| 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 | ||
| Response curves for a series resonant circuit |
| Since XL = wL = 2pfL and XC = 1/wC = 1/2pfC for resonance set XL = XC | ||
| Solve for the series resonant frequency fs | ||
| At resonance | ||
| Impedance of a series resonant circuit is small and the current is large | ||
| I = E/ZT = E/R | ||
| At resonance | |
| VR = IR | |
| VL = IXL | |
| VC = IXC |
| At resonance, average power is P = I2R | |
| Reactive powers dissipated by inductor and capacitor are I2X | |
| Reactive powers are equal and opposite at resonance |
| Q = reactive power/average power | ||
| Q may be expressed in terms of inductor or capacitor | ||
| For an inductor, Qcoil= XL/Rcoil | ||
| Q is often greater than 1 | ||
| Voltages across inductors and capacitors can be larger than source voltage | ||
| This is true even though the sum of the two voltages algebraically is zero |
Impedance of a Series Resonant Circuit
| Impedance of a series resonant circuit varies with frequency |
| 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 | ||
| A circuit with a narrow bandwidth | ||
| High selectivity | ||
| If the bandwidth is wide | ||
| Low selectivity | ||
| Cutoff frequencies | ||
| Found by evaluating frequencies at which the power dissipated by the circuit is half of the maximum power | ||
| From BW = f2 - f1 | ||
| BW = R/L | ||
| When expression is multiplied by w on top and bottom | ||
| BW = ws/Q (rad/sec) or BW = fs/Q (Hz) | ||
| For analysis of parallel resonant circuits | ||
| Necessary to convert a series inductor and its resistance to a parallel equivalent circuit | ||
| 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 | ||
| Parallel resonant circuit | ||
| Has XC and equivalents of inductive reactance and its series resistor, XLP and RS | ||
| At resonance | ||
| XC = XLP | ||
| Two reactances cancel each other at resonance | ||
| Cause an open circuit for that portion | ||
| ZT = RP at resonance | ||
| Response curves for a parallel resonant circuit |
| From XC = XLP | ||
| Resonant frequency is found to be | ||
| If (L/C) >> R | ||
| Term under the radical is approximately equal to 1 | ||
| If (L/C) ³ 100R | ||
| Resonant frequency becomes | ||
| 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 | ||
| Circuit currents are | |
| Magnitudes of currents through the inductor and capacitor | ||
| May be much larger than the current source | ||
| Cutoff frequencies are | |
| BW = w2 - w1 = 1/RC | ||
| If Q ³ 10 | ||
| Selectivity curve becomes symmetrical around wP | ||
| Equation of bandwidth becomes | |
| Same for both series and parallel circuits |