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Chapter 21

Chapter 21. Resonance. Series Resonance. Simple series resonant circuit Has an ac source, an inductor, a capacitor, and possibly a resistor Z T = R + jX L – jX C = R + j(X L – X C ) Resonance occurs when X L = X C At resonance, Z T = R. Series Resonance.

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Chapter 21

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  1. Chapter 21 Resonance

  2. Series Resonance • Simple series resonant circuit • Has an ac source, an inductor, a capacitor, and possibly a resistor • ZT = R + jXL – jXC = R + j(XL – XC) • Resonance occurs when XL = XC • At resonance, ZT = R

  3. Series Resonance • Response curves for a series resonant circuit

  4. Series Resonance

  5. Series Resonance • Since XL = L = 2fL and XC = 1/C = 1/2fC for resonance set XL = XC • Solve for the series resonant frequency fs

  6. Series Resonance • At resonance • Impedance of a series resonant circuit is small and the current is large • I = E/ZT = E/R

  7. Series Resonance • At resonance VR = IR VL = IXL VC = IXC

  8. Series Resonance • At resonance, average power is P = I2R • Reactive powers dissipated by inductor and capacitor are I2X • Reactive powers are equal and opposite at resonance

  9. The Quality Factor,Q • Q = reactive power/average power • Q may be expressed in terms of inductor or capacitor • For an inductor, Qcoil= XL/Rcoil

  10. The Quality Factor,Q • Q is often greater than 1 • Voltages across inductors and capacitors can be larger than source voltage

  11. The Quality Factor,Q • This is true even though the sum of the two voltages algebraically is zero

  12. Impedance of a Series Resonant Circuit • Impedance of a series resonant circuit varies with frequency

  13. Bandwidth • Bandwidth of a circuit • Difference between frequencies at which circuit delivers half of the maximum power • Frequencies, f1 and f2 • Half-power frequencies or the cutoff frequencies

  14. Bandwidth • A circuit with a narrow bandwidth • High selectivity • If the bandwidth is wide • Low selectivity

  15. Bandwidth • Cutoff frequencies • Found by evaluating frequencies at which the power dissipated by the circuit is half of the maximum power

  16. Bandwidth

  17. Bandwidth • From BW = f2 - f1 • BW = R/L • When expression is multiplied by  on top and bottom • BW = s/Q (rad/sec) or BW = fs/Q (Hz)

  18. Series-to-Parallel Conversion • For analysis of parallel resonant circuits • Necessary to convert a series inductor and its resistance to a parallel equivalent circuit

  19. Series-to-Parallel Conversion • If Q of a circuit is greater than or equal to 10 • Approximations may be made • Resistance of parallel network is approximately Q2 larger than resistance of series network • RP Q2RS • XLP  XLS

  20. Parallel Resonance • Parallel resonant circuit • Has XC and equivalents of inductive reactance and its series resistor, XLP and RS • At resonance • XC = XLP

  21. Parallel Resonance • Two reactances cancel each other at resonance • Cause an open circuit for that portion • ZT = RP at resonance

  22. Parallel Resonance • Response curves for a parallel resonant circuit

  23. Parallel Resonance • From XC = XLP • Resonant frequency is found to be

  24. Parallel Resonance • If (L/C) >> R • Term under the radical is approximately equal to 1 • If (L/C)  100R • Resonant frequency becomes

  25. Parallel Resonance • Because reactances cancel • Voltage is V = IR • Impedance is maximum at resonance • Q = R/XC • If resistance of coil is the only resistance present • Circuit Q will be that of the inductor

  26. Parallel Resonance • Circuit currents are

  27. Parallel Resonance • Magnitudes of currents through the inductor and capacitor • May be much larger than the current source

  28. Bandwidth • Cutoff frequencies are

  29. Bandwidth • BW = 2 - 1 = 1/RC • If Q  10 • Selectivity curve becomes symmetrical around P

  30. Bandwidth • Equation of bandwidth becomes • Same for both series and parallel circuits

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