(scheduled) derivation of the solution of the quadratic equation

 about (scheduled) :: every now and then (not too often) i set myself to re-figure it out (the history has proven there exists a variance of what i come up with each time ...)

... so -- Def.-s , etc. ... ::

\[\begin{array}{lcl}
\left({x-a}\right)\left({x-b}\right)=0 &\ &\\
x^2-\left({a-b}\right)x+ab=0 &\ &\\
\begin{array}{l}
x^2+px+q=0\qquad \qquad \qquad \qquad \rightarrow\\
x^2+2px+p^2=px-q+p^2\ |×3 &\\
x^2-2px+p^2=-3px-q+p^2\ |+\ \uparrow &\\
\hline
4x^2+4px+4p^2=0-4q+4p^2\ |-3p^2\\
4x^2+4px+p^2=p^2-4q\ |÷4\\
\mathbf{x^2+2\frac p2x+{\left({\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q}
\end{array} &\ &
\begin{array}{l}
\mathbf{x^2+2\frac p2x+{\left({\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q}\\
{\left({x+\frac p2}\right)}^2={\left({\frac p2}\right)}^2-q\\
\boxed{x=-\frac p2±\sqrt{{\left({\frac p2}\right)}^2-q}}\\{}\\{}
\end{array}
\end{array}\]

... (it) came out double ((at) this time) -- the short and the long -- way to (the solution) F;T
// is likely ↑↑ why ↑↑ in many blogs folks do not get a thing what i say
// (as an old school programmer i always live-compact my code(read: text))

see also the inner properties of @ About the Quadratic Equation

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[Eop]

Dif. Hi-Gain Amp. (not very practical)

\[V_b=V_Z+\left({V_n-V_Z}\right)\frac{R_g}{R_g+R_S}\]

\[V_a=V_S+\left({V_p-V_S}\right)\frac{R_g}{R_g+R_S}\]

\[Def\ :\ \frac1{\mathbf{R_X}}=\frac1{R_g+R_S}+\frac1{R_P}=\\ =\frac{R_g+R_S+R_P}{\left({R_g+R_S}\right)R_P}\]

\[V_n=V_Z+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}\]

\[V_p=V_S+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}\]

\[V_b=V_Z+\left[{\cancel{V_Z}+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}\cancel{-V_Z}}\right]\frac{R_g}{R_g+R_S}=V_Z+\left({V_O-V_Z}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}\]

\[V_a=V_S+\left[{\cancel{V_S}+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}\cancel{-V_S}}\right]\frac{R_g}{R_g+R_S}=V_S+\left({V_R-V_S}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}\]

\[V_S-V_Z=\left({V_O-V_R+V_S-V_Z}\right)\frac{R_X}{R_X+R_f}·\frac{R_g}{R_g+R_S}\]

\[\boxed{\quad A_V\quad}=\frac{V_O-V_R}{V_S-V_Z}=\left({1+\frac{R_f}{R_X}}\right)\left({1+\frac{R_S}{R_g}}\right)-1=\cancel{1}+\frac{R_f}{R_X}+\frac{R_S}{R_g}+\frac{R_fR_S}{R_XR_g}\cancel{-1}=\]

\[=\frac{R_gR_f+R_XR_S}{R_XR_g}+\frac{R_fR_S}{R_XR_g}=\frac{R_fR_S}{R_XR_g}\left({\frac{R_g}{R_S}+\frac{R_X}{R_f}+1}\right)=\]

\[=\frac{R_fR_S}{R_g}·\frac{R_g+R_S+R_P}{\left({R_g+R_S}\right)R_P}\left({\frac{R_g}{R_S}+1+\frac1{R_f}·\frac{\left({R_g+R_S}\right)R_P}{R_g+R_S+R_P}}\right)=\]

\[=R_f·\frac{R_S}{R_g}\left({\frac1{R_P}+\frac1{R_g+R_S}}\right)\left({\frac{R_g+R_S}{R_S}+\frac1{R_f}·\frac1{\frac1{R_P}+\frac1{R_g+R_S}}}\right)=\]

\[=R_f\left({\frac1{R_P}+\frac1{R_g+R_S}}\right)\frac{R_g+R_S}{R_g}+\frac{R_S}{R_g}=\boxed{\quad\frac1{R_g}\left[{R_f\left({\frac{R_g+R_S}{R_P}+1}\right)+R_S}\right]\quad}=\]

\[=R_f\left({\frac{1+\frac{R_S}{R_g}}{R_P}+\frac1{R_g}}\right)+\frac{R_S}{R_g}\]

\[\frac{A_V-\frac{R_S}{R_g}}{R_f}-\frac1{R_g}=\frac{1+\frac{R_S}{R_g}}{R_P}\]

\[\boxed{\quad R_P\quad}=\frac{1+\frac{R_S}{R_g}}{\frac{A_V-\frac{R_S}{R_g}}{R_f}-\frac1{R_g}}=\boxed{\quad \frac{R_g+R_S}{\frac{A_VR_g-R_S}{R_f}-1}\quad}\]

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about :

Uses the LM324 transistor model (the simulation)

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[Eop]

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

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

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

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

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

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

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

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

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

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

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

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

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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]