# Class 03

**Non Linear functions with OAs**

**Multivibrator**
Has negative and positive feedback, but net positive


**Log and AntiLog**
Amplifier with negative Feedback, but also 

**Logarithmic Converters**
Logarithmic / AntiLog (Exponential) Converter

Sound level is given in Decibels, as it is always compared to the sound threshold.

$$
20\log{\frac{P}{P_{0}}}
$$


$V_out = -V_{T}*\ln{}$

$V_o = R_1Ie^{\frac{-V_i}{V_T}}$

![[Pasted image 20220503133103.png]]
Expression of the above system

$V_in = -R_in * I_in = -Rin*I_sat e^{\frac{E_o}{V_T}}$
$\displaystyle E_O = V_T \ln(-\frac{V_in}{R_in*I_sat})$

![[Class_03 2022-05-03 13.39.51.excalidraw]]
$$
\begin{flalign}
&V_1 = -V_T\ln{\frac{Vi}{R1*I_{ES}}}&\\\
&V_2 = -V_T\ln{\frac{Vi}{R2*I_{ES}}}&\\
&V_O = -(V_1+V_2)&\\
&V_O = V_T\ln{(\frac{V_{i1}V_{i2}}{(RI_{ES})^2})}&\\
&V_O' = -RI_{es}e^{\frac{V_O}{V_T}}&\\
&V_O' = -\frac{V_{i1}V_{i2}}{RI_{es}}&\\
\\
&V_O'' = V_{i1}V_{i2}&\\
\end{flalign}
$$****

```circuitjs
$ 1 0.000005 10.20027730826997 50 5 43 5e-11
i -192 208 -112 208 0 0.01
a -112 224 16 224 8 15 -15 1000000 10.624172192589858 0 100000
g -112 240 -112 288 0 0
w -112 208 -112 128 0
t 64 176 64 128 0 1 14.983134150394081 15.001062402219262 100 default
t -16 176 -16 128 0 -1 -0.63704611232159 24.98818848248753 100 default
w -112 128 -32 128 0
w -16 176 -16 272 0
w -16 272 96 272 0
r 96 272 96 336 0 1000
g 96 336 96 368 0 0
w 96 272 208 272 0
r 208 272 208 144 0 1000
a 112 144 208 144 8 15 -15 1000000 -14.983134150394081 0 100000
w 0 128 16 128 0
w 16 128 48 128 0
w 16 128 16 224 0
g 64 176 64 192 0 0
i 192 48 112 48 0 0.01
w 112 48 112 128 0
w 80 128 112 128 0
g 112 160 112 208 0 0
g 192 48 224 48 0 0
g -192 208 -224 208 0 0
O 208 144 240 144 1 0
x 235 217 265 220 4 24 R1
x 124 308 154 311 4 24 R2
```
$$
\begin{flalign}
&I_1 = I_{es}e^{\frac{V_{BE_1}}{V_T}} \longrightarrow V_{BE_1} = V_T \ln{\frac{I_1}{I_{es}}}&\\\
&I_1 = I_{es}e^{\frac{V_{BE_2}}{V_T}} \longrightarrow V_{BE_2} = V_T \ln{\frac{I_2}{I_{es}}}&\\\
&V_1 = V_O*\frac{R2}{R1+R2}&\\
&V_O = (1+\frac{R1}{R2})V_T\ln{\frac{I_1}{I_2}}
\end{flalign}
$$

![[Pasted image 20220503142156.png]]

$$
\begin{flalign}
&V_{ref} = R_{ref}I_1 = R_{ref}I_{es}e^{\frac{V_{BE_1}}{V_T}} \longrightarrow V_T\ln{\frac{V_{ref}}{R_{ref}I_{es}}} = V_{BE_1}&\\\
&V_O = RI_2 = RI_{es}e^{\frac{V_{BE_2}}{V_T}} \longrightarrow V_T\ln{\frac{V_O}{R_{ref}I_{es}}} = V_{BE_2} &\\
&V_1 = V_i \frac{R_2}{R_1+R_2}&\\
&V_i = (1+\frac{R_1}{R_2})V_T \ln{\frac{V_{ref}}{V_O}\frac{R}{R_{ref}}}&\\
&V_O = \frac{V_{ref}}{}
\end{flalign}
$$