Resonance Circuits DR. GYURCSEK ISTVÁN
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1 DR. GYURCSEK ISTVÁN Resonance Circuits Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 016, ISBN: Ch. Alexander, M. Sadiku: Fundamentals of Electric Circuits, 6th Ed., McGraw Hill NY 016, ISBN: Simonyi K.: Villamosságtan. AK Budapest 1983, ISBN: Dr. Selmeczi K. Schnöller A.: Villamosságtan 1. MK Budapest 00, TK szám: 4903/I Dr. Selmeczi K. Schnöller A.: Villamosságtan. TK Budapest 00, ISBN: Zombory L.: Elektromágneses terek. MK Budapest 006, ( gyurcsek.istvan@mik.pte.hu
2 Series Resonance Circuits Parallel Resonance Circuits Free Resonance
3 Series Resonance Circuit Z eq = R + jωl j 1 ωc Resonance Im Z eq = 0 X L = X C ω r L = 1 ω r C ω r = 1 LC or f 1 r = π LC William Thomson (Lord Kelvin ) 3 gyurcsek.istvan@mik.pte.hu
4 Series RLC Circuit at Resonance Z = R + jx L jx C = R + j X L X C Z = R + ωl 1 ωc 4 gyurcsek.istvan@mik.pte.hu
5 Series RLC Circuit at Resonance I = V Z = V R + j X L X C I = V R + ωl 1 ωc 5 gyurcsek.istvan@mik.pte.hu
6 Phase of Series Resonance Circuit Z = R + j X L X C Z = R + ωl 1 ωc φ Z = tan 1 X L X C R (can be 0!) 6 gyurcsek.istvan@mik.pte.hu
7 Q-factor and Wave Impedance at Series Resonance Q π peak energy stored in the circuit dissipated energy in a period at resonance = π 1 I p L 1 I p RT = ω rl R = X r R = 1 ω r RC Q = ωl R ω = RQ L Q = 1 ωcr ω = 1 QCR RQ L = 1 QCR Q = 1 R L C = Z 0 R Z 0 = L C 7 gyurcsek.istvan@mik.pte.hu
8 Bandwidth at Series Resonance BW = f H f L ω r = 1 LC Z = min; I S = max P max I max Z = R X L X C = R Z = R + X L X C = R ቐ X C X L = R ω L = R L + R L + 1 LC ω H = + R L + R L + 1 LC 8 gyurcsek.istvan@mik.pte.hu
9 BW and Q-factor at Series Resonance ω L = R L + R L + 1 LC ω H = + R L + R L + 1 LC a + b a b = a b ቑ ω r = 1 LC ω r = ω L ω H Q = ω rl R R L = ω r Q BW = f H f L = 1 R π L = f r Q 9 gyurcsek.istvan@mik.pte.hu
10 Voltage Resonance Series resonance voltage resonance V L = V C = QV S V L = IX L = V S R X L = ωl R V S = QV S V C = IX C = V S R X C = 1 ωcr V S = QV S 10 gyurcsek.istvan@mik.pte.hu
11 Series Resonance Circuits Parallel Resonance Circuits Free Resonance
12 Parallel Resonance Circuit Resonance Im Y eq = 0 Y = 1 R + 1 jωl + jωc Y = G jb L + jb C Admittance, Y = 1 Z = G + B Conductance, G = 1 R Inductive Susceptance, B L = 1 πfl Capacitive Susceptance, B C = πfc Lord William Thomson Kelvin ( ) X L = X C ω r L = 1 ω r C ω r = 1 LC or f 1 r = π LC 1 gyurcsek.istvan@mik.pte.hu
13 Impedance at Parallel Resonance Inductive Capacitive
14 Susceptance at Parallel Resonance Y = G + jb C jb L = G + j B C B L Y = G + ωc 1 ωl 14 gyurcsek.istvan@mik.pte.hu
15 Current in a Parallel Resonance Circuit I R = V R I T = I R + 0 = I R I L = V X L = V ωl = V πfl I C = V X C = VπfC ഥI T = ഥI R + ഥI L + ഥI C V/R I T = I R + I L I C 15 gyurcsek.istvan@mik.pte.hu
16 Bandwidth & Selectivity (Q-factor) of a Parallel Resonance Circuit ω L = G C + G C + 1 LC B C B L = G ω H = + G C + G C + 1 LC Q π max. energy stored power loss = V B C V G = ω r C G = 1 ω r L G = 1 G C L = Y 0 G Q = ω r C G G C = ω r Q BW = f H f L = 1 G π C = f r Q 16 gyurcsek.istvan@mik.pte.hu
17 Current Resonance Parallel resonance current resonance I L = I C = QI I L = V S = IR = R I = QI X L X L ωl I C = V S X C = IR X C = ωcri = QI 17 gyurcsek.istvan@mik.pte.hu
18 Series Resonance Circuits Parallel Resonance Circuits Free Resonance
19 Free resonance (LC, RLC) W L = 1 L I, W C = 1 C V 1 L I = 1 C V V I = L C = Z 0 i t = I 0 e δt sin ωt (δ: damping factor) ω = ω 0 δ, ω 0 = 1 LC, δ = R L undamped (free oscillation) δ = 0 underdamped (slowly dampening osc.) δ < ω 0 critical damping (no oscillation) δ = ω 0 overdamped (no oscillation) δ > ω 0 19 gyurcsek.istvan@mik.pte.hu
20 Effect of Impure Components ω r = ω L ω H Resonance Im Z eq = 0 ω 0 ω L + ω H ω r ω 0 ω r = ω 0 π BW ω 0 + π BW = ω 0 ω 0 4Q = ω Q Q 1 ቑ R 0, δ 0 ω r ω 0 ω r = 1 LC R L = ω 0 δ, ω 0 = 1 LC, δ = R L ω r ω L + ω H IN PRACTICE coil and capacitor contains some resistance. Series RLC pure / impure components no affect the resonance frequency Parallel RLC with impure components affect the resonance frequency 0 gyurcsek.istvan@mik.pte.hu
21 App. Tesla Coil Air-core resonant transformer (N. Tesla 1891) high-voltage, high frequency Transmission of electrical energy without wires Spark gap radio transmitters for wireless telegraphy until the 190s (Titanic 191) Today: transistor / thyristor instead of spark gap Primer and seconder circuits tuned for same resonance freq ω P0 = ω S0 L P C P = L S C S 1 C PU P = 1 C SU S U S = U P C P C S = U P L S L P 1 gyurcsek.istvan@mik.pte.hu
22 Questions
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