ESONANCE SEIES ESONANT CICUITS 5/007 Enzo Pateno
ESONANT CICUITS A vey impotant cicuit, used in a wide vaiety o electical and electonic systems today (i.e. adio & television tunes), is called the esonant / tuned cicuit whose equency esponse chaacteistic is shown below: The esponse is a maximum @... is called the esonant equency. A tuning cicuit will be tuned o maximum esponse so to eceive the signal at its maximum enegy ( @ ). In mechanical systems, this equency is called the natual equency (i.e. The Tacoma Naows Bidge). 5/007 Enzo Pateno
ESONANT CICUITS The esonant electical cicuit must have both inductance and capacitance. When esonance occus due to the application o the pope equency (), the enegy absobed by one eactive element is the same as that eleased by anothe eactive element within the system. Enegy pulsates om one eactive element to the othe. Once an ideal (pue L, C) system has eached a state o esonance, it equies no uthe eactive powe since it is sel-sustaining (i.e. mechanical system pepetual motion). In a pactical cicuit, howeve, thee is some esistance associated with the eactive elements that will esult in the eventual damping o the oscillations between eactive elements. 5/007 Enzo Pateno 3
A esistive element will always be pesent due to the intenal esistance o the souce ( S ), the intenal esistance o the inducto ( L ), and any added esistance to contol the shape o the esponse cuve ( d ). Thus, @ Z T esonance will occu when X L X C We can calculate. Since X L X C and 5/007 Enzo Pateno 4
The cuent though the cicuit at esonance is the maximum cuent with the input voltage and the cuent in phase: The voltage acoss the inducto and the voltage acoss the capacito at esonance ae equal magnitude and ae 80º out o phase: Equal Magnitude The powe acto o the cicuit at esonance is: F P cos θ cos 0 5/007 Enzo Pateno 5
X X C πc [ Ω] X L πl [ Ω] Xc [ Ω ] X L [ Ω ] 0 X L < X C π LC X L > X C 5/007 Enzo Pateno 6 Capacitive netwok Inductive netwok esistive netwok [ Hz ]
Let us plot Z T : The minimum impedance occus at the esonant equency and is equal to the esistance. Note that the cuve is not symmetical about the esonant equency. 5/007 Enzo Pateno 7
SELECTIVITY - S The plot o I E/Z T, (E K,) vesus equency, (called the selectivity cuve), is shown below and is the invese o the impedance-vesus-equency cuve. Z T is a minimum @ and P ½ P max P HPF The ange o equencies between and is eeed to the bandwidth (BW) o the the esonant cicuit. BW BW S BW S The equencies and (i.e. @ 0.707 I max ) ae called the band equencies, cuto equencies, hal-powe equencies o -3db equencies. 5/007 Enzo Pateno 8
The quality acto Q o a seies esonant cicuit is deined as the atio o the eactive powe o eithe the inducto o the capacito to the aveage powe o the esisto at esonance: eactive powe Q Aveage powe The quality acto is an indication o how much enegy is stoed (continual tanse om one eactive element to the othe) compaed to that dissipated. The lowe the level o dissipation, the lage the Q acto and the moe intense the egion o esonance. A highe Q is desiable. Using inductive eactance the quality acto becomes: Q I I X L X L ωl 5/007 Enzo Pateno 9 L C I is only that o the coil, we speak o the Q o the coil (given by the manuactue ). Q Q (), as X L
5/007 Enzo Pateno
Fo cicuits whee Q > 0, a widely accepted appoximation is that the esonant equency bisects the bandwidth and that the esonant cuve is symmetical about the esonant equency. When designing a BPF, a Design ule o thumb is to Design o a Q > 0. The geometic mean o ω : ω ω ω BW a b + 5/007 Enzo Pateno BW BW + BW
BW ca be expessed in tems o & L: BW S BW S Q can be expessed in tems o BW: Q BW S Q BW S Q BW 5/007 Enzo Pateno
4 a. Detemine the Q and bandwidth o the esponse cuve below b. Fo C 0.5 nf, ind L and o the seies esonant cicuit. c. Detemine the applied voltage. a. 800 Hz, BW 00 Hz b. Q Q BW 800 00 L π LC L 3.83 mh 40 Ω L C 4 4π Q C L C E 8 v 5/007 Enzo Pateno 3
Detemine the equency esponse o the voltage Vo o the cicuit below. 50.3 (5.6 / ) 47.5 khz 50.3 + (5.6 / ) 53. khz V o 0.707 V omax 3.34 mv 5/007 Enzo Pateno 4
PSPICE SIMULATION 5/007 Enzo Pateno 5
8.85 mv 50.7 khz 5/007 Enzo Pateno 6
- Fomulas + BW BW Q ωl L C BW Q ω ω ω BW G out db 0 log Vin Enzo Pateno 7 V P 0log P out in