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Table 2 Analytical results for oscillator period and phase

From: Analytical approximations for the amplitude and period of a relaxation oscillator

Times

Hysteresisa

Delayb

τ1 (A > R)

n = 1: (β A /α A β R ) + K A /β A +(K A /β A ) ln(K A / β a / k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaHYoGydaWgaaWcbaGaemyyaegabeaakiabc+caViabdUgaRbWcbeaaaaa@3157@ )

α-1 ln [(β1 - β0)/(β1 - αK)]

 

n ≠ 1: (β A /α A β R ) + K A /β A + [β A (n- 1)]-1 K A n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4saS0aa0baaSqaaiabdgeabbqaaiabd6gaUbaaaaa@2F91@ (β a /k)1-n

 

τ2 (A <R)

α A = α R : α-1 ln [(β A /α)/ β a / k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaHYoGydaWgaaWcbaGaemyyaegabeaakiabc+caViabdUgaRbWcbeaaaaa@3157@ ]

α-1 ln [(β1 - αK - β0)/(αK - β0)]

 

α A α R : α min 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqySde2aa0baaSqaaiGbc2gaTjabcMgaPjabc6gaUbqaaiabgkHiTiabigdaXaaaaaa@33A0@ ln [(β A /|α A - α R |)/ β a / k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaHYoGydaWgaaWcbaGaemyyaegabeaakiabc+caViabdUgaRbWcbeaaaaa@3157@ ]

 

τ tot

τ1 + τ2

3(τ1 + τ2)

τtot, limiting

α-1 [β A /β R + ln(β A /α β a / k MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqaHYoGydaWgaaWcbaGaemyyaegabeaakiabc+caViabdUgaRbWcbeaaaaa@3157@ )]

3α-1 ln [β1/αK]

Concentrations

  

A max

(β A /α A )- (β R /α A ) ln(β A /β R )

(β1/α) - K

R max

α A = α R : [(β A - β a )/α-K A ]/e

 
 

α A α R : ( β A / α A ) ( α R / α A ) α R / α A MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqySde2aaSbaaSqaaiabdgeabbqabaGccqGHGjsUcqaHXoqydaWgaaWcbaGaemOuaifabeaakiabcQda6iabcIcaOiabek7aInaaBaaaleaacqWGbbqqaeqaaOGaei4la8IaeqySde2aaSbaaSqaaiabdgeabbqabaGccqGGPaqkcqGGOaakcqaHXoqydaWgaaWcbaGaemOuaifabeaakiabc+caViabeg7aHnaaBaaaleaacqWGbbqqaeqaaOGaeiykaKYaaWbaaSqabeaacqaHXoqydaWgaaadbaGaemOuaifabeaaliabc+caViabeg7aHnaaBaaameaacqWGbbqqaeqaaaaaaaa@4C3C@

 

C max

β A /α A

 

A min

β a /(α A + kRmax)

β0/α

Amin, limiting

β a /( A /αe)

β0/α

  1. aLimiting expressions for the hysteresis oscillator are for α A = α R = α, β A > β R , β A > αK A .
  2. bLimiting expressions for the delay oscillator are for β1/α > K > β0/α.