12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary 12: Resonance

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 1 / 11 Quadratic Factors +

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors +

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors +

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 X (jω) = 6R2C2(jω)2+7RCjω+1

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 X (jω) = 6R2C2(jω)2+7RCjω+1 1 = (6jωRC+1)(jωRC+1)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 X (jω) = 6R2C2(jω)2+7RCjω+1 1 = (6jωRC+1)(jωRC+1) 0.17 1 ωc = RC , RC

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ω = , ω c RC RC

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ωc = , = p1 , p2 ω RC RC | | | |

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ωc = , = p1 , p2 ω RC RC | | | | Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave it as a quadratic.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ωc = , = p1 , p2 ω RC RC | | | | Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave it as a quadratic. Sometimes called a quadratic resonance.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ωc = , = p1 , p2 ω RC RC | | | | Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave it as a quadratic. Sometimes called a quadratic resonance.

Any polynomial with real coefficients can be factored into linear and quadratic factors

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Quadratic Factors ++

12: Resonance 2 • QuadraticFactors + A quadratic factor in a transfer function is: F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC 2 • Behaviour at Resonance Case 1: If b 4ac then we can factorize it: • Away from resonance ≥ • Bandwidth and Q F (jω) = a(jω p1)(jω p2) • Power and Energy at − − Resonance + b √b2 4ac • Low Pass Filter where p = − ± − . • Resonance Peak for LP i 2a filter • Summary Y 1 0 X (jω) = 6R2C2(jω)2+7RCjω+1 -20 1 = (6jωRC+1)(jωRC+1) -40 0.17 1 0.1/RC 0.3/RC 1/RC 3/RC ωc = , = p1 , p2 ω RC RC | | | | Case 2: If b2 < 4ac, we cannot factorize with real coefficients so we leave it as a quadratic. Sometimes called a quadratic resonance.

Any polynomial with real coefficients can be factored into linear and quadratic factors a quadratic factor is as complicated as it gets. ⇒

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 2 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c • Behaviour at Resonance LF • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP p filter b • Summary We define the damping factor, “zeta”, to be ζ = 2aωc

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c  

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor. (b) ω just scales the frequency axis since F (jω) is a function of ω . c ωc

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor. (b) ω just scales the frequency axis since F (jω) is a function of ω . c ωc (c) The shape of the F (jω) graphs is determined entirely by ζ.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor. (b) ω just scales the frequency axis since F (jω) is a function of ω . c ωc (c) The shape of the F (jω) graphs is determined entirely by ζ. (d) The quadratic cannot be factorized b2 < 4ac ζ < 1. ⇔ ⇔| |

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor. (b) ω just scales the frequency axis since F (jω) is a function of ω . c ωc (c) The shape of the F (jω) graphs is determined entirely by ζ. (d) The quadratic cannot be factorized b2 < 4ac ζ < 1. ⇔ ⇔| | (e) At ω = ω , asymptote gain = c but F (jω) = c 2jζ. c ×

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Damping Factor and Q

12: Resonance 2 2 • QuadraticFactors + Suppose b < 4ac in F (jω) = a (jω) + b (jω) + c. • Damping Factor and Q • Parallel RLC Low/High freq asymptotes: F (jω) = c, F (jω) = a (jω)2 • Behaviour at Resonance LF HF • Away from resonance • Bandwidth and Q The asymptote magnitudes cross at the corner frequency: • Power and Energy at 2 c Resonance + a (jω ) = c ω = . • c c a Low Pass Filter | | ⇒ • Resonance Peak for LP filter p b bωc b sgn(a) • Summary We define the damping factor, “zeta”, to be ζ = = = 2aωc 2c √4ac 2 ω ω F (jω) = c j ω + 2ζ j ω + 1 ⇒  c   c   Properties to notice in this expression: (a) c is just an overall scale factor. (b) ω just scales the frequency axis since F (jω) is a function of ω . c ωc (c) The shape of the F (jω) graphs is determined entirely by ζ. (d) The quadratic cannot be factorized b2 < 4ac ζ < 1. ⇔ ⇔| | (e) At ω = ω , asymptote gain = c but F (jω) = c 2jζ. c × Alternatively, we sometimes use the quality factor, Q 1 = aωc . ≈ 2ζ b

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 3 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p 1 • Bandwidth and Q Asymptotes: jωL and jωC . • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p 1 • Bandwidth and Q Asymptotes: jωL and jωC . • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

Power absorbed by resistor Y 2. It peaks quite ∝ sharply at ω = 1000.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p 1 • Bandwidth and Q Asymptotes: jωL and jωC . • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

Power absorbed by resistor Y 2. It peaks quite ∝ sharply at ω = 1000. The resonant frequency, ωr, is when the impedance is purely real: Y at ωr = 1000, ZRLC = I = R.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p 1 • Bandwidth and Q Asymptotes: jωL and jωC . • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

Power absorbed by resistor Y 2. It peaks quite ∝ sharply at ω = 1000. The resonant frequency, ωr, is when the impedance is purely real: Y at ωr = 1000, ZRLC = I = R. A system with a strong peak in power absorption is a resonant system.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Parallel RLC

12: Resonance Y 1 jωL • 1 1 QuadraticFactors + I = + +jωC = LC(jω)2+ L jω+1 • Damping Factor and Q R jωL R • c b Parallel RLC ωc = = 1000, ζ = = 0.083 • Behaviour at Resonance a 2aωc • Away from resonance p 1 • Bandwidth and Q Asymptotes: jωL and jωC . • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

Power absorbed by resistor Y 2. It peaks quite ∝ sharply at ω = 1000. The resonant frequency, ωr, is when the impedance is purely real: Y at ωr = 1000, ZRLC = I = R. A system with a strong peak in power absorption is a resonant system.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 4 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary Z = Z I = I L − C ⇒ L − C

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = I R = 600∠0◦ = 56 dBV ⇒ R

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = IRR = 600∠0◦ = 56 dBV ⇒ Y 600 IL = = = 6j ⇒ ZL 100j −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = IRR = 600∠0◦ = 56 dBV ⇒ Y 600 IL = = = 6j ⇒ ZL 100j −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = IRR = 600∠0◦ = 56 dBV ⇒ Y 600 IL = = = 6j ⇒ ZL 100j −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = IRR = 600∠0◦ = 56 dBV ⇒ Y 600 IL = = = 6j ⇒ ZL 100j −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Behaviour at Resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 1000 ZL = 100j, ZC = 100j. filter ⇒ − • Summary ZL = ZC IL = IC I =−I +⇒I + I −= I = 1 ⇒ R L C R Y = IRR = 600∠0◦ = 56 dBV ⇒ Y 600 IL = = = 6j ⇒ ZL 100j −

Large currents in L and C exactly cancel out I = I and Z = R (real) ⇒ R

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 5 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ − • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV × −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV × −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× − I = = 0.11∠ 84◦ R R −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× ∠ − IR = R = 0.11 84◦ Y − IL = = 0.33∠ 174◦ ZL −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× ∠ − IR = R = 0.11 84◦ Y − IL = = 0.33∠ 174◦, IC = 1.33∠ + 6◦ ZL −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× ∠ − IR = R = 0.11 84◦ Y − IL = = 0.33∠ 174◦, IC = 1.33∠ + 6◦ ZL −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× ∠ − IR = R = 0.11 84◦ Y − IL = = 0.33∠ 174◦, IC = 1.33∠ + 6◦ ZL −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Away from resonance

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP ω = 2000 ZL = 200j, ZC = 50j filter ⇒ 1 − • Summary 1 1 1 − ∠ Z = R + Z + Z = 66 84◦  L C  − Y = I Z = 66∠ 84◦ = 36 dBV Y× ∠ − IR = R = 0.11 84◦ Y − IL = = 0.33∠ 174◦, IC = 1.33∠ + 6◦ ZL −

Most current now flows through C, only 0.11 through R. E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 6 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 1 R 1 2 1 2 = ( /R) +(ω3dBC /ω3dBL) 2 −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  − 

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  −  ω RC R/ω3dBL = 1 3dB − ±

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  −  2 ω RC R/ω3dBL = 1 ω RLC ω L R = 0 3dB − ± ⇒ 3dB ± 3dB −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  −  2 ω RC R/ω3dBL = 1 ω RLC ω L R = 0 3dB − ± ⇒ 3dB ± 3dB − L+√L2+4R2LC Positive roots: ω = ± = 920, 1086 rad/s 3dB 2RLC { }

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  −  2 ω RC R/ω3dBL = 1 ω RLC ω L R = 0 3dB − ± ⇒ 3dB ± 3dB − L+√L2+4R2LC Positive roots: ω = ± = 920, 1086 rad/s 3dB 2RLC { } Bandwidth: B = 1086 920 = 167 rad/s. −

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Bandwidth and Q

12: Resonance Y 1 • QuadraticFactors + I = 1/R+j(ωC 1/ωL) • Damping Factor and Q − • Parallel RLC Bandwidth is the range of frequencies for • Behaviour at Resonance Y 2 • Away from resonance which I is greater than half its peak. • Bandwidth and Q • Power and Energy at Also called half-power bandwidth or 3dB Resonance + • Low Pass Filter bandwidth. • Resonance Peak for LP filter Y 2 1 = 2 2 • Summary I (1/R) +(ωC 1/ωL) − Y 2 2 Peak is I (ω0) = R @ ω0 = 1000

Y 2 1 Y 2 At ω3dB: I (ω3dB) = 2 I (ω0) 2 2 1 R R 1 2 1 2 = 1 + ω3dBRC = 2 ( /R) +(ω3dBC /ω3dBL) 2 ω3dBL − ⇒  −  2 ω RC R/ω3dBL = 1 ω RLC ω L R = 0 3dB − ± ⇒ 3dB ± 3dB − L+√L2+4R2LC Positive roots: ω = ± = 920, 1086 rad/s 3dB 2RLC { } Bandwidth: B = 1086 920 = 167 rad/s. − Q factor ω0 = 1 = 6.(Q = “Quality”) ≈ B 2ζ E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 7 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q PL and PC opposite and PR. • Parallel RLC ≫ • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance sloshes between and . • Bandwidth and Q L C • Power and Energy at Resonance + • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance sloshes between and . • Bandwidth and Q L C • Power and Energy at Q , ω W P R Resonance + × stored ÷ • Low Pass Filter @ω = 1000: Y = 600, • Resonance Peak for LP filter IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance sloshes between and . • Bandwidth and Q L C • Power and Energy at Q , ω W P R Resonance + × stored ÷ • Low Pass Filter 1 2 1 2 @ω = 1000: Y = 600, • Resonance Peak for LP = ω 2 C IR 2 I R= ωRC filter × | | ÷ | | IR =1, IL = −6j, IC = +6j • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Power and Energy at Resonance +

12: Resonance • QuadraticFactors + Absorbed Power =v(t)i(t): • Damping Factor and Q P and P opposite and P . • L C R Parallel RLC 1 2 1 ≫ 2 • Behaviour at Resonance Stored Energy = 2 LiL + 2 Cy : • Away from resonance sloshes between and . • Bandwidth and Q L C • Power and Energy at Q , ω W P R Resonance + × stored ÷ • Low Pass Filter 1 2 1 2 @ω = 1000: Y = 600, • Resonance Peak for LP = ω 2 C IR 2 I R= ωRC filter × | | ÷ | | IR =1, IL = −6j, IC = +6j • Summary

Q , ω peak stored energy average power loss. × ÷ E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 8 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • Away from resonance LC • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ , X • Resonance Peak for LP ωc : ZL = ZC = 100j I = R filter − • Summary

C L

R

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary

C L

R

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot:

20 R=20, ζ=0.1 0 C L -20

R -40 100 1k 10k ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒

R=5, ζ=0.03 20 R=20, ζ=0.1 0 C L -20

R -40 100 1k 10k ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

R=5, ζ=0.03 20 R=20, ζ=0.1 R=60, ζ=0.3 0 C L -20

R -40 100 1k 10k ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

R=5, ζ=0.03 20 R=20, ζ=0.1 R=60, ζ=0.3 0 R=120, ζ=0.6 C L -20

R -40 100 1k 10k ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

Phase Plot:

R=5, ζ=0.03 0 20 ζ R=20, =0.1 ζ R=60, ζ=0.3 π R=20, =0.1 0 R=120, ζ=0.6 C L -0.5 -20

R -40 -1 100 1k 10k 100 1k 10k ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

Phase Plot: Small ζ fast phase change: π over 2ζ decades. ⇒

R=5, ζ=0.03 0 ζ 20 ζ R=5, =0.03 R=20, =0.1 ζ R=60, ζ=0.3 π R=20, =0.1 0 R=120, ζ=0.6 C L -0.5 -20

R -40 -1 100 1k 10k 100 1k 10k ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

Phase Plot: Small ζ fast phase change: π over 2ζ decades. ⇒

R=5, ζ=0.03 0 ζ 20 ζ R=5, =0.03 R=20, =0.1 ζ ζ π R=20, =0.1 R=60, =0.3 ζ 0 R=120, ζ=0.6 R=60, =0.3 C L -0.5 -20

R -40 -1 100 1k 10k 100 501 1k 2.00k 10k ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

Phase Plot: Small ζ fast phase change: π over 2ζ decades. ⇒

R=5, ζ=0.03 0 ζ 20 ζ R=5, =0.03 R=20, =0.1 ζ ζ π R=20, =0.1 R=60, =0.3 ζ 0 R=120, ζ=0.6 R=60, =0.3 ζ C L -0.5 R=120, =0.6 -20

R -40 -1 100 1k 10k 100 251 1k 3.98k 10k ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Low Pass Filter

12: Resonance Y 1/jωC 1 • QuadraticFactors + X = R+jωL+ 1 = LC(jω)2+RCjω+1 • Damping Factor and Q jωC • Parallel RLC 1 2 • Behaviour at Resonance Asymptotes: 1 and (jω)− . • LC Away from resonance c b R • Bandwidth and Q ωc = = 1000, ζ = = • Power and Energy at a 2aωc 200 Resonance + p • Low Pass Filter @ω : Z = Z = 100j, I = X , Y = 1 = 1 , ∠ Y = π • Resonance Peak for LP c L − C R X RCω 2ζ X − 2 filter • Summary Magntitude Plot: Small ζ less loss, higher peak, smaller bandwidth. ⇒ Large ζ more loss, smaller peak at a lower ω, larger bandwidth.

Phase Plot: Small ζ fast phase change: π over 2ζ decades. ⇒ ∠ Y π 1 ω ζ ω +ζ X −2 1 + ζ log10 ω for 10− < ω < 10 ≈  c  c R=5, ζ=0.03 0 ζ 20 ζ R=5, =0.03 R=20, =0.1 ζ ζ π R=20, =0.1 R=60, =0.3 ζ 0 R=120, ζ=0.6 R=60, =0.3 ζ C L -0.5 R=120, =0.6 -20

R -40 -1 100 1k 10k 100 251 1k 3.98k 10k ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 9 / 11 Resonance Peak for LP filter

12: Resonance Y 1 • QuadraticFactors + X = LC(jω)2+RCjω+1 • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 • QuadraticFactors + X = LC(jω)2+RCjω+1 • Damping Factor and Q • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q p • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q p • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − p 10 0

-10 0.7 0.8 0.9 1 1.2 1.4 ω (krad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − p 10 ζ 0.71 no peak, 0 ≥ ⇒ -10 0.7 0.8 0.9 1 1.2 1.4 ω (krad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − p 10 ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s)

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − p 10 ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 999, 26dB R=5, ζ=0.03 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − p 10 ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 999, 26dB R=5, ζ=0.03 2 990, 14dB ζ ω = ω 1 2ζ 20 R=20, =0.1 p c − 906, 5dB R=60, ζ=0.3 p 10 ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 999, 26dB R=5, ζ=0.03 ζ 2 20 990, 14dB R=20, =0.1 ωp = ωc 1 2ζ 906, 5dB R=60, ζ=0.3 − ζ ζ 0.5 passesp under corner, 10 529, 4dB R=120, =0.6 ≥ ⇒ ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 999, 26dB R=5, ζ=0.03 ζ 2 20 990, 14dB R=20, =0.1 ωp = ωc 1 2ζ 906, 5dB R=60, ζ=0.3 − ζ ζ 0.5 passesp under corner, 10 529, 4dB R=120, =0.6 ≥ ⇒ ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈ Three frequencies: ωp= peak, ωc= asymptotes cross, ωr= real impedance For ζ < 0.3, ω ω ω . All get called the resonant frequency. p ≈ c ≈ r

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Resonance Peak for LP filter

12: Resonance Y 1 1 • = 2 = 2 QuadraticFactors + X LC(jω) +RCjω+1 j ω +2ζj ω +1 • Damping Factor and Q ( ωc ) ωc • Parallel RLC c b bωc R • Behaviour at Resonance ωc = = 1000, ζ = = = • Away from resonance a 2aωc 2c 200 • Bandwidth and Q Y p ω • Power and Energy at is a function of so ω just scales frequency axis (= shift on log axis). Resonance + X ωc c • Low Pass Filter The damping factor, ζ, (“zeta”) determines the shape of the peak. • Resonance Peak for LP filter • Summary Peak frequency: 30 999, 26dB R=5, ζ=0.03 ζ 2 20 990, 14dB R=20, =0.1 ωp = ωc 1 2ζ 906, 5dB R=60, ζ=0.3 − ζ ζ 0.5 passesp under corner, 10 529, 4dB R=120, =0.6 ≥ ⇒ ζ 0.71 no peak, 0 ≥ ⇒ -10 ζ 1 can factorize 0.7 0.8 0.9 1 1.2 1.4 ≥ ⇒ ω (krad/s) 1 1 Gain relative to asymptote: @ ωp: @ ωc: Q 2ζ√1 ζ2 2ζ − ≈ Three frequencies: ωp= peak, ωc= asymptotes cross, ωr= real impedance For ζ < 0.3, ω ω ω . All get called the resonant frequency. p ≈ c ≈ r The exact relationship between ωp, ωc and ωr and the gain at these frequencies is affected by any other corner frequencies in the response.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 10 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q • Parallel RLC • Behaviour at Resonance • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q • Power and Energy at Resonance + • Low Pass Filter • Resonance Peak for LP filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter • Summary

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c 

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c  2 c b b sgn(a) a (jω) + b (jω) + c ωc = and ζ = = ◦ ⇒ a 2aωc √4ac p

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c  2 c b b sgn(a) a (jω) + b (jω) + c ωc = and ζ = = ◦ ⇒ a 2aωc √4ac 40 dB/decade slope change in magnitudep response ◦ ±

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c  2 c b b sgn(a) a (jω) + b (jω) + c ωc = and ζ = = ◦ ⇒ a 2aωc √4ac 40 dB/decade slope change in magnitudep response ◦ ± ζ phase changes rapidly by 180◦ over ω = 10∓ ω ◦ c

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c  2 c b b sgn(a) a (jω) + b (jω) + c ωc = and ζ = = ◦ ⇒ a 2aωc √4ac 40 dB/decade slope change in magnitudep response ◦ ± ζ phase changes rapidly by 180◦ over ω = 10∓ ω ◦ c Gain error in asymptote is 1 Q at ω ◦ 2ζ ≈ 0

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11 Summary *

12: Resonance Resonance is a peak in energy absorption • QuadraticFactors + • • Damping Factor and Q Parallel or series circuit has a real impedance at ωr • Parallel RLC ◦ • Behaviour at Resonance ⊲ peak response may be at a slightly different frequency • Away from resonance • Bandwidth and Q The quality factor, Q, of the resonance is • Power and Energy at ◦ Resonance + ω0 stored energy ω0 1 • , × Low Pass Filter Q power in R 3 dB bandwidth 2ζ • Resonance Peak for LP ≈ ≈ filter 1 1 • Summary 3 dB bandwidth is where power falls by or voltage by ◦ 2 √2 The stored energy sloshes between L and C ◦ 2 jω jω Quadratic factor: ω + 2ζ ω + 1 •  c   c  2 c b b sgn(a) a (jω) + b (jω) + c ωc = and ζ = = ◦ ⇒ a 2aωc √4ac 40 dB/decade slope change in magnitudep response ◦ ± ζ phase changes rapidly by 180◦ over ω = 10∓ ω ◦ c Gain error in asymptote is 1 Q at ω ◦ 2ζ ≈ 0 For further details see Hayt Ch 16 or Irwin Ch 12.

E1.1 Analysis of Circuits (2017-10213) Resonance: 12 – 11 / 11