14. RCL Circuits + Resonance

• RCL Circuits and resonance

  • Series RCL
  • Resonance
  • Parallel RCL
  • Parallel and series parallel RCL
  • Parallel resonance
  • Bandwidth
  • Applications

  
• Series RCL Circuits


  • Opposite effects total reactance is less than individual reactances
  • XL ® current lag; XC current leads Vs
  • VXL, XC offset each other ® cancel reactance = 0
  • XT = XL – XC
• Rectangular Form


  Z = R + jXL - jXC
• Polar Form


  • Magnitude (Z)
  • Phase angle between total current and VS
• Total impedance in rectangular and polar form

  • Circuit more inductive
  • Capacitive
  • Current leading

  
• Analysis of Series RCL Circuits

  1. Two reactances cancel series resonance

  2. Find impedance in polar form at f=1kHz, f=3.5 kHz, f=5kHz

  
• In series RCL VCL is always less than the larger VC and VL
• • VL and VC effectively subtract
  • VC and VL always 180° out of phase
• Find current and voltages across each element

  1. Total impedance

  2. Ohm's Law

  3. Ohm's Law for voltages

  
• Series Resonance


  • XC and XL at resonant frequency (fr)
  • Purely resistive
  • Cancellation of XL and XC at Resonance
  • Equal magnitude, 180° out of phase, series (same current flows)

  
• At resonance:


  • fr?
  • RCL impedance
  • f<fr capacitive
  • f>fr inductive
  • Resultant impedance resistive
• Determine impedance magnitude at resonance

  1. 1000 Hz below resonance

  2. 1000 Hz above resonance

  3. Resonant frequency
• Current and Voltages in Series RCL Circuits

  • At resonance VC = VS
  • At f=0 Max at resonant frequency
  • Phase angle of a series RCL circuit

  
• Impedance of Parallel RCL Circuits

  • Rectangular
  • Polar
  • Conductive, Susceptance and Admittance

  
• Current relationships in RCL parallel circuits

  1. Smaller reactance dominates ® largest current

  2. Always subtract

  3. Find each branch current and total current

  4. Ohm's Law

  5. IT = phasor sum of branch currents (Kirchhoff's Law)

  6. Or in polar form

  
• Find VC in polar form is circuit predominantly inductive or capacitor?

  Find the voltage at B wrt ground
• Conversion of Series Parallel to Parallel


  • Important series-parallel circuit
  • Q=XL/RW
  • Quality factor
  • Parallel L and C + winding resistance RW
  • Helpful to view equivalent form as parallel
  • Simplifies analysis of parallel resonance
  • Same IT and same phase angles
• Parallel Resonance: Ideal LC


  • Stores energy in magnetic field of coil and electric field of capacitor
  • Stored energy is passed back and forth between L and C on alternative half cycles
• Non ideal parallel resonance


  • At resonance
  • Results in Rp(eq) in parallel with ideal coil and capacitor
  • Tank circuit
  • Z®¥at resonance
  • Zr = RW (Q2+ 1)

  
• Derivation impedance of non ideal tank circuit at resonance


  At resonance Zr is purely resistive
• Non ideal parallel resonance


  At resonance the parallel LC portion appears open and the source sees only RP(eq)
• Variation of impedance with frequency

  • Max at resonance
  • XL dominates at low f
  • XC dominates
• • At parallel resonance Total current at resonance IT = VS
  • Phase Angle = 0°
  • Impedance is purely resistive
• Parallel resonant frequency in a non ideal circuit

  1. At resonance

  2. 1 for Q > 10, same fr as for series resonance

  3. More precise expression

  
• Find fr, Zr and total current at resonance


  For Zr find XL and Q
• Bandwidth (BW) of resonant circuits

  • Series resonant circuit
  • Parallel Resonant Circuits
  • Ideally fr = f1 + f2/2
  • Half power frequencies
  • Upper and lower critical frequencies f1 f2
  • Power = (Power fr)
  • Pmax = I2 max R
  • Power at f1 , f2
  • Selectivity – smaller BW greater selectivity
  • Higher Q results in a smaller BW
  • BW = fr/Q
  • Slope affects selectivity 'faster' slope ® greater selectivity
• Double tuned transformer coupling in a receiver

  • Band pass filters
  • Parallel resonant
  • Coupled to increase amplification + wider bandwidth and greater selectivity
• Signal reception and separation in a TV receiver

  • Tuned amplifier: amplify signals within a specific band
  • Desired fr
  • selectivity
  • Antenna input to a receiver
  • em waves ® small induce voltages
  • Extract a limited band of frequencies
• Superheterodyne Receiver


  • Audio signal
  • Intermediate frequency (IF)
  • Heterodyning of beating to Produce 455kHz AM signal
  • Gang tuned capacitors (mechanically linked)
  • am (amplitude modulation) receiver eg, 535kHz – 1605 kHz station have a narrowband within the range
  • 3 parallel resonant band pass filters