By Kuo-Chang and Marie-Eve Carre
When Do We Have an Amplifier or an Oscillator?
When designing an amplifier, we all have faced the risk of seeing it unstable and getting an oscillator instead. The opposite is true as well: when trying to design an oscillator, one observes we can get something completely "silent". There is obviously an invisible barrier between those 2 functions. And it is correct to say an amplifier can, sometimes or always behave as an oscillator and vice-versa. That barrier is often dictated by the kind of feedback loop. Its intensity can determine which behavior, amplifier, or oscillator. It is up to the designer's skill to ensure which one they will generate. In this blog, we will opt for the angle of designing an oscillator starting from an amplifier "correctly" feedbacked.
What is the Barkhausen Criterion?
To refresh our high school or university lessons, the Barkhausen criterion is used to design the right quantity of feedback to ensure a circuit will be an oscillator and give the wanted frequency.
To illustrate the Barhausen criterion, let’s take a classic amplifier (of pure gain A) with a phase shift feedback β (complex number). The elements in A and β blocks are chosen in order to ensure a n*2*π (n being an integer value) phase shift between output and input for a full positive loop. In such cases, the produced output is enforcing the original input and makes the system unstable. The Barkhausen criterion gives the frequency and the minimum gain required. It’s very easy to establish that condition:
One can observe Xo = A.Xi = A. (Xs+Xf) = A. (Xs + β.Xo) and therefore :
Xo = A . Xs / (1-A.β)
Since there is no input in an oscillator: Xs=0 and Xo can exist (eventually) if 1-A.β is also 0 (we have a typical case of 0/0 which is “undetermined” (could be 0, ∞ or hopefully something in between…)
1 – Aβ = 0 is the Barkhausen Criterion or condition.
Or 1 – A (β_{r} + j β_{i}) = 0 ; since β is a complex number
There are thus 2 equations hidden in the Barkhausen relation :
How to Apply the Barkhausen Criterion?
Let's consider the following phase-shift sinewave oscillator; one can see the amplifier part made by an opamp mounted as the classic inverting scheme with gain = -R2/R1. We have then the feedback part made by 3 buffered RC circuits in series. Each cell ensures a 0° to 90° phase shift. Since the amplifier is already ensuring a -180° shift, we will end up with a global shift that can be more than 360°; assuring a strong "instability".
The feedback circuit (β) is then immediately computed as :
β = [1/(1 +jωRC)]. [1/(1 +jωRC)]. [1/(1 +jωRC)] = 1/(1+jωRC)^{3}
By developing the cube of a sum (i.e.( a+b)^{3} = a^{3 }+3a²b + 3ab² + b^{3}) :
β = 1 / (1 + 3 (jωRC) + 3(jωRC)² + (jωRC)^{3})
β = 1 / (1 + 3jωRC - 3ω²R²C² - jω^{3}R^{3}C^{3})
The first equation of the Barkhausen criterion says the imaginary part of β must be 0; therefore the oscillator frequency can be extracted:
3ωRC - ω^{3}R^{3}C^{3} = 0 or ω²R²C² = 3 or ω = √3/RC radian/s = √3/(2πRC) Hz; knowing R and C, the oscillator frequency is obtained
The second equation of the Barkhausen criterion says: 1 – Aβr = 0 or βr = 1/A or A=1/βr
In our circuit, βr = 1/(1-3ω²R²C²) and since from previous equation, ω²R²C² = 3, we will have :
βr = 1/(1-3x3) = 1/(-8) = -1/8 And the gain A is -8 independently of values R and C
To summarize, the Barkhausen criterion applied to the circuit is just: ω²R²C² = 3 and A = -8; that's the minimum (absolute) gain the amplifier has to provide in order to ensure and maintain the oscillations.
Simulation Results:
Thanks to the support from Marie-Eve Carre, the first attempt is just to quickly select an opamp (AD8613) and simulate it with LT-Spice. We have the oscillations but the frequency obtained is not exactly the one calculated. A more precise analysis of the opamp characteristics influence will be made.
Conclusion:
The Barkhausen criterion can be used to design an oscillator having the structure of a gain stage and a feedback stage. The feedback equation will give the oscillating frequency and the forward stage will fix the minimum gain the amplifier has to provide.
You are invited to discuss, share, and ask questions about this blog!