From Newsgroup: sci.electronics.design

"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? 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?

These days you just use LTSpice and you see that your ramps become parabolas as expected.
Also look at what happens without R2.

Version 4.1
SHEET 1 880 680
WIRE 208 -96 160 -96
WIRE 352 -96 288 -96
WIRE 160 32 160 -96
WIRE 224 32 160 32
WIRE 352 32 352 -96
WIRE 352 32 288 32
WIRE -304 128 -304 80
WIRE -256 128 -304 128
WIRE -256 144 -256 128
WIRE -64 144 -96 144
WIRE 32 144 -64 144
WIRE 160 144 160 32
WIRE 160 144 112 144
WIRE 240 144 160 144
WIRE 352 160 352 32
WIRE 352 160 304 160
WIRE 400 160 352 160
WIRE 240 176 192 176
WIRE -304 192 -304 128
WIRE 192 224 192 176
WIRE -96 256 -96 144
WIRE -96 384 -96 336
FLAG -256 144 0
FLAG -304 0 +15
FLAG -304 272 -15
FLAG 272 128 +15
FLAG 272 192 -15
FLAG 400 160 Parabola
FLAG -64 144 Ramp
FLAG 192 224 0
FLAG -96 384 0
SYMBOL OpAmps\\UniversalOpAmp 272 160 R0
SYMATTR InstName U1
SYMATTR Value2 Avol=100k GBW=8Meg Vos=0
SYMBOL voltage -304 -16 R0
WINDOW 123 0 0 Left 0
WINDOW 39 0 0 Left 0
SYMATTR InstName V1
SYMATTR Value 15V
SYMBOL voltage -304 176 R0
WINDOW 123 0 0 Left 0
WINDOW 39 0 0 Left 0
SYMATTR InstName V2
SYMATTR Value 15V
SYMBOL voltage -96 240 R0
WINDOW 123 24 124 Left 2
WINDOW 39 0 0 Left 0
WINDOW 3 76 51 Left 2
SYMATTR Value2 AC 1
SYMATTR Value PULSE(0 1 0 1ms 0ms 0 0 10)
SYMATTR InstName V3
SYMBOL res 128 128 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R1
SYMATTR Value 10k
SYMBOL cap 288 16 R90
WINDOW 0 0 32 VBottom 2
WINDOW 3 32 32 VTop 2
SYMATTR InstName C1
SYMATTR Value 10nF
SYMBOL res 304 -112 R90
WINDOW 0 0 56 VBottom 2
WINDOW 3 32 56 VTop 2
SYMATTR InstName R2
SYMATTR Value 100k
TEXT -464 416 Left 2 !.tran 10ms



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From Newsgroup: sci.electronics.design

On Mon, 11 Aug 2025 09:07:01 -0800, Christopher Howard <christopher@librehacker.com> wrote:

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? Is that the same as
calculating the gain for a sinusoidal input of a particular amplitude
and frequency?

No, you can calculate the frequency response by setting s=jw, or the
dc responce by setting to 0.

(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?

The transfer function is the ratio of the Laplace transform of the
system output to the Laplace transform of the systen input. So to
find the system output y(t) to an arbitrary input x(t) you:

1 - calculate L[x(t)]
2 - multiply above by the transfer fuction
3 - calculate the inverse transform of the above

For example for a heaviside input (x = 1 for t > 0 otherwise x = 0)
L[x] = 1/s. For an integrator with a transfer function 1/s the output
would therefore be 1/s^2 (ie L[y]). Taking the inverse transform of
that would yield y = t (ie a ramp).
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From Newsgroup: sci.electronics.design

JM <sunaecoNoChoppedPork@gmail.com> wrote:

On Mon, 11 Aug 2025 09:07:01 -0800, Christopher Howard <christopher@librehacker.com> wrote:

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? Is that the same as
calculating the gain for a sinusoidal input of a particular amplitude
and frequency?

No, you can calculate the frequency response by setting s=jw, or the
dc responce by setting to 0.

(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?

The transfer function is the ratio of the Laplace transform of the
system output to the Laplace transform of the systen input. So to
find the system output y(t) to an arbitrary input x(t) you:

1 - calculate L[x(t)]
2 - multiply above by the transfer fuction
3 - calculate the inverse transform of the above

For example for a heaviside input (x = 1 for t > 0 otherwise x = 0)
L[x] = 1/s. For an integrator with a transfer function 1/s the output
would therefore be 1/s^2 (ie L[y]). Taking the inverse transform of
that would yield y = t (ie a ramp).

**Two-sided** Laplace, i.e. Fourier with a change of variable.

ThatrCOs one of the many sources of confusion when EEs talk to other
technical folk.

WDDNS two- sided Laplace.

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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