SaM - Introduction to Oscillators

Remeber:

• The basis of an oscillator is a circuit like this:
  
• Not only that but we want it to have a AC output even if $V_s=0\text{V}$.
• To do this we first define the input-output transfer function:$\frac{V_0}{V_S}=\frac{A}{1+\beta A}$As we can see from this formula to have a “diverging output”, so that the transient part does not decay, we need two things, we will later see more in detail in the Barkhausen Conditions:$\begin{array}{l}|\beta A|\ge 1\\ \angle \beta A=180°\end{array}$

• In an oscillator, I want a $V_{out}$ which is unstable or marginally stable.
  So with a couple of poles which are complex and conjugate, and contrary to stable poles, we actually need:$\mathrm{Re}(p_1)=\mathrm{Re}(p_2)\ge 0$Actually, we can make a distinction between two outcomes:
  
  • We can have: $\mathrm{Re}(p_1)=\mathrm{Re}(p_2)>0$ : unstable poles ⇒ The output will become a square wave do to the $V_{\text{SAT}}$ voltage source.
  • Or instead: $\mathrm{Re}(p_1)=\mathrm{Re}(p_2)=0$ : marginally stable poles ⇒ The output will become a sine wave.

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We have said that the AC excitation was given by a voltage generator, sometimes I called it oscillator.

• The base of oscillating circuit is a feedback circuit.
• So we have to refer to the simple schematic of an amplifier $A$, and a feedback network $\beta$, which is sent to the input with a negative feedback
• Iwant also to put here this two terminals, ($\pm V_{cc}$), because this is a network where we have an active device, an amplifier, which is provided by a continuous DC power.
• Then we have a passive network, which is the feedback network, which brings back the output voltage to input, with this node here.
• Every time we have used an operational amplifier, we use this kind of structure here.
• Usually the feedback loop is used to control the output, make sure that it converges to a certain value, assurign that the response is stable and the transient vanishes.
  For oscillators is the opposite.
• NOTE: $A$ in this case is the gain of the amplifier, for definition it is:$A=\frac{V_0}{V_i}$

Now we want to have a circuit which we don’t provide the input for, so $V_S=0$ ==We want to have a circuit which provides a variable output with no input==.

• What we want is therefore a circuit which is not stable since we want that with the signal input equal to zero, the output signal is a wave, an AC wave.
• Two types I can accept:
  • A sine wave.
  • Or a square wave (actually I’ll only have this if I take into account the nonlinear behavior, so I’m out from the normal modeling).
• I want a $V_{out}$ like this, which is unstable or marginally stable with a couple of poles which are complex and conjugate.
• So I need a couple of complex conjugated poles with real part, larger or equal to zero.
• ==When I take into account this kind of situation, I will have that the response to the initial condition, or the transient response in any case will never finish, will be persistent and this is what I need from a oscillator==.

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