New Realizations of Single OTRA-Based Sinusoidal Oscillators

Table 2

Key equations for the proposed oscillators.

Configuration	Oscillation condition and frequency	Active and passive sensitivities	Frequency stability factor
Figure 4(a)	Ideal: 𝐺2𝐺1𝐶−1=^2𝐶1; 𝜔𝑜=𝐺1𝐺2𝐶1𝐶2; Nonideal: 𝐺2𝐺𝑥(1+𝑘)𝐺1𝐺2+𝐺𝑥=𝐶1+𝐶2𝐶1−2𝑘𝐶2; 𝜔^′𝑜=𝐺1𝐺2𝐺𝑥𝐶1𝐶2𝐺2+𝐺𝑥	𝑆^𝜔′𝑜𝐺1=𝑆^𝜔′𝑜𝐺2=−𝑆^𝜔′𝑜𝐶1=−𝑆^𝜔′𝑜𝐶2=12; 𝑆^𝜔′𝑜𝐺𝑥=12𝐺2𝐺2+𝐺𝑥; 𝑆^𝜔′𝑜𝛽=𝑆^𝜔′𝑜𝐺𝑧=0	√𝑛; For conditions: 𝐶1=𝐶2𝐺=𝐶1=𝐺and𝐺2=𝑛𝐺
Figure 4(b)	Ideal: 𝐺1+^2𝐺1+𝐺3𝐺1=𝐶2𝐶1; 𝜔𝑜=𝐺1𝐺2𝐶1𝐶2; Nonideal: (1+𝑘)𝐺1+𝐺2+𝐺3𝐺1−2𝑘𝐺2=𝐶2𝐶1; 𝜔^′𝑜=𝐺2(𝐺1+(𝑘𝐺3/(1+𝑘))𝐶1𝐶2	𝑆^𝜔′𝑜𝐺2=−𝑆^𝜔′𝑜𝐶1=−𝑆^𝜔′𝑜𝐶2=12; 𝑆^𝜔′𝑜𝐺𝑧=−𝑆^𝜔′𝑜𝛽=12𝑘𝐺3(1+𝑘)^2𝐺1+𝑘(1+𝑘)𝐺3; 𝑆^𝜔′𝑜𝐺𝑥1=−2𝑘𝐺3/(1+𝑘)(1+𝑘)𝐺1+𝑘𝐺3; 𝑆^𝜔′𝑜𝐺1=121+𝑘𝑘1+𝐺3+1; 𝑆^𝜔′𝑜𝐺3=12𝑘𝐺3(1+𝑘)𝐺1+𝑘𝐺3	2√2𝑛2𝑛+1; For conditions: 𝐶2=𝐶=2𝐶1𝐺1=𝐺2=𝐺and𝐺3=𝐺𝑛
Figure 4(c)	Ideal: 𝐺1+^2𝐺1=𝐶2𝐶1; 𝜔𝑜=𝐺1𝐺2𝐶1𝐶2+𝐶3; Nonideal: ()𝐺1+𝑘1+^2𝐺1+𝑘𝐺2𝐶3𝐺1𝐶1=𝐶2𝐶1; 𝜔^′𝑜=(1−𝑘)𝐺1𝐺2(1+𝑘)𝐶1𝐶2+𝐶3	𝑆^𝜔′𝑜𝐺1=𝑆^𝜔′𝑜𝐺2=−𝑆^𝜔′𝑜𝐶1=12; 𝑆^𝜔′𝑜𝐺𝑥=−𝑆^𝜔′𝑜𝐺𝑧=𝑆^𝜔′𝑜𝛽=𝐺𝑧1−𝑘^2; 𝑆^𝜔′𝑜𝐶21=−2𝐶2𝐶2+𝐶3; 𝑆^𝜔′𝑜𝐶31=−2𝐶3𝐶2+𝐶3	8(1+𝑛)3; For conditions: 𝐶2=𝐶=3𝐶1; 𝐶3=𝑛𝐶2=𝑛𝐶; 𝐺1𝐺=𝐺=^22