First Order Filter

A first order filter, with cutoff frequency R, is used to identify the dynamic component, x , of the input signal, X. The transfer functions of the signal sent to the dynamic and the static models, assuming X0=0, are:

TF of signal to dynamic model =

x( jω ) X( jω ) = ω 2 R 2 + ω 2 +j Rω R 2 + ω 2  , j= −1 , ω=2πf=angular frequency MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4bWaaeWaaeaacaWGQbGaeqyYdChacaGLOaGaayzkaaaabaGaamiw amaabmaabaGaamOAaiabeM8a3bGaayjkaiaawMcaaaaacqGH9aqpda WcaaqaaiabeM8a3naaCaaaleqabaGaaGOmaaaaaOqaaiaadkfadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcqaHjpWDdaahaaWcbeqaaiaaik daaaaaaOGaey4kaSIaamOAamaalaaabaGaamOuaiabeM8a3bqaaiaa dkfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaHjpWDdaahaaWcbe qaaiaaikdaaaaaaOGaaGjbVlaacYcacaaMe8UaamOAaiabg2da9maa kaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcacaaMe8UaeqyYdCNaey ypa0JaaGOmaiabec8aWjaadAgacqGH9aqpcaWGHbGaamOBaiaadEga caWG1bGaamiBaiaadggacaWGYbGaaGjbVlaadAgacaWGYbGaamyzai aadghacaWG1bGaamyzaiaad6gacaWGJbGaamyEaaaa@753C@

TF of signal to dynamic model = 1

The following are Bode plots for these transfer functions:



Figure 1.
The Bode plots show the magnitude and loss angle of the transfer functions over a range of operating frequencies:
  • The top figure plots the magnitude of the transfer function against Log 10 ( ω/R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaab+ gacaqGNbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGaeqyY dCNaai4laiaadkfaaiaawIcacaGLPaaaaaa@3F2C@ .
  • The bottom figure plots the loss angle of the transfer function against Log 10 ( ω / R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeitaiaab+ gacaqGNbWaaSbaaSqaaiaaigdacaaIWaaabeaakmaabmaabaGaeqyY dCNaai4laiaadkfaaiaawIcacaGLPaaaaaa@3F2C@ .
  • Plots of the signal sent to the dynamic model are gray-blue.
  • Plots of the signal sent to the static model are brick-red.

The log scale used for the x-axis lets you view a wide range of frequencies and filter behavior as follows:

  1. When ( ω / R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHjpWDcaGGVaGaamOuaaGaayjkaiaawMcaaaaa@3AD6@ ≪ 1, that is at low frequencies, then:
    • The magnitude of the signal sent to the dynamic model is close to 0.
    • The loss angle of the signal sent to the dynamic model is close to 90°.
    • The magnitude of the signal sent to the static model is 1.
    • The loss angle of the signal sent to the static model is 0°.

    The bushing essentially behaves as the static model. The loss angle of the signal sent to the dynamic model is close to 90°, but this is not important since the magnitude of the signal is close to zero.

  2. When ( ω / R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHjpWDcaGGVaGaamOuaaGaayjkaiaawMcaaaaa@3AD6@ ≫ 1, that is at high frequencies, then:
    • The magnitude of the signal sent to the dynamic model is close to 1.
    • The loss angle of the signal sent to the dynamic model is close to 0° .
    • The magnitude of the signal sent to the static model is 1.
    • The loss angle of the signal sent to the static model is 0°.

    The bushing essentially behaves as a dynamic model superimposed on top of a static model.

  3. When ( ω / R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHjpWDcaGGVaGaamOuaaGaayjkaiaawMcaaaaa@3AD6@ ≫ 1, that is at cut-off frequency, then:
    • The magnitude of the signal sent to the dynamic model is 1 / 2 ≈ 0.701 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+ cadaGcaaqaaiaaikdaaSqabaGccqGHijYUcaaIWaGaaiOlaiaaiEda caaIWaGaaGymaaaa@3D98@ .
    • The loss angle of the signal sent to the dynamic model is 45°.
    • The magnitude of the signal sent to the static model is 1.
    • The loss angle of the signal sent to the static model is 0°.

    The bushing essentially behaves as a dynamic model superimposed on top of a static model.