Resonant and cut-off frequencies Tuned network quality, bandwidth, and power levels Quality factor

Transcription

1 Chapter 20 Resonant and cut-off frequencies Tuned network quality, bandwidth, and power levels Quality factor ECET 207 AC Circuit Analysis, PNC 2 1

2 A condition established by the application of a particular frequency to a series or parallel RLC network. The transfer of power is at maximum, and power drops off for frequencies above and below this frequency. ECET 207 AC Circuit Analysis, PNC 4 2

3 ECET 207 AC Circuit Analysis, PNC 5 Must have resistive elements Internal resistances (reality) Helps shape curve Must have reactive elements Both capacitive and inductive required of equal impedance Energy level absorbed by one is released by another FIG Resonance curve. ECET 207 AC Circuit Analysis, PNC 6 3

4 Must have capacitive and inductive element Resistive elements R - source internal resistance R - inductor internal resistance R - added resistance to shape the response curve FIG Series resonant circuit. ECET 207 AC Circuit Analysis, PNC 7 R = R + R + R Z = R + j(x X ) Z = F = F = FIG Series resonant circuit. ECET 207 AC Circuit Analysis, PNC 8 4

5 AT RESONANCE E and I in phase Power triangle P= Real power Q= Reactive power S= Apparent power FIG Phasor diagram for the series resonant circuit at resonance. FIG Power triangle for the series resonant circuit at resonance. ECET 207 AC Circuit Analysis, PNC 9 P (Real Power, Watts) Power delivered to resistive elements in circuit P = I R or R Resistances only Q (Reactive Power, Volt-amps Reactive VAR) Q = VI sin θ Reactances only S (Apparent Power) S Accounts for phase angle I Z or Z or P ± jq Q P ECET 207 AC Circuit Analysis, PNC 10 5

6 p counters p FIG Power curves at resonance for the series resonant circuit. ECET 207 AC Circuit Analysis, PNC 11 Level of power stored (L or C) compared to the level of power dissipated (R) Q = ECET 207 AC Circuit Analysis, PNC 12 6

7 High Q systems (communication) V or V may be higher than source Requires special insulation for Q rise FIG High-Q series resonant circuit. ECET 207 AC Circuit Analysis, PNC 13 Total Z function of frequency ECET 207 AC Circuit Analysis, PNC 14 7

8 FIG Resistance versus frequency. ECET 207 AC Circuit Analysis, PNC 15 FIG Inductive reactance versus frequency. FIG Capacitive reactance versus frequency. ECET 207 AC Circuit Analysis, PNC 16 8

9 FIG Frequency response of the inductive and capacitive reactance of a series R-L-C circuit on the same set of axes. FIG Z T versus frequency for the series resonant circuit. At F Z = R ECET 207 AC Circuit Analysis, PNC 17 FIG Phase plot for the series resonant circuit. ECET 207 AC Circuit Analysis, PNC 18 9

10 Find X L I T V R, V L, V C Q L and C at 5 khz ECET 207 AC Circuit Analysis, PNC 19 Bandwidth (-3 db) Freq. where P is near max f f At f 0.707I 0.707V P FIG I versus frequency for the series resonant circuit. ECET 207 AC Circuit Analysis, PNC 20 10

11 Freq. must be selected to fall within BW Large BW, small selectivity Component values shape curve Higher R, L, C, tighter the curve FIG Effect of R, L, and C on the selectivity curve for the series resonant circuit. ECET 207 AC Circuit Analysis, PNC 21 Quality factor (Q ) proportional to BW Small BW, high Q Q 10 Resonance Freq bisects BW Curve is symmetrical on RF Ideal situation FIG Approximate series resonance curve for Q s 10. ECET 207 AC Circuit Analysis, PNC 22 11

12 f = + + f = + BW = f f = + = Fractional Bandwidth = ECET 207 AC Circuit Analysis, PNC 23 For Ex. 20.1, Find I V at resonance V at resonance V at resonance Q BW when f = 5 khz P () at P + FIG Example ECET 207 AC Circuit Analysis, PNC 24 12

13 For Ex. 20.4, find Q L and R when C=100nF E FIG Example 20.4 ECET 207 AC Circuit Analysis, PNC 26 13

14 FIG Ideal parallel resonant network. ECET 207 AC Circuit Analysis, PNC 27 Practical More realistic R l -inductor internal resistance R = (no phase angles) X = (no phase angles) FIG Equivalent parallel network for a series R-L combination. ECET 207 AC Circuit Analysis, PNC 28 14

15 FIG Substituting the equivalent parallel network for the series R-L combination in Fig ECET 207 AC Circuit Analysis, PNC 29 Final result Same look as ideal circuit Correct, realistic values R = R s R p FIG Substituting R = R s R p for the network in Fig ECET 207 AC Circuit Analysis, PNC 30 15

16 Unity Power Factor, f p Maximum Impedance, f m ECET 207 AC Circuit Analysis, PNC 31 Unity power factor (f ) Total reactive element = 0 X = X or f = f f > f 1 = X FIG Z T versus frequency for the parallel resonant circuit. ECET 207 AC Circuit Analysis, PNC 32 16

17 Maximum Impedance (f ) Actual freq. input Z is the highest Highest power output Based on R f = f 1 Z = R X X f = f m FIG Z T versus frequency for the parallel resonant circuit. ECET 207 AC Circuit Analysis, PNC 33 Q = = = When R R Q = = Q BW = f f = f = + + f = + + ECET 207 AC Circuit Analysis, PNC 34 17

18 FIG Effect of Rl, L, and C on the parallel resonance curve. ECET 207 AC Circuit Analysis, PNC 35 FIG Phase plot for the parallel resonant circuit. ECET 207 AC Circuit Analysis, PNC 36 18

19 Inductive Reactance, X LP X X X Resonant Frequency, f p (Unity Power Factor) f f = Resonant Frequency, f m (Maximum V C ) f f f ECET 207 AC Circuit Analysis, PNC 37 R p R Q R Z Tp R R = R Q R If R = Ω or R R Z = Q R Q p If R = Ω or R R Q Q ECET 207 AC Circuit Analysis, PNC 38 19

20 FIG Approximate equivalent circuit for Q l 10. ECET 207 AC Circuit Analysis, PNC 39 BW BW = f f I = Q I I = Q I FIG Establishing the relationship between I C and I L and the current I T. ECET 207 AC Circuit Analysis, PNC 40 20

21 ECET 207 AC Circuit Analysis, PNC 41 Use the summery tables Multiple versions of equations listed If no value given, it is skipped Ex- No Rs given, exclude it from equation Given Prob. 15 (Pg. 909), find Resonance frequency V tank Power delivered by source at resonance Power loss in the tank coil ECET 207 AC Circuit Analysis, PNC 42 21