SaM - MEMORIZE (Amplifiers)

List of things to memorize:

SaM - Amplifiers

• SaM - Operational Amplifier
  • SaM - Operational Amplifier as a Voltage Comparator
  • SaM - Non-Inverting Operational Amplifier
  • SaM - Inverting Operational Amplifier
  • SaM - Non-Ideal Operational Amplifier • Input and Output Resistances
  • SaM - Non-Ideal Operational Amplifier • OZE (Out of Zero Error) • Special Kinds of Operational Amplifiers • Precision Amplifier • Low Noise • Low Input Bias Current • Zero Drift • Low Noise Operational Amplifier
• SaM - Single Input Amplifier
• SaM - Differential Amplifier
  • SaM - Simple Differential Amplifier (Made from an O.A.)
  • SaM - Instrumentational Amplifier
  • SaM ~ Example of a Real Instrumentational Amplifier • INA128
• SaM - Thevenim and Norton Equivalent
  • SaM - Voltage Amplifier
  • SaM - Current Amplifier
  • SaM - Passive Sensor
• SaM - AC-Coupled Amplifier
• SaM - Carrier Amplifier
  • SaM - AM Modulation with 2 Sensors • Carrier Amplifier
• SaM - Charge Amplifier
• SaM - Logarithmic Amplifier

Link to original

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SaM - Operational Amplifier

• Open Loop Frequency Response Formula:$V_{out}=(V_2-V_1)\cdot \frac{A_{DC}}{1+j\frac{f}{f_H}}$
• Non-Idealities of Operational Amplifier:
  
  • $Z_{in}$ : input resistance, $Z_{in}\in [100\text{k}\text{ohm}÷1\text{M}\text{ohm}]$, usually we consider it $\simeq \infty$.
  • $Z_{out}$ : output resistance, $Z_{out}\in [100\text{ohm}÷20\text{k}\text{ohm}]$, we usually consider it $\simeq 0$.
  • $V_{io}$ : voltage input offset, $V_{io}\in [\mu \text{V}÷\text{mV}]$.
  • $I_{\text{bias}}$ : input bias current, $I_{\text{bias}}\in [\text{fA}÷\mu \text{A}]$ (Remember: $\text{f}=10^{-15}$, $\mu =10^{-6}$).
• Terminology:
  • $A_{DC}$ : open loop gain (ideally infinite, in reality $\in [10^5÷10^6]$).
  • $f_H$ : cut-off frequency $(f_H\in [1\text{Hz}÷1\text{MHz}])$.

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SaM - Non-Inverting Operational Amplifier

• Formula:$V_{out}=V_{in}\frac{1}{\beta }$Where: $\beta =\frac{R_1}{R_1+R_2}$
• Terminology:
  • $\beta$ : feedback factor.

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SaM - Inverting Operational Amplifier

• Formula:$V_{out}=V_{in}\cdot -\frac{R_2}{R_1}$

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SaM - Non-Ideal Operational Amplifier • Input and Output Resistances

• Formula:$V_{out}^{\prime }=A_d{V}_d+A_c{V}_c$Where:
  • $V_d$ : “differential voltage” $(V_d=V_2-V_1)$.
    It’s the voltage on $R_D$
  • $V_C$ : “common mode voltage” $\left(V_C=\frac{V_2+V_1}{2}\right)$
    It’s the voltage on $R_{C1}$ plus the voltage on $R_{C2}$
  • $A_d$ : “*differential gain” $\left(A_d=\frac{A_2-A_1}{2}\right)$
  • $A_C$ : “*common mode gain” $(A_C=A_2+A_1)$
  • $A_1={V_{out}|}_{V_2=0}$
  • $A_2={V_{out}|}_{V_1=0}$
• $R_{C1}$ and $R_{C2}$ are the “common mode resistances”. (Ideally they are both $\infty$NOT_SURE_ABOUT_THIS )
• $R_D$ is the “differential mode resisitance”. (Ideally we consider it $\infty$)
• $R_0$ is the “output resistance, ideally it’s $0$.
• ==Normally, an ideal op.amp. would have only $A_d=A=\infty$ for the output, however in reality, the ouput also depends on $A_c{V}_c$==.

Plus a resistive bridge:

• Where:$V_{TH}=V\left(\frac{R_1}{R_1+R_4}-\frac{R_2}{R_2+R_3}\right)$ Combining the two circuits:
  
• $R_C\gg R_{TH}$
• The output:$V_{out}^{\prime }=A_d{V}_d+A_c\left(V_{CC}+\frac{V_d}{2}\right)$

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SaM - Non-Ideal Operational Amplifier • OZE (Out of Zero Error) • Special Kinds of Operational Amplifiers • Precision Amplifier • Low Noise • Low Input Bias Current • Zero Drift • Low Noise Operational Amplifier

Plus the Thevenim equivalent of a measurement system:

• OZE (Out of Zero Error):${V_{out}|}_{V_s=0}=\left(1+\frac{R_2}{R_1}\right)\left(I_{b2}{R}_S-I_{b1}\frac{R_1{R}_2}{R_1+R_2}+V_{io}\right)$
• IZE (Input out of Zero Error):$\text{IZE}=I_{b2}{R}_S-I_{b1}\frac{R_1{R}_2}{R_1+R_2}+V_{io}$

	$I_b$	$V_{io}$	TCV	Other
Low input bias current op.amp.	$<100\text{pA}$
Precision amplifiers		$<1\text{mV}$	$<2\frac{\mu \text{V}}{°\text{C}}$
Zero drift op. amp		$\sim 1000$ times lower then precision amplifiers	$\sim 1000$ times lower then precision amplifiers	Slow devices (high rise time)
Low noise amplifier				Minimize the white noise interferance.

• Rule of thumb:$V_{\text{DRIFT}}=\frac{1}{10}{V}_{io}$
• Terminology:
  • $V^{+}$ : $(+)$ terminal of the amplifier
  • $V^{-}$ : $(-)$ terminal of the amplifier
  • $V_{io}$ : input voltage offset
  • $I_{b1}$ & $I_{b1}$ : input bias current on each of the the two terminals.
  • $V_S$, $R_S$ : source voltage and source resistance (thevenim equivalent of the source).
  • TCV: Temperature Coefficient of Voltage (similar to TCR)

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SaM - Single Input Amplifier

• Representation:
  
• Usually an A.O. with one of the two inputs connected to ground:
  

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SaM - Differential Amplifier

A differential amplifier output functions depends only on the difference: $V^{+}-V^{-}$Here’s an example:

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SaM - Simple Differential Amplifier (Made from an O.A.)

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SaM - Instrumentational Amplifier

• Formula:$V_{out}=(V_2-V_1)\cdot \left(1+\frac{2R_1}{R_{\text{gain}}}\right)\frac{R_3}{R_2}$

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Don’t know if it is a requested knowledge for the exam, here’s some information on the instrumentational amplifier “INA128”

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SaM - Thevenim and Norton Equivalent

Passages:

1. Disconnect A-B (“the load”).
2. Find the voltage $V_{TH}$.
3. Reconect the load, disconnect all indipendent sources, and calculate the Thevenim resistance ($R_{TH}$) between A and B.

• Terminology:
  • $Z_L$ : load impedance.
  • $Z_{TH}$ : thevenim impedance.

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SaM - Voltage Amplifier

• Very high input impedance, very low output impedance
• Formula:$V_{out}=V_i\cdot \frac{1}{\beta }$

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SaM - Current Amplifier

• Very low input impedance, very low output impedance
• The output is:$V_{out}=R\cdot I_f$And $I_f$ is:$I_f=\frac{Z_N}{\frac{R}{A(f)}+Z_N}{I}_N$If $A_f$ is really big, as it usually is for DC signals, we find that:$I_f=I_N$
• Terminology:
  • $I_f$ : feedback current.

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SaM - AC-Coupled Amplifier

Used to remove from the output signal the DC components.

• Singol input amplifier:
  
• Differential amplifier:
  

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SaM - Carrier Amplifier

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SaM - AM Modulation with 2 Sensors • Carrier Amplifier

• Sensors:$C_1=\frac{C_0}{1+x},C_2=\frac{C_0}{1-x}$
• Output:$V_{out}=V_g\left(\frac{Z_f}{Z_1}-\frac{Z_f}{Z_2}\right)$Where:
  • $Z_f=R_f\parallel \frac{1}{j\omega C_f}=\frac{R_f}{1+j\omega R_f{C}_f}$
  • $Z_1=\frac{1}{j\omega C_1}$
  • $Z_2=\frac{1}{j\omega C_2}$
• Output after calculations:$V_{out}=Vg\frac{R_f}{1+j\omega R_f{C}_f}(j\omega 2C_{0}x)$Where:
  • $x\in [0,1]$, also $x\ll 1$
  • $C_1=\frac{C_0}{1+x}$
  • $C_2=\frac{C_0}{1-x}$
• Plot:
  
  • Simplified output for $f_0\gg \frac{1}{2\pi R_f{C}_f}$ and $f_0\gg f_{S_{MAX}}$:$V_0\simeq V\cdot \frac{2C_0}{C_f}\cdot |x|$
• Complete circuit with sign reovery:
  
• Terminology:
  • $V_g$ : sinusoidal (or alternating) source voltage.
  • $Z_f=R_f\parallel \frac{1}{j\omega C_f}$ : capacitive feedback impedance.
  • $x$ : measured quantity (in this case we can imagine to meaure the proximity, so in $m$).
  • TODO Why $x\in [0,1]$, also $x\ll 1$??, which capacitive sensor does this $C=\frac{C_0}{1+x}$?

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SaM - Charge Amplifier

• Formula:$f_{\min }=\frac{1}{2\pi R_f{C}_f}$

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SaM - Logarithmic Amplifier

• Formula: $V_{out}=-V_T\ln \left(\frac{V_{in}}{I_{S}R}\right)$Where:$V_T=\frac{KT}{q}$
• BJT Formula:$V_{BE}=V_T\cdot \ln \left(\frac{I_C}{I_S}\right)$
• Terminology:
  • $V_T$ : thermal voltage $\left(V_T=\frac{KT}{q}\right)$
  • $K$ : Boltzman’s constant $\left(K=1.38\cdot 10^{-23}\frac{{\text{m}}^2\cdot \text{kg}}{°\text{K}\cdot \text{s}}\right)$

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