Series and Parallel Resonances

Transcription

1 Series and Parallel esnances Series esnance Cnsider the series circuit shwn in the frequency dmain. The input impedance is Z Vs jl jl I jc C H s esnance ccurs when the imaginary part f the transfer functin is zer, r ImZ L 0 C The value f that satisfies this cnditin is called the resnant frequency. Thus, the resnance cnditin is Then Z rad/sec The series resnant circuit.

2 Nte that at resnance: Series esnance. The impedance is purely resistive, thus, Z =. In ther wrds, the series cmbinatin acts like a shrt circuit, and the entire vltage is acrss. (Therefre, an ideal series resnance circuit with 0 has zer impedance at resnance.). The vltage V s and the current I are in phase, s that the pwer factr is unity. 3. The magnitude f the transfer functin H() = Z() is minimum. 4. The inductr vltage and capacitr vltage can be much higher than the surce vltage.

3 Series esnance The frequency respnse f the circuit s current magnitude is. The average pwer dissipated by the circuit is V V m P P P P P I Hence, and are called the half-pwer frequencies. 3

4 Series esnance Since the impedance is lwest at resnance, the half-pwer frequencies are btained by setting the magnitude f Z equal L C L C L C L L 0 C C C C 0 0 C C 4 L L L L Similarly L L fr >0 fr >0 4

5 Series esnance L L L L L L B esnant Frequency Bandwidth L L L L L 5

6 Series esnance The sharpness f the resnance in a (single) resnant circuit is measured quantitatively by the quality factr Q. At resnance, the reactive energy in the circuit scillates between the inductr and the capacitr. The quality factr relates the maximum r peak energy stred t the energy dissipated in the circuit per cycle f scillatin: Q Peak energy stred in the circuit Energy dissipated by the circuit in ne perid at resnance In the series circuit, the peak energy stred is LI /, while the energy dissipated in ne perid is (I /)(/f ). Hence, LI L L Q = where I / f C C At resnance: Vm VL LQVm Vm VC QV C m 6

7 Series esnance Q B L L C C = C L Q The quality factr f a single resnant circuit is the rati f its resnant frequency t its bandwidth. The higher the circuit Q, the smaller the bandwidth. 7

8 Series esnance The selectivity f a single circuit is its ability t respnd t a certain frequency and discriminate against all ther frequencies ( filter applicatins). If the band f frequencies t be selected r rejected is narrw, the quality factr f the single resnant circuit must be high. If the band f frequencies is wide, the quality factr is lw. A resnant circuit is designed t perate at r near its resnant frequency. It is said t be a high-q circuit when its quality factr is equal t r greater than 0. Fr high-q circuits (Q 0), the half-pwer frequencies are, fr all practical purpses, symmetrical arund the resnant frequency and can be apprximated as, B L L Q B 8

9 Example In the circuit, let =, L = mh, and C = 0.4F. a) Find the resnant frequency and the half-pwer frequencies. b) Calculate the bandwidth and the quality factr. c) Determine the amplitude f the current at 0,, and. f f khz 9

10 Example (b) Methd : 0

11 (b) Methd : Example Vm 0 V (c) At, I 0 A Vm 0 V At, I 7.07A

12 Parallel esnance The parallel circuit is the dual f the series circuit when vltage is exchanged with current and impedance with admittance. Y Is jc jc V jl L H s esnance ccurs when the imaginary part f the transfer functin is zer, r ImY C 0 L The value f that satisfies this cnditin is called the resnant frequency. Thus, the resnance cnditin is The parallel resnant circuit. rad/sec Then Y, Z (Therefre, an ideal parallel resnance circuit with has infinite impedance at resnance.)

13 Parallel esnance C C, B Q B C C C L = L C Q The vltage amplitude versus frequency fr the parallel resnant circuit. Im At resnance: I QI, I I C QI L L m C m m 3

14 The characteristics f resnant circuits Summary f the characteristics f circuits. L L r r C C L r C r 4

15 Let v 0cs( t) V. s a) Calculate, Q and B. b) Find and. 0 Example c) Determine the pwer dissipated in the 45-k resistr at resnance. a) Assume Q 0 : krad/s 60mH μf C μf Q k 45k L 60mH krad/s B rad/s Q

16 Example B b) Since Q 0 : krad/s B krad/s c) At resnance, an ideal parallel circuit has infinite impedance. Thus I 0V k45k ma P 45k (35.09mA) I 45k.77 mw 6

17 Example 3 f f.7khz 7

18 Example 3 8

19 Example 4 f f 5.9 khz 9

20 Practical Cnsideratins In practical resnant circuits, the highest lsses are usually attributed t the inductr and are represented as a series resistr. Hw can this be translated int a parallel resnant circuit? f(, L, C)? S P S P Z S ( j) ZP ( j) jc jc S j L P jl S jl jl js C ( j) L j ( j) P 0

21 Practical Cnsideratins. ZS( j) ZP( j) if S 0 and P. This is the ideal (lssless) parallel resnant circuit.. In practice, we assume that the resnance frequencies f the tw circuits are identical and that the Q s are greater than 0. Then L L L Q Q P S P P S L S S P L Z C where Z Example: Let L=0mH, C=5nF and S =40, then L L 00krad/s, QS 50 0 P 00k C S S S S L C

22 Example 5 Determine the resnant frequency. j Yin j0. j0. 0 j At resnance: Im Y ( 0) Q Q L C in rad/s.36 rad/s L 0 0 s (series resnance) C 00. (parallel resnance) p Qeq (nt a gd resnant circuit - lsses are t high!) Q Q Q eq L C p s Why is there a difference f mre than 0 % in resnant frequency?

23 A Mre Practical Example LmH, 0, C nf, 500k s jl Yin jc jc j L L s p s p s L ImYin0 C 0 ( 0) L s 0 s 5 0 Ls C rad/s L p.50 Q L 50, Q C 500 L 0 s C 0 p Qeq Q Q Q eq L C 0 8 C p s rad/s, f khz 3 L

24 elatinship between and Q vs vc H V V C s jc jl jc ( ) j C j ecall quadratic ple: H 0 H C j ( j ) 0 0 C C since Q L L Q C Q j / / j 0 0 4