op amp equation

754 days ago by jay

The Op Amp Equation

For an inverting op amp, the general equation for the output voltage is

V_{out}=\frac{-{\left(a r_{i} r_{o} v_{p} + {\left(a^{2} r_{i} + a r_{o}\right)} r_{2} v_{p} + {\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}
and in the usual case when V_p is ground, it reduces to
V_{out}=\frac{-{\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}}{{\left({\left(r_{2} + r_{i}\right)} r_{o} + {\left({\left(a + 1\right)} r_{2} + r_{1}\right)} r_{i} + r_{1} r_{2}\right)}}.
Further, the ideal op amp has A_{OL}=R_{in}=\infty and R_{out}=0, from which we get
V_{out}=  \frac{-r_{1} v_{s}}{r_{2}}.

 

The process:

First, set up all our variables.
v1,i1,r1,v2,i2,r2,v_s,a,v_o,i_o,r_o,v_n,r_i,i_i,v_p=var('v1,i1,r1,v2,i2,r2,v_s,a,v_o,i_o,r_o,v_n,r_i,i_i,v_p') 
       

Here I define the system of equations and variables to solve for.  For each equation in eqns, there must be a variable from that equation listed in vars.
eqns=[ i_i == i1+i2, i2 == (v_s - v_n)/r2, i1 == (a*(v_p - v_n) - v_n)/(r_o + r1), i_i == v_n/r_i, v_o == -a*v_n - i1*r_o ] vars=[v_o,v_n,i1,i2,i_i] 
       

Solve now for each var in vars!
solns=solve(eqns,vars, solution_dict=True) solns[0] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left\{i_{i}:\: \frac{{\left(a r_{2} v_{p} + {\left(r_{1} + r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, i_{1}:\: \frac{{\left(a r_{2} v_{p} + a r_{i} v_{p} - {\left(a r_{i} + r_{i}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, v_{n}:\: \frac{{\left(a r_{2} r_{i} v_{p} + {\left(r_{1} r_{i} + r_{i} r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, v_{o}:\: \frac{-{\left(a r_{i} r_{o} v_{p} + {\left(a^{2} r_{i} + a r_{o}\right)} r_{2} v_{p} + {\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, i_{2}:\: \frac{-{\left(a r_{i} v_{p} - {\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}\right\}
\newcommand{\Bold}[1]{\mathbf{#1}}\left\{i_{i}:\: \frac{{\left(a r_{2} v_{p} + {\left(r_{1} + r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, i_{1}:\: \frac{{\left(a r_{2} v_{p} + a r_{i} v_{p} - {\left(a r_{i} + r_{i}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, v_{n}:\: \frac{{\left(a r_{2} r_{i} v_{p} + {\left(r_{1} r_{i} + r_{i} r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, v_{o}:\: \frac{-{\left(a r_{i} r_{o} v_{p} + {\left(a^{2} r_{i} + a r_{o}\right)} r_{2} v_{p} + {\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}, i_{2}:\: \frac{-{\left(a r_{i} v_{p} - {\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} v_{s}\right)}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}\right\}

Set Vo to be a reference to the solution to v_o.
Vo=solns[0][v_o] 
       

Calculate a specific output voltage with the stated parameters.
Vo.substitute(v_p=0,a=50000,v_s=150e-3,r2=60e3,r1=240e3,r_i=500e3,r_o=750) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}-0.599934041747493
\newcommand{\Bold}[1]{\mathbf{#1}}-0.599934041747493

What is the equation when V_p is tied to ground?
Vo.substitute(v_p=0) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-{\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-{\left(a r_{1} r_{i} - r_{i} r_{o}\right)} v_{s}}{{\left({\left(a r_{i} + r_{1} + r_{i} + r_{o}\right)} r_{2} + r_{1} r_{i} + r_{i} r_{o}\right)}}

Take a limit to deal with the really big and small - here is the classic gain equation.
limit(Vo.substitute(v_p=0,r_o=0),a=oo) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-r_{1} v_{s}}{r_{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-r_{1} v_{s}}{r_{2}}