13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 13: Filters • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 1 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. [Wikipedia] • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. [Wikipedia] • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. [Wikipedia] • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 Filters 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary A filter is a circuit whose gain varies with frequency. Often a filter aims to allow some frequencies to pass while blocking others. • Radio/TV: a “tuning” filter blocks all frequencies except the wanted channel • Loudspeaker: “crossover” filters send the right frequencies to different drive units • Sampling: an “anti-aliasing filter” eliminates all frequencies above half the sampling rate ◦ Phones: Sample rate = 8 kHz : filter eliminates frequencies above 3.4 kHz. [Wikipedia] • Computer cables: filter eliminates interference E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 2 / 13 1st Order Low-Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = 1/jωC R+1/jωC = 1 jωRC+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = Transformations (A) • Conformal Filter Transformations (B) • Summary 1 jω p +1 1 RC 0 |Gain| (dB) 13: Filters -10 -20 -30 E1.1 Analysis of Circuits (2015-7256) 0.1p p ω (rad/s) 10p Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = 1 jω p +1 1 RC p Asymptotes: 1 and jω • Summary 0 |Gain| (dB) 13: Filters -10 -20 -30 E1.1 Analysis of Circuits (2015-7256) 0.1p p ω (rad/s) 10p Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = 1 jω p +1 1 RC p Asymptotes: 1 and jω Very low ω : Capacitor = open circuit • Summary 0 |Gain| (dB) 13: Filters -10 -20 -30 E1.1 Analysis of Circuits (2015-7256) 0.1p p ω (rad/s) 10p Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = 1 jω p +1 1 RC p Asymptotes: 1 and jω Very low ω : Capacitor = open circuit Very high ω : Capacitor short circuit 0 |Gain| (dB) 13: Filters -10 -20 -30 E1.1 Analysis of Circuits (2015-7256) 0.1p p ω (rad/s) 10p Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = 1 jω p +1 1 RC p Asymptotes: 1 and jω Very low ω : Capacitor = open circuit Very high ω : Capacitor short circuit 0 |Gain| (dB) 13: Filters -10 -20 -30 0.1p p ω (rad/s) 10p A low-pass filter because it allows low frequencies to pass but attenuates (makes smaller) high frequencies. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 3 / 13 1st Order Low-Pass Filter • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = 1/jωC R+1/jωC = 1 jωRC+1 = b Corner frequency: p = a = 1 jω p +1 1 RC p Asymptotes: 1 and jω Very low ω : Capacitor = open circuit Very high ω : Capacitor short circuit 0 |Gain| (dB) 13: Filters -10 -20 -30 0.1p p ω (rad/s) 10p A low-pass filter because it allows low frequencies to pass but attenuates (makes smaller) high frequencies. The order of a filter: highest power of jω in the denominator. Almost always equals the total number of L and/or C . E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 3 / 13 Low-Pass with Gain Floor 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R+1/jωC 4R+1/jωC = jωRC+1 jω4RC+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 4 / 13 Low-Pass with Gain Floor 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R+1/jωC 4R+1/jωC = jωRC+1 jω4RC+1 = jω q +1 jω p +1 1 1 , q = RC Corner frequencies: p = 4RC Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -5 -10 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 4 / 13 Low-Pass with Gain Floor 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R+1/jωC 4R+1/jωC = jωRC+1 jω4RC+1 = jω q +1 jω p +1 1 1 , q = RC Corner frequencies: p = 4RC Asymptotes: 1 and 14 Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -5 -10 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 4 / 13 Low-Pass with Gain Floor 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = R+1/jωC 4R+1/jωC = jωRC+1 jω4RC+1 = jω q +1 jω p +1 1 1 , q = RC Corner frequencies: p = 4RC Asymptotes: 1 and 14 Very low ω : Capacitor = open circuit Resistor R unattached. Gain = 1 0 -5 -10 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 4 / 13 Low-Pass with Gain Floor 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = R+1/jωC 4R+1/jωC = jωRC+1 jω4RC+1 = jω q +1 jω p +1 1 1 , q = RC Corner frequencies: p = 4RC Asymptotes: 1 and 14 Very low ω : Capacitor = open circuit Resistor R unattached. Gain = 1 0 -5 -10 Very high ω : Capacitor short circuit 0.1q p ω q 10q Circuit is potential divider with gain 20 log10 14 = −12 dB. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 4 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Inverting amplifier so Y X /jωC ) = − 3R||(R+ R 1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × 1 R+1/jωC 4R+1/jωC Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Same transfer function as before except × − 3 = +9.5 dB. Transformations (A) • Conformal Filter Transformations (B) • Summary 10 5 0 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Same transfer function as before except × − 3 = +9.5 dB. Advantages of op-amp crcuit: • Summary 10 5 0 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Same transfer function as before except × − 3 = +9.5 dB. Advantages of op-amp crcuit: 1. Can have gain > 1. 10 5 0 0.1q E1.1 Analysis of Circuits (2015-7256) p ω q 10q Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Same transfer function as before except × − 3 = +9.5 dB. Advantages of op-amp crcuit: 1. Can have gain > 1. 10 5 0 2. Low output impedance - loading does not affect filter E1.1 Analysis of Circuits (2015-7256) 0.1q p ω q 10q Filters: 13 – 5 / 13 Opamp filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Inverting amplifier so Y X /jωC ) 3R(R+ /jωC ) = − 3R||(R+ = − R R×(3R+R+1/jωC ) 1 = −3 × R+1/jωC 4R+1/jωC 1 = −3 × jωRC+1 jω4RC+1 Same transfer function as before except × − 3 = +9.5 dB. Advantages of op-amp crcuit: 1. Can have gain > 1. 10 5 0 2. Low output impedance - loading does not affect filter 0.1q p ω q 10q 3. Resistive input impedance - does not vary with frequency E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 5 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X =− 1/jωC R Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X =− Transformations (A) • Conformal Filter Transformations (B) 1/jωC R 1 = − jωRC 20 0 • Summary -20 0.1 E1.1 Analysis of Circuits (2015-7256) 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X =− 1/jωC R 1 = − jωRC C Capacitor: i = C dv dt Transformations (A) • Conformal Filter Transformations (B) 20 0 • Summary -20 0.1 E1.1 Analysis of Circuits (2015-7256) 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X =− 1/jωC R 1 = − jωRC C Capacitor: i = C dv dt i= x R Transformations (A) • Conformal Filter Transformations (B) = −C dy dt 20 0 • Summary -20 0.1 E1.1 Analysis of Circuits (2015-7256) 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X =− 1/jωC R 1 = − jωRC C Capacitor: i = C dv dt i= dy dt x R = Transformations (A) • Conformal Filter Transformations (B) = −C dy dt −1 RC x 20 0 • Summary -20 0.1 E1.1 Analysis of Circuits (2015-7256) 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) Y X =− 1/jωC R 1 = − jωRC C Capacitor: i = C dv dt i= dy dt x R = Rt = −C dy dt −1 RC x dy dt 0 dt = −1 RC Rt 0 xdt 20 0 • Summary -20 0.1 E1.1 Analysis of Circuits (2015-7256) 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− R 1 = − jωRC C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = E1.1 Analysis of Circuits (2015-7256) = −1 RC −1 RC Rt 0 Rt 0 xdt xdt + y(0) 20 0 -20 0.1 1 ω RC 10 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− R 1 = − jωRC C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 0 xdt + y(0) ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ E1.1 Analysis of Circuits (2015-7256) 20 -20 0.1 1 ω RC 10 1 ω. Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− R 1 = − jωRC C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 20 0 xdt + y(0) ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ -20 0.1 1 ω RC 10 1 ω. We can limit the LF gain to 20 dB: E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− 1 = − jωRC R C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 20 0 xdt + y(0) ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ -20 0.1 1 ω RC 10 1 ω. We can limit the LF gain to 20 dB: Y X = − 10R||R/jωC E1.1 Analysis of Circuits (2015-7256) 1 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− 1 = − jωRC R C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 20 0 xdt + y(0) -20 0.1 ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ 1 ω RC 10 1 ω. We can limit the LF gain to 20 dB: Y X 10R× /jωC = − 10R||R/jωC = − R(10R+ 1/jωC ) E1.1 Analysis of Circuits (2015-7256) 1 1 Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− 1 = − jωRC R C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 20 0 xdt + y(0) -20 0.1 ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ 1 ω RC 10 1 ω. We can limit the LF gain to 20 dB: Y X 10R× /jωC = − 10R||R/jωC = − R(10R+ 1/jωC ) 1 1 10 = − jω10RC+1 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 6 / 13 Integrator 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X 1/jωC =− 1 = − jωRC R C Capacitor: i = C dv dt i= dy dt = −C dy dt x R = −1 RC x Rt dy dt 0 dt y(t) = = −1 RC −1 RC Rt 0 Rt 0 xdt 20 0 xdt + y(0) -20 0.1 ωt RNote: if x(t) = cos cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝ 1 ω RC 10 1 ω. We can limit the LF gain to 20 dB: Y X 10R× /jωC = − 10R||R/jωC = − R(10R+ 1/jωC ) 0.1 10 = − jω10RC+1 ωc = RC E1.1 Analysis of Circuits (2015-7256) 1 1 Filters: 13 – 6 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R R+1/jωC Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R R+1/jωC = jωRC jωRC+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Asymptotes: jωRC and 1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Asymptotes: jωRC and 1 Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -10 -20 -30 0.1p E1.1 Analysis of Circuits (2015-7256) ω p 10p Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Asymptotes: jωRC and 1 Very low ω : C open circuit: gain = 0 Very high ω : C short circuit: gain = 1 • Summary 0 -10 -20 -30 0.1p E1.1 Analysis of Circuits (2015-7256) ω p 10p Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Asymptotes: jωRC and 1 Very low ω : C open circuit: gain = 0 Very high ω : C short circuit: gain = 1 We can add an op-amp to give a low-impedance output. 0 -10 -20 -30 0.1p E1.1 Analysis of Circuits (2015-7256) ω p 10p Filters: 13 – 7 / 13 High Pass Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary Y X = R R+1/jωC = jωRC jωRC+1 1 Corner Freq: p = RC Asymptotes: jωRC and 1 Very low ω : C open circuit: gain = 0 Very high ω : C short circuit: gain = 1 We can add an op-amp to give a low-impedance output. Or add gain: Z X = 1+ RB RA × jωRC jωRC+1 0 -10 -20 -30 0.1p E1.1 Analysis of Circuits (2015-7256) ω p 10p Filters: 13 – 7 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X = R2 +jωL 1/jωC +R1 +R2 +jωL Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Asymptotes: jωR2 C and 1 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Asymptotes: jωR2 C and 1 Corner frequencies: +20 dB/dec at p = RL2 = 100 rad/s −40 dB/dec at q= √1 LC E1.1 Analysis of Circuits (2015-7256) = 1000 rad/s Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Asymptotes: jωR2 C and 1 Corner frequencies: +20 dB/dec at p = RL2 = 100 rad/s −40 dB/dec at q= √1 LC E1.1 Analysis of Circuits (2015-7256) = 1000 rad/s 0 -20 -40 p 100 q ω 1k 10k Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Asymptotes: jωR2 C and 1 Corner frequencies: +20 dB/dec at p = RL2 = 100 rad/s −40 dB/dec at q= √1 LC = 1000 rad/s 0 -20 -40 p 100 q ω 1k 10k q = 2 (R1 + R2 ) C = 0.6. 4ac = Q = 0.83 = −1.6 dB (+0.04 dB due to p) Damping factor: ζ = √ b 1 Gain error at q is 2ζ E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 2nd order filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Y X R2 +jωL 1/jωC +R1 +R2 +jωL = = LC(jω)2 +R2 Cjω LC(jω)2 +(R1 +R2 )Cjω+1 = jωC(jωL+R2 ) LC(jω)2 +(R1 +R2 )Cjω+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Asymptotes: jωR2 C and 1 Corner frequencies: +20 dB/dec at p = RL2 = 100 rad/s −40 dB/dec at q= √1 LC = 1000 rad/s 0 -20 -40 p 100 q ω 1k 10k q = 2 (R1 + R2 ) C = 0.6. 4ac = Q = 0.83 = −1.6 dB (+0.04 dB due to p) Damping factor: ζ = √ b 1 Gain error at q is 2ζ Compare with 1st order: 2nd order filter attenuates more rapidly than a 1st order filter. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 8 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 • Summary E1.1 Analysis of Circuits (2015-7256) [assume V+ = V− = Z ] Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC Sub Y : Z E1.1 Analysis of Circuits (2015-7256) (1+jωmRC) jωmRC (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) jωmRC m(jωRC)2 Z X = m(jωRC)2 +2jωRC+1 Sub Y : Z ⇒ E1.1 Analysis of Circuits (2015-7256) − Z (1 + jωRC) = XjωRC Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 = 996 rad/s Corner freq: p = √mRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 1 = pRC = √1m = 0.6 = 996 rad/s, ζ = 2Q Corner freq: p = √mRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -20 -40 100 ω 1k 10k −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 1 = pRC = √1m = 0.6 = 996 rad/s, ζ = 2Q Corner freq: p = √mRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -20 -40 100 Asymptotes: ω 1k jω p 2 10k and 1 −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 1 = pRC = √1m = 0.6 = 996 rad/s, ζ = 2Q Corner freq: p = √mRC E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -20 -40 100 Asymptotes: ω 1k jω p 2 10k and 1 −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 1 = pRC = √1m = 0.6 = 996 rad/s, ζ = 2Q Corner freq: p = √mRC Sallen-Key: 2nd order filter without inductors. Can easily have gain >1. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Sallen-Key Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary 0 -20 -40 100 Asymptotes: ω 1k jω p 2 10k and 1 −X Y −Z Y −Z KCL @ Y : Y 1/jωC + 1/jωC + R =0 [assume V+ = V− = Z ] ⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC Z + 1Z−Y KCL @ V+ : mR /jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC (1+jωmRC) (1 + 2jωRC) − Z (1 + jωRC) = jωmRC m(jωRC)2 (jω/p)2 Z X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1 Sub Y : Z ⇒ XjωRC 1 1 = pRC = √1m = 0.6 = 996 rad/s, ζ = 2Q Corner freq: p = √mRC Sallen-Key: 2nd order filter without inductors. Can easily have gain >1. ζ Designing: Choose m = ζ −2 ; C any convenient value; R = pC . E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 9 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter After much algebra: Z X = (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter After much algebra: (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 Transformations (A) • Conformal Filter Transformations (B) • Summary Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 • Summary Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Z Non-inverting amp, X =1+m Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Z Non-inverting amp, X =1+m At ω = p, jω p 2 = −1: numerator = zero resulting in infinite attenuation. Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Z Non-inverting amp, X =1+m At ω = p, jω p 2 m+1 = 5.6dB 2ζ p 0 -20 -40 200 300 ω (rad/s) 500 = −1: numerator = zero resulting in infinite attenuation. Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Z Non-inverting amp, X =1+m At ω = p, jω p m+1 = 5.6dB 2ζ p 0 -20 -40 200 300 ω (rad/s) 500 2 = −1: numerator = zero resulting in infinite attenuation. The 3 dB notch width is approximately 2ζp = 2(1 − m)p. Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Twin-T Notch Filter 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary After much algebra: p= (1+m)((2jωRC)2 +1) (2jωRC)2 +4(1−m)jωRC+1 Z X = = (1+m)((jω/p)2 +1) (jω/p)2 +2ζ(jω/p)+1 1 2RC = 314, ζ = 1 − m = 0.1 Very low ω : C open circuit Z Non-inverting amp, X =1+m Very high ω : C short circuit Z Non-inverting amp, X =1+m At ω = p, jω p m+1 = 5.6dB 2ζ p 0 -20 -40 200 300 ω (rad/s) 500 2 = −1: numerator = zero resulting in infinite attenuation. The 3 dB notch width is approximately 2ζp = 2(1 − m)p. Used to remove one specific frequency (e.g. mains hum @ 50 Hz) Do not try to memorize this circuit E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 10 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: Transformations (A) • Conformal Filter Transformations (B) ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 ♠ -10 -20 -30 10 100 1k ω rad/s 10k • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, Impedance scaling: ZL ZR = jωL ZL R , ZC 0 ♠ -10 Scale all impedances by k : Transformations (A) • Conformal Filter Transformations (B) = −ω 2 LC . -20 -30 10 100 1k ω rad/s 10k • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL -20 -30 10 100 1k ω rad/s 10k • Summary k = 20 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. -20 -30 10 100 1k ω rad/s 10k k = 20 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) -20 -30 10 100 1k ω rad/s 10k k = 20 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) Frequency Shift: Scale reactive components by k : E1.1 Analysis of Circuits (2015-7256) -20 -30 10 100 1k ω rad/s 10k k = 20 Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) Frequency Shift: Scale reactive components by k : -20 -30 10 100 1k ω rad/s 10k k = 20 R′ = R, C ′ = kC, L′ = kL k=5 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC = −ω 2 LC . 0 Impedance scaling: ♠ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) Frequency Shift: Scale reactive components by k : -20 -30 10 100 1k ω rad/s 10k k = 20 R′ = R, C ′ = kC, L′ = kL ⇒ Z ′ (k−1 ω) ≡ Z(ω) k=5 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC 0 Impedance scaling: ♥ -10 Scale all impedances by k : ′ ′ R = kR, C = k −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) Frequency Shift: Scale reactive components by k : R′ = R, C ′ = kC, L′ = kL ⇒ Z ′ (k−1 ω) ≡ Z(ω) Graph shifts left by a factor of k . E1.1 Analysis of Circuits (2015-7256) = −ω 2 LC . ♠ -20 -30 10 100 1k ω rad/s 10k k = 20 k=5 Filters: 13 – 11 / 13 Conformal Filter Transformations (A) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary VY A dimensionless gain, V , can always be written using dimensionless X impedance ratio terms: ZR ZC = jωRC, ZL ZR = jωL ZL R , ZC 0 Impedance scaling: ♥ -10 Scale all impedances by k : ′ ′ R = kR, C = k = −ω 2 LC . −1 ′ C, L = kL Impedance ratios are unchanged so graph stays the same. (k is arbitrary) Frequency Shift: Scale reactive components by k : R′ = R, C ′ = kC, L′ = kL ⇒ Z ′ (k−1 ω) ≡ Z(ω) Graph shifts left by a factor of k . ♠ -20 -30 10 100 1k ω rad/s 10k k = 20 k=5 Must scale all reactive components in the circuit by the same factor. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 11 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Change LR circuit to RC: 0 ♦ -10 -20 -30 1k 10k ω rad/s 100k 1M Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter 0 Change LR circuit to RC: ♦ -10 ′ ′ Change R = kL, C = 1 kR -20 -30 1k 10k ω rad/s 100k 1M Transformations (A) • Conformal Filter Transformations (B) • Summary k = 106 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter 0 Change LR circuit to RC: ♦ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k 10k ω rad/s 100k 1M Transformations (A) • Conformal Filter Transformations (B) • Summary k = 106 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♦ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. 10k ω rad/s 100k 1M • Summary k = 106 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♦ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. (k is arbitrary) 10k ω rad/s 100k 1M • Summary k = 106 E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♦ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. (k is arbitrary) 10k ω rad/s 100k 1M • Summary Reflect frequency axis around ωm : k = 106 Change R′ = ω k C , C ′ = ω 1kR m m E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♦ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. (k is arbitrary) 10k ω rad/s 100k 1M • Summary Reflect frequency axis around ωm : k = 106 Change R′ = ω k C , C ′ = ω 1kR ∗m 2 m ⇒ ZR′ ZC ′ ωm ω = ZC ZR (ω) k = 0.1, ωm = 20 k E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♣ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. (k is arbitrary) ♦ 10k ω rad/s 100k 1M • Summary Reflect frequency axis around ωm : k = 106 Change R′ = ω k C , C ′ = ω 1kR ∗m 2 m ⇒ ZR′ ZC ′ ωm ω = ZC ZR (ω) (a) Magnitude graph flips E1.1 Analysis of Circuits (2015-7256) k = 0.1, ωm = 20 k Filters: 13 – 12 / 13 Conformal Filter Transformations (B) 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) 0 Change LR circuit to RC: ♣ -10 ′ Change R = kL, C ⇒ ZR′ ZC ′ ′ = jωR′ C ′ = 1 = kR jωL ZL = R ZR -20 -30 1k Impedance ratios are unchanged at all ω so graph stays the same. (k is arbitrary) ♦ 10k ω rad/s 100k 1M • Summary Reflect frequency axis around ωm : k = 106 Change R′ = ω k C , C ′ = ω 1kR ∗m 2 m ⇒ ZR′ ZC ′ ωm ω = ZC ZR (ω) (a) Magnitude graph flips k = 0.1, ωm = 20 k (b) Phase graph flips and negates since ∠z ∗ = −∠z . (k is arbitrary) E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 12 / 13 Summary 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter • The order of a filter is the highest power of jω in the transfer function denominator. Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 13 / 13 Summary 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter • The order of a filter is the highest power of jω in the transfer function denominator. • Active filters use op-amps and usually avoid the need for inductors. ◦ Sallen-Key design for high-pass and low-pass. ◦ Twin-T design for notch filter: gain = 0 at notch. Transformations (A) • Conformal Filter Transformations (B) • Summary E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 13 / 13 Summary 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary • The order of a filter is the highest power of jω in the transfer function denominator. • Active filters use op-amps and usually avoid the need for inductors. ◦ Sallen-Key design for high-pass and low-pass. ◦ Twin-T design for notch filter: gain = 0 at notch. • For filters using R and C only: ◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales −1 frequency by (pk) . E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 13 / 13 Summary 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary • The order of a filter is the highest power of jω in the transfer function denominator. • Active filters use op-amps and usually avoid the need for inductors. ◦ Sallen-Key design for high-pass and low-pass. ◦ Twin-T design for notch filter: gain = 0 at notch. • For filters using R and C only: ◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales −1 frequency by (pk) . ◦ Interchange R and C : Substituting R′ = ω0kC and C ′ = kω10 R flips the frequency response around ω0 (∀k ). Changes a low-pass filter to high pass and vice-versa. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 13 / 13 Summary 13: Filters • Filters • 1st Order Low-Pass Filter • Low-Pass with Gain Floor • Opamp filter • Integrator • High Pass Filter • 2nd order filter • Sallen-Key Filter • Twin-T Notch Filter • Conformal Filter Transformations (A) • Conformal Filter Transformations (B) • Summary • The order of a filter is the highest power of jω in the transfer function denominator. • Active filters use op-amps and usually avoid the need for inductors. ◦ Sallen-Key design for high-pass and low-pass. ◦ Twin-T design for notch filter: gain = 0 at notch. • For filters using R and C only: ◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales −1 frequency by (pk) . ◦ Interchange R and C : Substituting R′ = ω0kC and C ′ = kω10 R flips the frequency response around ω0 (∀k ). Changes a low-pass filter to high pass and vice-versa. For further details see Hayt et al. Chapter 16. E1.1 Analysis of Circuits (2015-7256) Filters: 13 – 13 / 13