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.