 Hz. (a) current in circuit, (b) voltmeter (RMS) reading across each element, (c) phase angle and (d) power dissipated.

3/31/2020USF Physics (b) RMS voltage across elements Note that these do not add up to 90 V. They are not in phase. Instantaneous voltages do add up to source voltage at that instant. (c) Phase angle (d) Power dissipated

3/31/2020USF Physics Parallel LCR Circuit Phases differ by 90 ° Here the voltage across each element is just the source voltage at any time t with no phase differences. The currents are not in phase but must obey the node rule at any point in time to preserve continuity.

3/31/2020USF Physics I R0 I C0 I L0 tt I L0 – I C0 I R0 I0I0  Pythagorean Theorem 

3/31/2020USF Physics (for a parallel circuit} Full expression for current at any time t I L0 – I C0 I R0 I0I0  Note that here (parallel LCR)  is actually a retardation rather than an advancement (series LCR).

3/31/2020USF Physics Resonance in LCR Circuits (revisited): DD Series

3/31/2020USF Physics DD Parallel Now Z is a maximum at resonance and the current goes through a minimum.

3/31/2020USF Physics Series Parallel Band pass filter “notch” filters Band block filter

3/31/2020USF Physics Example: 1040 kHz radio receiver. L = 4.0 mH, what C is needed Use a series LCR circuit, Want maximum or resonance at 1040 kHz