SaM - MEMORIZE (Oscillators)

List of things to memorize:

SaM - Oscillators

• SaM - Introduction to Oscillators
• SaM - Square Wave Oscillator
• SaM - Barkhausen Conditions
• SaM - FM (Frequency Modulation) Based on Oscillators
• SaM - AM (Amplitude Modulation) Based on Oscillators
• SaM - Oscillators based on Wien Bridge
• SaM - Phase Shift Oscillator
• SaM - High Frequency Oscillator • 3 Point Oscillator • Colpits and Hartley Oscillators
  • SaM - Quartz in a 3 Point Oscillator
  • SaM - AT-Cut Quartz
  • SaM - Behavior of the Quartz Ocillator at High Frequencies
• SaM - Quartz Oscillator

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SaM - Introduction to Oscillators

• Feedback structure:
  
• Transfer function:$\frac{V_0}{V_S}=\frac{A}{1+\beta A}$
• Barkhausen Conditions:$\begin{array}{l}|\beta A|\ge 1\\ \angle \beta A=180°\end{array}$
• To have an unstable or marginally stable (oscillating) output, i will a couple of poles which are complex and conjugate with: $\mathrm{Re}(p_1)=\mathrm{Re}(p_2)\ge 0$.
• Difference between unstable and marginally unstable:
  

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SaM - Square Wave Oscillator

• Structure:
  
• Formula:$T=2RC\cdot \ln \left(\frac{1-\beta }{1+\beta }\right)$

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SaM - Barkhausen Conditions

• To create an oscillating output:

1. Transfer function: $\frac{V_0}{V_S}=\frac{A}{1+\beta A}$
2. I need a (preferably) marginally stable output, so: $1+\beta (j\omega )A(j\omega )=0$
3. Solution: “Barkausen conditions”:$\begin{array}{l}|\beta A|\ge 1\\ \angle \beta A=180°\end{array}$(If $|\beta A|$ is exactly $1$, we will have a marginally stable output)

• Frequency stability: the two parameters ($\beta$ and $A$), which are the gain of the amplifier and the gain of the feedback network, have to be stable in time, not changing.
  ⇒ We can “take a big slope” when crossing the $-180°$ degrees line, so around ${\omega }_0$ (where I need to oscillate).
• “Big slope solution:
  

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SaM - FM (Frequency Modulation) Based on Oscillators

• Using a square wave oscillator:
  
• Formula:$T=2RC\cdot \ln \left(\frac{1+\beta }{1-\beta }\right)$
• Frequency of an FM modulated signal:$f(t)=f_0\left(1+\frac{f_A}{f_0}x(t)\right)$
• FM modulation using an integrator:
  (For instance if I have $x(t)$ which is a slowly growing invariant signal, then the frequency would increase a litte over time.)
  
• Bandwidth of the FM modulated signal:
  
  Where:
  • $\Delta f_{MAX}=f_A\cdot x_{MAX}$.
  • And $f_A$ comes from: $f(t)=f_0+f_{A}x(t)$.
• Since FM transforms a the physical input variation into a variation of frequency, it is a very robust solution to reject noise.
• Complete FM Circuit:
  
  • The sensor is part of the oscillator.
  • The FM Demodulator gives as output a voltage related to the frequency in input.
• Real World Measures:
  • FM is used for signals which can be static but also they have a large bandwidth up to $100\text{kHz}$
  • The typical value for $f_0$ is $2\text{MHz}$ (much larger than the maximum signal frequency).

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SaM - AM (Amplitude Modulation) Based on Oscillators

• Simplest circuit for AM (Amplitude Modulation):
  
• Output:$V_o(t)=\frac{C_0}{C_R}(1+x)\cdot V_g$Where: $V_g=V\sin (2\pi f_{0}t+\phi )$
• General formula of the AM modulation:$V_0(t)=V_R[1+mx(t)]\sin (2\pi f_{0}t+\phi )$Where: $V_R=V\cdot \frac{C_0}{C_R}$
• Spectrum analysis:
  
• Good circuits for AM modulation:
  • Current or charge amplifiers.
  • AC bridges.
  • Voltage dividers.

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SaM - Oscillators based on Wien Bridge

• Circuit:
  
• Formulas:
  • $A\triangleq \frac{V_o}{V_i}=1+\frac{R_2}{R_1}$
  • $\beta =-\frac{Z_1}{Z_1+Z_2}$
• Applying the Barkhausen conditions, after the calculations, we can satisfy them if we take:
  1. ${\omega }_0=RC$
  2. $R_2\ge 2\cdot R_1$
• Plot:
  Oscillators based on Wien Bridge have a constant slope around ${\omega }_0$:${\frac{d\phi (\beta )}{d\omega }|}_{\omega ={\omega }_0}=\frac{2}{3}$
• Variation of the Wein Bridge:
  (The zener diodes ensures to reduce the distortion due to an unstable behavior of the circuit)
  

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SaM - Phase Shift Oscillator

• Circuit:
  

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SaM - High Frequency Oscillator • 3 Point Oscillator • Colpits and Hartley Oscillators

• High frequency device: $f\in \left[100\text{kHz}÷30\text{MHz}\right]$
• Circuit:
  
• Equivalent model:
  
• Formulas:
  ($Z_1$, $Z_2$, $Z_3$, are all pure immaginaries: $Z_i=j{X}_i$).$\begin{array}{l}A=\frac{V_0}{V_i}=A_v\cdot \frac{Z_3\parallel (Z_1+Z_2)}{R_0+Z_3\parallel (Z_1+Z_2)}\\ \beta =\frac{V_f}{V_i}=-\frac{X_1}{X_1+X_2}\end{array}$
• Typical open loop formula of an amplifier:
  
• For $f\ll f_h$ we have that*:$A=A_v\frac{j{X}_3(j{X}_1+j{X}_2)}{j{R}_0(X_3+X_1+X_2)+j{X}_3(j{X}_1+j{X}_2)}$If we impose $X_1+X_2+X_3=0$, or more in general:$X_i+X_j=-X_z$We will find: $A=A_v$
  • Hartely oscillator: $2$ inductances and $1$ capacitance.
  • Colpits oscillator: $2$ capacitances and $1$ inductances.
• Barkausen Conditions:
  (Also considering $X_1+X_2=-X_3$)
  $\beta =\frac{X_1({\omega }_0)}{X_3({\omega }_0)}\Rightarrow |\beta A{|}_{\omega ={\omega }_0}=A_v\frac{X_1}{X_3}$So:
  • If $A_v>0$ (non-inverting amplifier configuration) ⇒ $X_1$ and $X_3$ must have OPPOSITE types:
    • if $X_1$ is a capacitance, $X_3$ must be an inductance.
    • if $X_1$ is an inductance, $X_3$ must be a capacitance.
  • If $A_v<0$ (inverting operational amplifier configuration) ⇒ $X_1$ and $X_3$ must have the SAME types, meaning:
    • if $X_1$ is a capacitance, $X_3$ must be a capacitance.
    • if $X_1$ is an inductance, $X_3$ must be an inductance.
• Terminology:
  • $A_v$ is the open loop of the amplifier, $A_v\in \mathbb{R}$.
  • $Z_1=j{X}_1$.
  • $Z_2=j{X}_2$.
  • $Z_3=j{X}_3$.

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SaM - Quartz in a 3 Point Oscillator

• Using an AT-cut quartz in a Colpits oscillator:
  (This circuit will oscillate only at frequency in which the quartz acts as an inductance, so for $f_0$ in the interval $[{\omega }_s,{\omega }_p]$)
  
• $Q$-factor value:
  (Adding a capacitance in series or in parallel to our quartz, will reduce this value, not goood)$Q=\frac{1}{{\omega }_s{R}_{eq}{C}_{eq}}$
• Adding a capacitance in series or in parallel to the quartz:
  (Adding a capacitance in series or in parallel can reduce the frequencies $f_s$ and $f_p$, giving us more control, good)
  • No added capacitance, base graph:
    
  • Added capacitance:
    
    • In series: we increase $f_s$.
    • In paralle: we reduce $f_p$.
    • In both cases we reduce the range $[f_s,f_p]$

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SaM - Behavior of the Quartz Ocillator at High Frequencies

• AT-cut quartz’s impedance:$Z_{eq}=C_E\parallel (L_{eq}+R_{eq}+C_{eq})$
• Special frequencies:
  (They depend on the type of cut of the quartz)$\begin{array}{l}{\omega }_s=\sqrt{\frac{1}{L_{eq}{C}_{eq}}}\\ {\omega }_p=\sqrt{\frac{C_E+C_{eq}}{C_E{L}_{eq}{C}_{eq}}}\end{array}$
• Reactance formula $\left(X_{eq}:Z_{eq}=j{X}_{eq}\right)$:$X_{eq}=-\frac{1}{\omega C_E}\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$
• Plot of $\frac{1-\frac{{\omega }^2}{{\omega }_s^2}}{1-\frac{{\omega }^2}{{\omega }_p^2}}$:
  
• More realistic impedance model:
  (If we work at a high frequency $(f\in [20\text{MHz}÷30\text{MHz}])$, we need to consider this additional impedaces, after the first).
  (If we work at lower frequencies $(<10\text{MHz})$ we can consider just the first $Z_{eq}$).
  
• Formula for each component:$\begin{array}{l}{L}_{m_i}=\frac{1}{8}\frac{m}{k^2{d}^2}\\ C_{m_i}=\frac{8}{(i\cdot \pi {)}^2}k{d}^2\\ R_{m_i}=\frac{(i\cdot \pi {)}^2}{8}\frac{\lambda }{k^2{d}^2}\end{array}$And the value of ${\omega }_{s_i}$:${\omega }_{s_i}\approx i\cdot w_{s_1}$
• Real World Measures:
  • At $\omega =10\text{MHz}$, we gave some common values for the AT-cut quartz’s impedances:$\begin{array}{l}{L}_{eq}=100\text{mH}\\ C_{eq}=15\text{fF}\\ R_{eq}=20\text{ohm}\end{array}$
• Terminology:
  • $m,\lambda ,k$ : mass, dumping coefficient, elastic coefficient of the piezoelectric slice.
  • $d$ is the piezoelectric coefficient.

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SaM - Quartz Oscillator

• Lumped parameter system:
  
• Wavelenght-thickness real world relationship:$t\approx \frac{{\lambda }_w}{2}$
• AT-cut quartz as a sensor, equivalent electrical circuit:
  
  • Formulas:$\begin{array}{l}{L}_{eq}=\frac{m}{k^2{d}^2}\\ R_{eq}=\frac{\lambda }{k^2{d}^2}\\ C_{eq}=k{d}^2\end{array}$
  • If we want to consider the added “mechanical impedance of the medium”:$\frac{Z_M}{k^2{d}^2}$Usually negligible.
  • The piezoelectric behavior of this structure depends on the surface orientation with respect to the crystallographic axis.
• Terminology:
  • $m,\lambda ,k$ : mass, dumping coefficient, elastic coefficient of the piezoelectric slice.
  • $d$ is the piezoelectric coefficient.
  • $Z_M$ : mechanical impedance of the medium, as an example when we talked about the ultrasonic transducer we said that it used an “added damper”, in that case $Z_M$ would represent this added dumper.

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