# Parallel Inductor-Resistor-Capacitor (RLC) Circuits

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1 Parallel Inductor-Resistor-Capacitor (RLC) Circuits Session 4b for Basic Electricity A Fairfield University E-Course Powered by LearnLinc 7/9/2002 Basic Electricity 1

2 Module: Basic Electronics (AC Circuits and Impedance: two parts) Text: Electricity One-Seven, Harry Mileaf, Prentice-Hall, 1996, ISBN (Covers much more material than this section) References: Digital Mini Test: Principles of Electricity Lessons One and Two, SNET Home Study Coordinator, (203) Electronics Tutorial (Thanks to Alex Pounds) Electronics Tutorial (Thanks to Mark Sokos) Basic Math Tutorial (Thanks to George Mason University) Vector Math Tutorial (Thanks to California Polytec at atom.physics.calpoly.edu ) Alternating Current and Impedance 5 on-line sessions plus one lab Resonance and Filters 5 on-line sessions plus one lab 7/9/2002 Basic Electricity 2

3 Section 4: AC, Inductors and Capacitors 0BJECTIVES: This section discusses AC voltage / current and their effects on parallel circuit components (resistors, inductors, transformers and capacitors). The concept of resonance and its use to produce filters is also described. 7/9/2002 Basic Electricity 3

4 Section 4 Schedule: Session 4a 07/08 Parallel L-C Circuits Text Session 4b 07/10 (break for a week) Session 4c 07/22 Session 4d 07/24 (lab - 07/27, Sat.) Session 4e 07/29 (Quiz 4 due 08/12) Session 4f 08/12 08/14 08/17 Parallel R-L-C Circuits (no class on 07/15 or 07/17) Parallel Resonance Tuning and Filters Transformers and Impedance Matching Review (Discuss Quiz 4) MT2 Review MT2 AC Circuits Text Text Text Text /9/2002 Basic Electricity 4

5 Session 3 (Parallel L-C) Review Capacitive reactance X C = 1/2πfC at -90º Inductive reactance X L = 2πfL at 90º Impedances in parallel add as inverses Adding Vectors Separately add their horizontal and vertical components Graphically: head-to-tail or parallelogram Here the vectors are in opposite directions; they just subtract. Inductive reactance points up (90º) Capacitive reactance points down (-90º) Multiplying Vectors Multiply their magnitudes (lengths) Add their phases Dividing Vectors Divide their magnitudes (lengths) Subtract their phases Ohm s and Kirchoff s laws still work with AC 7/9/2002 Basic Electricity 5

6 Parallel R-L-C Circuits Voltage (ref. phase) is the same across all parallel components Branch currents add (vectors) to produce I Line Impedances in parallel add (vectors) as inverses 7/9/2002 Basic Electricity 6

7 A Two-Step Solution Solve an RLC in two steps Combine the L and C branches (both vertical) Add the resistor branch to the result (as a vector) 7/9/2002 Basic Electricity 7

8 Parallel RLC- Current AC currents always add as vectors Voltage (ref. Phase) is the same across all parallel components Inductor - I L points down (Lags voltage by 90º) Capacitor - I C points up (Leads voltage by 90º) Add I L and I (they subtract) to get I C Line I L = 1-90º = -1 90º I C = 2 90º I LC = (2-1) 90º = 1 90º (capacitive circuit) Now add I R I Line = ( ) ½ = 10 ½ = I Line = arctan(1/3) = = Radians 7/9/2002 Basic Electricity 8

9 Parallel LC- Current Waveforms AC currents always add as vectors Voltage (ref. Phase) is the same across all parallel components Resistor current is in phase with the voltage Inductor - I L points down (Lags voltage by 90º) Capacitor - I C points up (Leads voltage by 90º) Line current - I Line has a phase between I LC and I R 7/9/2002 Basic Electricity 9

10 Parallel RLC - Impedance Impedances in parallel add as inverse vectors X L (up) and X C (down) are in opposite directions First combine the L and C branches 1/X L = 1/(10 90º )= º 1/X C = 1/(5-90º )= º 1/X LC = º 1/R = º 1/Z = ( ) ½ = ( ) ½ = (0.0125) ½ = /Z = arctan(0.1/0.05) = 63.4º Z = º (capacitive) 7/9/2002 Basic Electricity 10

11 The Effect of Frequency Z L = 2πfL (rises linearly with frequency) Z C = 1/2πfL (decreases with frequency) Resonance is when they are equal and cancel; the impedance is then just the resistance 7/9/2002 Basic Electricity 11

12 First find the Branch currents I C = 220/100 = º I R = 220/500 = º Example I LC = (2.2-1) 90º = º I Line = ( ) ½ = ( ) ½ = (1.6336) ½ = amps I Line = arctan(1.2/0.44) = arctan(2.72) = 69.9º 7/9/2002 Basic Electricity 12

13 First find the Inductor reactance Example (continued) X L = 220 volts/1 amp = º Ohms Now add the impedances as inverses 1/X LC = 1/220 90º + 1/100-90º = º º = º 1/ X t = ( ) ½ = ( ) ½ = ( ) ½ = ( X t = 172 ohms, I line = 220/172 = 1.28 amps) 1/ X t = arctan( /0.002) = arctan(-2.75) = -70º 7/9/2002 Basic Electricity 13

14 Resonance X L and X C cancel Parallel Resonance High Impedance Low line current (high current in the LC loop!) Series Resonance Low impedance High line current Resonant frequency 2πfL = 1/2πfC f = 1/2π(LC) ½ 7/9/2002 Basic Electricity 14

15 Section 4 Schedule: Session 4a 07/08 Parallel L-C Circuits Text Session 4b 07/10 (break for a week) Session 4c 07/22 Session 4d 07/24 (lab - 07/27, Sat.) Session 4e 07/29 (Quiz 4 due 08/12) Session 4f 08/12 08/14 08/17 Parallel L-R-C Circuits (no class on 07/15 or 07/17) Parallel Resonance Tuning and Filters Transformers and Impedance Matching Review (Discuss Quiz 4) MT2 Review MT2 AC Circuits Text Text Text Text /9/2002 Basic Electricity 15

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