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Re: Integrator transfer function and arbitrary continuous input signals

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Edward Rawde <invalid@invalid.invalid> wrote:
> "Christopher Howard" <christopher@librehacker.com> wrote in message
> news:877bz9zt1m.fsf@librehacker.com...
>> Hi, I'm trying to work slowly through the great Op Amp book by Roberge et al
>> that was recommended earlier. I downloaded the 2nd edition v. 1.8.1. I'm
>> finding it enlightening to slowly process the paragraphs and take notes
>> on the diagrams and equations.
>> 
>> Something I'm getting hung up on though is they dive early on into
>> transfer functions, and that hasn't been covered yet in my introductory
>> DE book. I've been trying to cram in some quick Internet research on
>> laplace transforms and such but it has been a bumpy ride.
>> 
>> In chapter one (equation 1.21) they gave the transfer function of an
>> integrator, i.e., an op amp with resistor and capacitor feedback
>> network, as -1/(RCs). I found that if I replaced s with 2 pi f, I could
>> predict the gain from a steady sinusoidal signal of matching frequency,
>> and when I tried this out with my real integrators, I got matching
>> results.
>> 
>> My questions:
>> 
>> (1) so, if I replace s with a complex number, one that has both a real
>> and an imaginary part, what does that mean?
> 
> It means that instead of just individual numbers, variables consist of pairs of numbers.
> This is so that both magnitude (length) and phase (angle) can be accommodated.
> For example (1, 0) is equivalent to the real number 1 and it has a length of 1 unit.
> It can be written 1 + j0 or just 1
> The number (0, 1) also has a length of one unit but it's not on the real
> number line, it's been rotated 90 degrees counterclockwise.
> It can be written 0 + j1 or just j
> The following may be worth watching. It's not specifically about electronics.
> If you don't want to watch it all, just watch the part at 19:10
> Note also that outside electronics j usually becomes i.
> Except in some of my old handed down textbooks where "operator j" is used
> and the complex plane is called the Argand diagram.
> https://en.wikipedia.org/wiki/Jean-Robert_Argand
> I think it remained j in electronics because i is used for current.
> 
>> Is that the same as
>> calculating the gain for a sinusoidal input of a particular amplitude
>> and frequency?
>> 
>> (2) How do I use/apply this transfer function if I've got some
>> nonsinusoidal continuous input, like say a steady voltage, or a linear
>> ramping voltage?
>> 
>> -- 
>> Christopher Howard 
>

A couple of points that may clarify issues. 

First, the complex numbers are _numbers_. That is, they allow the same
arithmetic operations as the reals. The only axiom of the reals that has to
be relaxed is the Archimedean order property—you can sort any list of reals
by sign and magnitude, but you can’t do that with complex numbers. 

Second, even a real-valued function will in general have a complex-valued
transform. 
Constraining the function values to be real is equivalent to constraining
the transform to be Hermitian, i.e. if h(t) is real-valued and has
transform H(f), then 

H(-f) = H^*(f),

the complex conjugate. 

Cheers 

Phil Hobbs 



-- 
Dr Philip C D Hobbs  Principal Consultant  ElectroOptical Innovations LLC /
Hobbs ElectroOptics  Optics, Electro-optics, Photonics, Analog Electronics 

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Integrator transfer function and arbitrary continuous input signals Christopher Howard <christopher@librehacker.com> - 2025-08-11 09:07 -0800
  Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-11 17:34 +0000
  Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 14:17 -0400
    Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 14:19 -0400
    Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-11 19:43 +0000
  Re: Integrator transfer function and arbitrary continuous input signals "Edward Rawde" <invalid@invalid.invalid> - 2025-08-11 16:56 -0400
  Re: Integrator transfer function and arbitrary continuous input signals JM <sunaecoNoChoppedPork@gmail.com> - 2025-08-15 23:14 +0100
    Re: Integrator transfer function and arbitrary continuous input signals Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2025-08-16 00:34 +0000

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