(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]