New KCC Quiz AQQ299 about Opamp Open Loop Gain and LTspice

Readily solved iteratively with Excel Goal Seek.  M = 29.35306 or 0.901423.

Botha

the solution for

20*log10(M) = M

can be found with the Lambert W function
which is a solution for x
of the function y = x*exp(x)
W(y) = x

with
x = -M*ln(10)/20

you get
y = -ln(10)/20

and with the Lambert W function
x = W(y)

you get finally
M = (-20/ln(10)) * W(y)
M = (-20/ln(10)) * W( -ln(10)/20 ) =~= 1.14028753461

which proves true with my calculator

20*log10(1.14028753461) = 1.14028753461

brgds
steve

PDF

There are 3 points where |T(s)| = |T(s)| (in dB units).  An analytical solution of this problem would be very cumbersome.  Graphing works to approximate a point then refine it to improve the answer.  If you don’t have any tools on your PC to solve it, there’s always the web (or A.I.) where an equation solver was found (mathway.com) and produced the following values and graphs.

Similar points for RF power (10logP) are 0.8266713…, 1.371288…, and 10.

I've just received steve12's additional answer (below).  I should have graphed the function to see that there are in fact three roots.  Excel Goal Seek results depend on the starting point!  Excel confirms steve12's result at M= 1.1402...

There is no analytical equation for all three roots so the solution must be graphical or iterattive.

branch -1 of lambert W function

gives another real solution

of M = 29.353070859629902

You're still missing the third root Steve.  I only got two on the first attempt.  Note that question is about magnitude so the solutions must be real.  Logs of negative numbers are like heading North from the North pole!

yes i did not see that it was asked a soution for

abs(20*log10(M))==M

the third solution is

M = 0.90142340431067646240910539747071

i calculated it with matlab

here is the script

clear;clc;
format long
syms M

k=-1:0;
SK=(-20*lambertw(k, -log(10)/20))/log(10);

eq = abs(20*log10(M))==M;
SM(1)=vpasolve(eq,M,0.9);
SM(2)=SK(2);
SM(3)=SK(1);

format compact
sprintf("Solutions of M = ")
double(SM')

here is the output

ans =
"Solutions of M = "
ans =
0.901423404310676
1.140287534610586
29.353070859629902

1.122 and 29.36

shouldn't it be ∣20log10​∣T(s)∣∣=∣T(s)∣  for the 3rd solution?