Dif. Pos.-fbk. op.-Amp.

see also : Pos. fbk. Op. Amp. - (spice) noise

 

\[V_a=V_S+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]

\[V_b=V_R+\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}\]

\[\left({V_O-V_R}\right)\frac{R_1}{R_1+R_3}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_0}{R_0+R_2}\]

\[\frac{V_O-V_R}{V_S-V_R}=
\frac{R_1+R_3}{R_1}\left({1+\frac{V_O-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}}\right)\]

\[A_V·\frac{R_1}{R_1+R_3}=1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)\frac{R_0}{R_0+R_2}\]

\[A_V\left({\frac{R_1}{R_1+R_3}-\frac{R_0}{R_0+R_2}}\right)=1-\frac{R_0}{R_0+R_2}\]

\[A_V=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{\left({R_0+R_2}\right)\left[{R_1\left({R_0+R_2}\right)-R_0\left({R_1+R_3}\right)}\right]}=\]

\[=\frac{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)}{R_2\left({R_0+R_2}\right)\left({R_1+R_3}\right)\left[{\frac{R_1}{R_2}·\frac{R_0+R_2}{R_1+R_3}-\frac{R_0}{R_2}}\right]}=\]

\[=\left[{Def.:\ R\ ,\ \ \frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}=\frac1{R}}\right]=\]

\[=\frac1{\frac{R_1R_0R_2}{R_2R_1R_3}-\frac{R_0}{R_2}}=\frac1{R_0\left({\frac1{R_3}-\frac1{R_2}}\right)}=\frac1{R_0\left({\frac1{R_0}-\frac1{R_1}}\right)}=\frac1{1-\frac{R_0}{R_1}}={A_V}^+\]

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What the above means - is that in case of the shown configuration - the non-common mode signal is extracted from common mode one , amplified ... and added back to the common mode one . . . shortly put :

\[V_O=V_R+A_V·\left({V_S-V_R}\right)\]

PS! : It is also possible to show - as for the above positive voltage gain derivation - that when we swap V_S and V_R then the negative/inverting voltage gain becomes \(A_V=\frac1{1-\frac{R_1}{R_0}}\) (see below) . . .

assuming the relation \(R_2=R_0·\left({A_V-1}\right)\) for the positive gain and the relation \(R_2=-A_V·R_0\) for the inverting gain -- the following applies :

the neg. gain case :

\[{A_V}^+=1-{A_V}^-=1-\frac1{1-\frac{R_1}{R_0}}=\frac{1-\frac{R_1}{R_0}-1}{1-\frac{R_1}{R_0}}=\frac{-\frac{R_1}{R_0}}{1-\frac{R_1}{R_0}}=\frac1{1-\frac{R_0}{R_1}}\]

e.g. \(\quad\left|{{A_V}^-}\right|={A_V}^+-1\quad\) ← that
for the same resistor values or for the same R₀ : R₁ ratio

about formulas :

parameter	non-inverting	inverting
\(A_V\quad\)	user set \(\displaystyle{\frac1{1-\frac{R_0}{R_1}}}\)	user set \(\displaystyle{\frac1{1-\frac{R_1}{R_0}}}\)
\(R_0\)	user set	user set
\(R_2\)	\(R_0·\left({A_V-1}\right)\)	\(-A_V·R_0\)
\(\frac1R\)	\(\displaystyle{\frac1{R_0}+\frac1{R_2}=\frac1{R_1}+\frac1{R_3}}\)
\(R_1\)	\(\displaystyle{\frac{R_0}{1-\frac1{A_V}}}\)	\(R_0\left({1-\frac1{A_V}}\right)\)
\(R_3\)	\(\displaystyle{\frac1{\frac1R-\frac1{R_1}}}\)
\(R_3\)	\(\displaystyle{R_1\left({\frac1{1-{\left({1-\frac1{A_V}}\right)}^2}-1}\right)}\)	\(\displaystyle{\frac{R_1}{{\left({1-\frac1{A_V}}\right)}^2-1}}\)

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\[V_b=V_S+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]

\[V_a=V_R+\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}\]

\[\left({V_O-V_R}\right)\frac{R_0}{R_0+R_2}=
\left({V_S-V_R}\right)+\left({V_O-V_S}\right)\frac{R_1}{R_1+R_3}\]

\[\frac{V_O-V_R}{V_S-V_R}·\frac{R_0}{R_0+R_2}=
1+\frac{V_O-V_R+V_R-V_S}{V_S-V_R}·\frac{R_1}{R_1+R_3}\]

\[A_V·\frac{R_0}{R_0+R_2}=1+\left({A_V-1}\right)·\frac{R_1}{R_1+R_3}\]

\[A_V·\left({\frac{R_0}{R_0+R_2}-\frac{R_1}{R_1+R_3}}\right)=1-\frac{R_1}{R_1+R_3}=\frac{R_3}{R_1+R_3}\]

\[A_V=\frac{R_3\left({R_1+R_3}\right)\left({R_0+R_2}\right)}{\left({R_1+R_3}\right)\left[{R_0\left({R_1+R_3}\right)-R_1\left({R_0+R_2}\right)}\right]}=\frac{\mathbb{Z}}{\mathbb{Z}\left[{\frac{R_0}{R_3}·\frac{R_1+R_3}{R_0+R_2}-\frac{R_1}{R_3}}\right]}=\]

\[=\frac1{\frac{R_0R_1R_3}{R_0R_2R_3}-\frac{R_1}{R_3}}=\frac1{R_1\left({\frac1{R_2}-\frac1{R_3}}\right)}=
\frac1{\frac{R_1}{R_1}-\frac{R_1}{R_0}}=\frac1{1-\frac{R_1}{R_0}}={A_V}^-\]

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 [Eop]

Re :: Pos. Feedback Op Amp

Definitions ::

\[\boxed{\frac1{R}=\frac1{R_1}+\frac1{R_2}=\frac1{R_3}+\frac1{R_4}\\ \frac1{R_{1_2}}{}^{↓}=^{↓}\frac1R-\frac1{R_{2_1}}\\ \frac1{R_{3_4}}{}^{↓}=^{↓}\frac1R-\frac1{R_{4_3}}}\]

\[{}^{↓↓}\ \frac{R_3}{R_1}+\frac{R_3}{R_2}-\frac{R_3}{R_3}=\frac{R_3}{R_4}=\\ =\mathbf{R_3·\frac{R_1+R_2}{R_1·R_2}-1}\]

\[V_B=V_X·\frac{R_1}{R_1+R_2}\\ V_A=V_S+\left({V_X-V_S}\right)·\frac{R_3}{R_3+R_4}\]

\[V_X·\frac{R_1}{R_1+R_2}=V_S+\left({V_X-V_S}\right)·\frac{R_3}{R_3+R_4}\\ V_X·\left({\frac{R_1}{R_1+R_2}-\frac{R_4}{R_3+R_4}}\right)=V_S·\left({1-\frac{R_4}{R_3+R_4}}\right)\]

\[A_V=N=\frac{V_X}{V_S}=\frac{\frac{\cancel{R_3}+R_4\cancel{-R_3}}{\bcancel{R_3+R_4}}}{\frac{\cancel{R_1·R_3}+R_1·R_4\cancel{-R_1·R_3}+R_2·R_3}{\left({R_1+R_2}\right)·\bcancel{\left({R_3+R_4}\right)}}}=\]

\[=\frac{R_1·R_4+R_2·R_4}{R_1·R_4-R_2·R_3}=\frac{1+\frac{R_2}{R_1}}{1-\frac{R_2}{R_1}·\frac{R_3}{R_4}}\]

\[N-N·\frac{R_2}{R_1}·\frac{R_3}{R_4}=1+\frac{R_2}{R_1}\]

\[\left({N-1}\right)·R_1·R_4=R_2·\left({N·R_3+R_4}\right)\]

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\[\frac{R_2}{R_1}=\frac{\left({N-1}\right·R_4)}{N·R_3+R_4}=\frac{N-1}{N·\frac{R_3}{R_4}+1}\]

\[{}^{↑↑}\ \frac{R_2}{R_1}·\left({N·\mathbf{\left({R_3·\frac{R_1+R_2}{R_1·R_2}-1}\right)}+1}\right)=N-1\]

\[\frac{R_2}{R_1}·\left({N·R_3·\frac{R_1+R_2}{R_1·R_2}-\left({N-1}\right)}\right)=N-1\]

\[\frac{R_2}{R_1}=\frac1{\frac N{N-1}·R_3·\frac{R_1+R_2}{R_1·R_2}-1}\]

\[\frac N{N-1}·R_3·\frac{\frac{R_1}{R_2}+1}{R_1}-1=\frac{R_1}{R_2}\]

 \[\frac N{N-1}·R_3·\frac{1+\frac{R_2}{R_1}}{R_1}-\frac{R_2}{R_1}=1\]

 \[\frac N{N-1}·R_3·\frac{\bcancel{\frac{R_2}{R_1}+1}}{R_1}=\bcancel{\frac{R_2}{R_1}+1}\]

 \[\frac N{N-1}=\mathbf{\frac{R_1}{R_3}}\]

---

 \[\frac{R_2}{R_1}=\frac{N-1}{N·\frac{R_3}{R_4}+1}=\frac1{\frac N{N-1}·\frac{R_3}{R_4}+\frac1{N-1}}=\]

\[=\frac1{\mathbf{\frac{R_1}{\cancel{R_3}}}·\frac{\cancel{R_3}}{R_4}+\frac1{N-1}}=\frac1{\frac{R_1}{R_4}+\frac1{N-1}}\]

\[\frac{R_2}{R_1}·\left({\frac{R_1}{R_4}+\frac1{N-1}}\right)=1\\ \frac1{R_4}+\frac1{R_1}·\frac1{N-1}=\frac1{R_2}\\ \frac1{R_1}·\frac1{N-1}=\frac1{R_2}-\frac1{R_4}\]

\[N-1=\frac1{R_1·\left({\frac1{R_2}-\frac1{R_4}}\right)}\]

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\[N=1+\frac1{R_1·\left({\frac1{R_2}-\frac1{R_4}}\right)}{}^{↑}=^{↑}1+\frac1{R_1·\left({\frac1{R_2}+\frac1{R_3}-\frac1R}\right)}=1+\frac1{R_1·\left({\frac1{R_3}-\frac1{R_1}}\right)}=\]

\[=\boxed{A_V=1+\frac1{\frac{R_1}{R_3}-1}}\]

Reminder :: \(\frac1{R_4}=\frac1{R_2}+\frac1{R_1}-\frac1{R_3} \) or \(\cases{\frac1{R_2}=\frac1R-\frac1{R_1}\\ \frac1{R_4}=\frac1R-\frac1{R_3}}\)

\({Def\ ::\ d=\frac{R_1}{R_3}\\ then\ :\ N-1=\frac1{d-1}\ and\ d=1+\frac1{N-1}}\)

if R is also given the rest can be computed ... as \(R_1=d·R_3\)

in "general" situation the LM308 likes the biasing resistance of the double 7k5 Ω --e.g.-- the R = 3.75 kΩ (for the noisy input the 100 kΩ in parallel with the 5 pF or less may be more suitable/stable however - so ... )

Related Post : More Op-Amp biasing schemes

The LTSpice Example ::

PS! the negative gain can result by the reached formula for the voltage gain -- but in practise it won't work !!!

a slew-rate versus a common mode input impedance ::

Note :: the realistic/conventional Op.-Amp.-s have the impedance value a bit lower than shown in the simulation

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[Eop]

Resistor Bridge

Simulation / Schematic :

\[U_R=\frac{U_LR_2R_4+U_UR_0R_4+U_DR_0R_2}{R_2R_4+R_0R_4+R_0R_2}=
\frac{U_L\left({R_1R_3+R_0R_3+R_0R_1}\right)-U_UR_0R_3-U_DR_0R_1}{R_1R_3}\]
\[U_L=\frac{U_RR_1R_3+U_UR_0R_3+U_DR_0R_1}{R_1R_3+R_0R_3+R_0R_1}=
\frac{U_R\left({R_2R_4+R_0R_4+R_0R_2}\right)-U_UR_0R_4-U_DR_0R_2}{R_2R_4}\]
\(U_RR_1R_3R_2R_4+U_UR_0R_3R_2R_4+U_DR_0R_1R_2R_4=U_R\sum_E^3\sum_O^3-U_UR_0R_4\sum_O^3-U_DR_0R_2\sum_O^3\)
\[U_R=R_0·\frac{\left({U_UR_3+U_DR_1}\right)R_2R_4+\left({U_UR_4+U_DR_2}\right)\sum_O^3}{\sum_E^3\sum_O^3-R_1R_2R_3R_4}\]
\(U_LR_1R_3R_2R_4+U_UR_0R_4R_1R_3+U_DR_0R_2R_1R_3=U_L\sum_O^3\sum_E^3-U_UR_0R_3\sum_E^3-U_DR_0R_1\sum_E^3\)
\[U_L=R_0·\frac{\left({U_UR_4+U_DR_2}\right)R_1R_3+\left({U_UR_3+U_DR_1}\right)\sum_E^3}{\sum_E^3\sum_O^3-R_1R_2R_3R_4}\]

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[Eop]

3 - Resistor Voltage Divider

\[I_0+I_1+I_2=0\qquad,\qquad\text{! Notice the directions of the currents !}\]
\[{\large \cases{I_0=\frac{V_0-V_X}{R_0}\\ {\ }\\
I_1=\frac{V_1-V_X}{R_1}\\ {\ }\\
I_2=\frac{V_2-V_X}{R_2}}}\]
\[\frac{V_0-V_X}{R_0}+\frac{V_1-V_X}{R_1}+\frac{V_2-V_X}{R_2}=0\]
\[\frac{V_0R_1R_2-V_XR_1R_2+V_1R_0R_2-V_XR_0R_2+V_2R_0R_1-V_XR_0R_1}{R_0R_1R_2}=0\qquad|\qquad×R_0R_1R_2\]
\[\frac{V_0R_1R_2+V_1R_0R_2+V_2R_0R_1}{R_1R_2+R_0R_2+R_0R_1}=V_X\]
Note: the \(V_X\) is the unmarked voltage point on the schematic !

It is easy to see the calculation scheme easily adjusts for N voltages all connecting to the single \(V_X\)
\[V_{X_N}=\frac{\displaystyle{\sum_{i=0}^{N-1}V_i\frac{\displaystyle{\prod_{k=0}^{N-1}R_k}}{R_i}}}{\displaystyle{\sum_{i=0}^{N-1}\frac{\displaystyle{\prod_{k=0}^{N-1}R_k}}{R_i}}}\]
thus BWD chek ::
for N=2 :
\[V_X=\frac{V_1R_0+V_0R_1}{R_0+R_1}\]
incase of the \(V_0=0\) it reduces to a trivial 2 resistor divider formula :
\[V_X=V_1\frac{R_0}{R_0+R_1}\]

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Simulation example/chk. for N=4 :

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[Eop]

INA formula chk

\[E_1+\left({V_X-E_1}\right)·\frac{R_2}{R_6+R_2}=E_2+\left({V_R-E_2}\right)·\frac{R_5}{R_7+R_5}\]
\[E_1+\left({E_2-E_1}\right)·\frac{R_1}{R_1+R_4+R_3}=V_1\qquad\frac{E_1-V_1}{R_1}=\frac{E_1-E_2}{R_{\Sigma3}}\]
\[E_2+\left({E_1-E_2}\right)·\frac{R_4}{R_1+R_4+R_3}=V_2\qquad\frac{V_2-E_2}{R_4}=\frac{E_1-E_2}{R_{\Sigma3}}\]
\[E_1=\frac{R_1}{R_4}\left({V_2-E_2}\right)+V_1\qquad E_2=\frac{R_4}{R_1}\left({V_1-E_1}\right)+V_2\]
\[E_2-E_1+\frac{\left({E_1-E_2}\right)R_4+\left({E_1-E_2}\right)R_1}{R_{\Sigma3}}=V_2-V_1\]

\[1-\frac{R_1+R_4}{R_{\Sigma3}}=\frac{V_2-V_1}{E_2-E_1}=\frac{V_2-V_1}{E_2-\frac{R_1}{R_4}\left({V_2-E_2}\right)-V_1}=\frac{V_2-V_1}{\frac{R_4}{R_1}\left({V_1-E_1}\right)+V_2-E_1}\]
\[E_2\left({1+\frac{R_1}{R_4}}\right)-\left({V_2\frac{R_1}{R_4}+V_1}\right)=\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}\]
 \[E_1\left({1+\frac{R_4}{R_1}}\right)-\left({V_2+V_1\frac{R_4}{R_1}}\right)=-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}\]
\[E_2=\frac{\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}+V_2\frac{R_1}{R_4}+V_1}{1+\frac{R_1}{R_4}}=\frac{\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}R_4+V_2R_1+V_1R_4}{R_1+R_4}\]
\[E_1=\frac{-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}+V_2+V_1\frac{R_4}{R_1}}{1+\frac{R_4}{R_1}}=\frac{-\frac{V_2-V_1}{1-\frac{R_1+R_4}{R_{\Sigma3}}}R_1+V_2R_1+V_1R_4}{R_1+R_4}\]
\[.\ .\ .\]
\[V_X\frac{R_2}{R_6+R_2}-V_R\frac{R_5}{R_7+R_5}=E_2\frac{R_7}{R_7+R_5}-E_1\frac{R_6}{R_6+R_2}\]

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\[IF\ :\ \cases{R_1=R_4=R_A\\ R_3=R_D\\ R_6=R_7=R_F\\ R_2=R_5=R_G}\]
\[E_1=\frac{\cancel{R_A}\left({V_2+V_1}\right)-\frac{V_2-V_1}{1-\frac{2R_A}{2R_A+R_D}}\cancel{R_A}}{2\ \cancel{R_A}}=\frac{V_2+V_1}2-\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}\]
\[E_2=\frac{\cancel{R_A}\left({V_2+V_1}\right)+\frac{V_2-V_1}{1-\frac{2R_A}{2R_A+R_D}}\cancel{R_A}}{2\ \cancel{R_A}}=\frac{V_2+V_1}2+\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}\]
\[.\ .\ .\]

\[\left({V_X-V_R}\right)\frac{R_G}{\cancel{R_F+R_G}}=\left[{\left({\cancel{\frac{V_2+V_1}2}+\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}}\right)-\left({\cancel{\frac{V_2+V_1}2}-\frac{V_2-V_1}2·\frac{2R_A+R_D}{R_D}}\right)}\right]\frac{R_F}{\cancel{R_F+R_G}}\]
\[\boxed{\frac{V_X-V_R}{V_2-V_1}=\frac{R_F}{R_G}·\frac{2R_A+R_D}{R_D}=\frac{R_6}{R_2}·\frac{2R_1+R_3}{R_3}}\]

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dd

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[Eop]

Another Diff. op.-Amp. circuit

\[\frac{V_X-V_R}{V_H-V_L}=\frac{R_AR_F}{R_AR_V+R_G\left({R_A+R_V}\right)}\]

\[V_D=\frac{V_CR_{11}R_3+V_XR_0R_3+V_LR_0R_{11}}{R_{11}R_3+R_0R_3+R_0R_{11}}\]
\[V_U=\frac{V_CR_{22}R_5+V_RR_4R_5+V_HR_4R_{22}}{R_{22}R_5+R_4R_5+R_4R_{22}}\]
\[FROM\ :\ V_X-V_R=\left({V_U-V_D}\right)\frac{R_F}{R_G}\]
\[IF\ :\ \cases{R_1=R_2=R_F\\ R_{10}=R_{20}=R_G\\ R_0=R_4=R_A\\ R_3=R_5=R_V}\]
\[\boxed{!\ note\ that\ the\ above\ condition\ makes\ the\ biasing\ invariant\ of\ the\ \mathbf{V_C}}\]
 \[\left({V_X-V_R}\right)\frac{R_G}{R_F}=
\frac{\underline{V_C\left({R_F+R_G}\right)R_V}+\boxed{V_RR_AR_V}+V_HR_A\left({R_F+R_G}\right)}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)}-\frac{\underline{V_C\left({R_F+R_G}\right)R_V}+\boxed{V_XR_AR_V}+V_LR_A\left({R_F+R_G}\right)}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)}\]
\[\left({V_X-V_R}\right)\left({\frac{R_G}{R_F}+\frac{R_AR_V}{R_{\Sigma3}}}\right)=\left({V_H-V_L}\right)\frac{R_A\left({R_F+R_G}\right)}{R_{\Sigma3}}\]
\[\frac{V_X-V_R}{V_H-V_L}=\frac{R_A\left({R_F+R_G}\right)}{\cancel{R_{\Sigma3}}}·\frac{R_F·\cancel{R_{\Sigma3}}}{R_GR_{\Sigma3}+R_AR_VR_F}=\]
\[=\frac{R_A\left({R_F+R_G}\right)\frac{R_F}{R_G}}{\left({R_F+R_G}\right)R_V+R_AR_V+R_A\left({R_F+R_G}\right)+R_AR_V\frac{R_F}{R_G}}=\]
\[=\frac{R_A\cancel{\left({R_F+R_G}\right)}\frac{R_F}{R_G}}{\cancel{\left({R_F+R_G}\right)}\left({R_A+R_V}\right)+R_AR_V\frac{\cancel{\left({R_F+R_G}\right)}}{R_G}}=\]

\[=\frac{R_AR_F}{R_AR_V+R_G\left({R_A+R_V}\right)}\]

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a simulation example ::

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[Eop]

Differential Op-Amp formulas check

the case for :

\[\cases{signal\ :\ U_S=U_1\\ reference\ :\ U_R=U_0\\ gain\ :\ R_G=R_0\\ feedback\ :\ R_F=R_1\\ output\ :\ U_O=U_X}\]

\[U_R=U_\overline{IN}=U_S+\left({U_O-U_S}\right)·\frac{R_G}{R_G+R_F}\]

\[U_S-U_R=-\left({U_O-U_R+U_R-U_S}\right)·\frac{R_G}{R_G+R_F}\]

\[\frac{U_O-U_R}{U_S-U_R}-1=A_V-1=-1-\frac{R_F}{R_G}\]

\[\boxed{A_V=-\frac{R_F}{R_G}}\]

\[\begin{align*}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \\ \hline \end{align*}\]

\[\cases{signal\ :\ U_S=U_0\\ reference\ :\ U_R=U_1}\]

\[U_S=U_\overline{IN}=U_R+\left({U_O-U_R}\right)·\frac{R_G}{R_G+R_F}\]

\[U_S-U_R=\left({U_O-U_R}\right)·\frac{R_G}{R_G+R_F}\]

\[\frac{U_O-U_R}{U_S-U_R}=\boxed{A_V=1+\frac{R_F}{R_G}}\]

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\[\frac{U_Y}{U_1}=\frac{R_0+R_1}{R_0}\qquad \frac{U_X-U_0}{U_Y-U_0}=-\frac{R_3}{R_2}\\ {\ }\]
\[\boxed{U_X=}\ U_0\left({\frac{R_2+R_3}{R_2}}\right)-\frac{R_3}{R_2}U_1\left({\frac{R_0+R_1}{R_0}}\right)\ {^?= {}_?}\ \mathbf{...}\ =\left({U_0-U_1}\right)\frac{R_2+R_3}{R_2}\ \boxed{=\Delta U_{IN}\left({1+\mathbf{M}}\right)}\\ {\ }\]
\[?\qquad \frac{R_2+R_3}{R_2}=\frac{R_3R_0+R_3R_1}{R_0R_0}\ {^?= {}_?}\ ...\ =\frac{R_3\frac{R_0}{R_1}+R_3}{\frac{R_0}{R_1}R_2}=\frac{R_3\frac{R_3}{R_2}+R_3}{\frac{R_3}{R_2}R_2}=\frac{R_2+R_3}{R_2}\\ {\ }\]
\[R_0=\frac{R_3\left({R_0+R_1}\right)}{R_3+R_2}\\ {\ }\]
\[\cases{\underline{R_0R_3}+R_0R_2=\underline{R_0R_3}+R_1R_3\\ {\ }\\ \boxed{R_0R_2=R_1R_3}}\qquad \frac{R_3}{R_2}=\frac{sR_0}{sR_1}=\mathbf{M}\qquad \mathbf{...} \uparrow\]

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some tests :

MAX input impedance test :

Max. frequency TEST :

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[Eop]