traderbam said:Well I don't wear a white coat...but I am constantly trying to hide from those who do.
I'm sure Nelson isn't suggesting that the DIY crowd are all ignorant of or incapable of understanding mathematics. Understanding how to calculate poles and zeros of a circuit is one of the core skills needed for audio circuit design and it really isn't all that difficult. Implying this is the preserve of white-coated boffins may lead the amateur to feel excluded and I reject this notion.
Explanation of complex notation can be found all over the internet. If you can understand Pythagoras' theorem (right-angled triangles) you can understand complex number notation. This notation is not intended to make it more difficult for the DIYer () but rather is intended to make the mathematics simpler for everyone to use.
By the way, "s" is shorthand for j.w (where J is root -1 and w is the signal frequency in radians per second. w is the same as 2.pi.f where f is the signal frequency in hertz).
Exactly. Ludwig von Boltzman said it well some 100+ years ago: "There's nothing so practical as a really good theory".
Jan Didden
What is the fault in my Calculator?
Now you can hear me.
My pocket calculator must have gone mad.
It is Casio Scientific Calculator.
When I take the square root of -1
something strange happen.
Display shows nothing but an "E".
I need some help in trying to figure out
what is wrong in the calculator.
I took out the batteries, but they seem to be okay.
I do not have any schematic.
Somebody help me please!
eLarson said:
Now you can hear me.
My pocket calculator must have gone mad.
It is Casio Scientific Calculator.
When I take the square root of -1
something strange happen.
Display shows nothing but an "E".
I need some help in trying to figure out
what is wrong in the calculator.
I took out the batteries, but they seem to be okay.
I do not have any schematic.
Somebody help me please!
Re: What is the fault in my Calculator?
It's not always easy on this forum to determine if someone is pulling your leg, but I take the chance.
Sqroot of -1 is correctly displayed as E, because it is an irrational number. There is, of course no number that you can square to get -1, which is what Sqroot -1 really means.
So, you'r calculator is OK.
Jan Didden
halojoy said:
Now you can hear me.
My pocket calculator must have gone mad.
It is Casio Scientific Calculator.
When I take the square root of -1
something strange happen.
Display shows nothing but an "E".
I need some help in trying to figure out
what is wrong in the calculator.
I took out the batteries, but they seem to be okay.
I do not have any schematic.
Somebody help me please!
It's not always easy on this forum to determine if someone is pulling your leg, but I take the chance.
Sqroot of -1 is correctly displayed as E, because it is an irrational number. There is, of course no number that you can square to get -1, which is what Sqroot -1 really means.
So, you'r calculator is OK.
Jan Didden
Another Stupid Question
Yes, I was joking.
But one question remains,
how do I use this formula
if my calculator will not accept irrational numbers?
Or would I be as well off
by using another formula.
Maybe I could save some money
by not having to buy me a more advanced calculator,
or should I call it a more "irrational calculator"?
Anyway, resoldering with silver might be a good idea.
Will probably give the figures of my calculations
with somewhat less 3rd harmonic distortion.
Yes, I was joking.
But one question remains,
how do I use this formula
if my calculator will not accept irrational numbers?
Or would I be as well off
by using another formula.
Maybe I could save some money
by not having to buy me a more advanced calculator,
or should I call it a more "irrational calculator"?
Anyway, resoldering with silver might be a good idea.
Will probably give the figures of my calculations
with somewhat less 3rd harmonic distortion.
Halojoy: a pencil and paper are cheaper than a new calculator.
"it seems that if you narrow the bandwidth of the amplifier to the audible range this would result in some sort of improvement... Can anyone wearing a white coat tell me if I'm wrong/right?"
Give this person a cigar. Your intuition is good...but the means used to limit the bandwidth needs care to avoid unwanted side-effects.
"it seems that if you narrow the bandwidth of the amplifier to the audible range this would result in some sort of improvement... Can anyone wearing a white coat tell me if I'm wrong/right?"
Give this person a cigar. Your intuition is good...but the means used to limit the bandwidth needs care to avoid unwanted side-effects.
I've been experimenting with opamps for quite a while now. I've tried the LM power amplifier series (LM1875/3875/3886) and also the BB OPA541/548/549. I tried a variety of configurations: with or without a compensating cap in the feedback leg, with or without a coupling cap at the input, with or without a zobel, a combination of these, inverting and non-inverting..... I reached a point where further experimentation requires a more scientific approach, meaning, instead of just trying the same things that everyone has tried already and shared the results over the internet and listen to the results, I am still obsessed enough about this topic that I need to take it a step a step further... So that generated my initial question. Please understand that I am limited by the fact that I am not an engineer and have no foundation to deal with some of the basic issues involved in most what I am doing as a hobby
traderbam, apologizing if my questions sound silly and repetitive, what is the best way to narrow an opamp bandwidth? I thought the answer was to explore the feature documented by National on the LM3875 specsheet. Is this not the case?
Thanks
Paul
traderbam, apologizing if my questions sound silly and repetitive, what is the best way to narrow an opamp bandwidth? I thought the answer was to explore the feature documented by National on the LM3875 specsheet. Is this not the case?
Thanks
Paul
roll off
Paul,
I think you must make a distinction between narrowing the opamp bandwidth and narrowing the circuit bandwidth.
Narrowing the opamp bandwidth can only be done with those opamps that have a compensation pin(s). This should be documented in the data sheet. Often, these opamps are internally compensated for a gain higher than 1, because with a gain higher than one the feedback is less, stability is better and the comp cap is smaller. Example is the good old 5534 which IIRC is compensated for gains of 5 or greater. So, using this opamp in a follower (g=1) mode, you need to increase the comp cap for stability.
But, you can also overcompensate these opamps, for instance use the 5534 in g=10 but still use an extra comp cap to narrow the bandwidth. Again, the data sheet should have a graph showing the comp cap value vs the bandwidth.
The other thing is to narrow the bandwidth of the circuit. For instance, if you have an opamp circuit with a feedback resistor from opamp output to inv input, you can parallel this resistor with a cap to limit the circuit bandwidth. If your feedback resistor is 10k and you want to limit the bandwidth to 50kHz, calculate a cap that has an impedance of 10k at 50kHz. That gives you a circuit bandwidth that rolls off 3dB at 50kHz to a gain of 1 at very high frequencies. You cannot roll off further, because the minimum impedance of the cap = 0, and at that point you have a follower.
So, the *best way* really depends on exactly what you want to achieve and what your circuit topology is. Does this help?
Jan Didden
Paul,
I think you must make a distinction between narrowing the opamp bandwidth and narrowing the circuit bandwidth.
Narrowing the opamp bandwidth can only be done with those opamps that have a compensation pin(s). This should be documented in the data sheet. Often, these opamps are internally compensated for a gain higher than 1, because with a gain higher than one the feedback is less, stability is better and the comp cap is smaller. Example is the good old 5534 which IIRC is compensated for gains of 5 or greater. So, using this opamp in a follower (g=1) mode, you need to increase the comp cap for stability.
But, you can also overcompensate these opamps, for instance use the 5534 in g=10 but still use an extra comp cap to narrow the bandwidth. Again, the data sheet should have a graph showing the comp cap value vs the bandwidth.
The other thing is to narrow the bandwidth of the circuit. For instance, if you have an opamp circuit with a feedback resistor from opamp output to inv input, you can parallel this resistor with a cap to limit the circuit bandwidth. If your feedback resistor is 10k and you want to limit the bandwidth to 50kHz, calculate a cap that has an impedance of 10k at 50kHz. That gives you a circuit bandwidth that rolls off 3dB at 50kHz to a gain of 1 at very high frequencies. You cannot roll off further, because the minimum impedance of the cap = 0, and at that point you have a follower.
So, the *best way* really depends on exactly what you want to achieve and what your circuit topology is. Does this help?
Jan Didden
Impedance of Caps
Is the impedance of a cap same as reactance?
I mean, you mention 10000 ohms at a certain frequency.
So the parallell resistance, cap//resistor becomes 5000 ohms.
Which will lower the gain the higher the frequency.
Reactance (Ohm)= 1/[2pi x f(Hz)x C(Farad)]
Capacitance (Farad)= 1/[2pi x f x R(Ohm)]
If you know the formula of the impedance
of the components,
it should be very easy to make a little program
for your PC, and plot the curves of freq response.
Or to program your Calculator.
My can not plot curves. But with paper and pen I can do it.
Is the impedance of a cap same as reactance?
I mean, you mention 10000 ohms at a certain frequency.
So the parallell resistance, cap//resistor becomes 5000 ohms.
Which will lower the gain the higher the frequency.
Reactance (Ohm)= 1/[2pi x f(Hz)x C(Farad)]
Capacitance (Farad)= 1/[2pi x f x R(Ohm)]
If you know the formula of the impedance
of the components,
it should be very easy to make a little program
for your PC, and plot the curves of freq response.
Or to program your Calculator.
My can not plot curves. But with paper and pen I can do it.
Re: Impedance of Caps
Yes, you can do this. You can also plot the phase shift of a signal as well.
The impedance of a cap is the reactance. The impedance of a resistor is called resistance, combine resistance and reactance in vector form, and you get impedance. For example, 10000 ohms R in parallel with 10000 ohms Xc does not equil 5000 ohms impedance, it actually equils 7071 ohms reactance. (Remember, that the -3db Point is where the voltage = 70% of the start value) This reactance also happens at an angle -45 degrees. (If you know nothing of vector or complex math, look on the web or for some math books... ...it is easy to learn)
What this means is that the frequency which causes the capacitors reactance to be 10000 ohms, will be shifted by -45 degrees, and will be reduced to 70.7% of what it should be.
Hope this helps
-Dan
halojoy said:
Yes, you can do this. You can also plot the phase shift of a signal as well.
The impedance of a cap is the reactance. The impedance of a resistor is called resistance, combine resistance and reactance in vector form, and you get impedance. For example, 10000 ohms R in parallel with 10000 ohms Xc does not equil 5000 ohms impedance, it actually equils 7071 ohms reactance. (Remember, that the -3db Point is where the voltage = 70% of the start value) This reactance also happens at an angle -45 degrees. (If you know nothing of vector or complex math, look on the web or for some math books... ...it is easy to learn)
What this means is that the frequency which causes the capacitors reactance to be 10000 ohms, will be shifted by -45 degrees, and will be reduced to 70.7% of what it should be.
Hope this helps
-Dan
s, j and other abstract weirdness
I hope "They" don't come to get me for blabbing
about it.
I had let my IEEE membership
lapse, too. I wonder if that makes me a reckless
fugitive!
Regarding the 'square-root-of-minus-1" try not to
think of it too directly; if you come at it from an
angle it helps...
if -1 is thought of as a 1 with an angle of 180
degress (in other words turned 'round backward
from a reg'lar "1", same magnitude, opposite direction on a line), and if you think of the angle
as being part of an exponent, what happens when
you raise the whole mess of it to the 1/2 power?
You get 1 and half that exponent.. 90 degrees, now.
'cept its not on that line anymore... its sticking up
in the air.
eL
(looks like someone spilled the beans already... hopefully 'They' won't be out to get him, too!)
Nelson Pass said:
S is a variable designed to keep the DIY crowd in
their place.
Only white coated engineers have mastered the
esoteric and secret meaning of S.
I hope "They" don't come to get me for blabbing
about it.
I had let my IEEE membershiplapse, too. I wonder if that makes me a reckless
fugitive!
Regarding the 'square-root-of-minus-1" try not to
think of it too directly; if you come at it from an
angle it helps...
if -1 is thought of as a 1 with an angle of 180
degress (in other words turned 'round backward
from a reg'lar "1", same magnitude, opposite direction on a line), and if you think of the angle
as being part of an exponent, what happens when
you raise the whole mess of it to the 1/2 power?
You get 1 and half that exponent.. 90 degrees, now.
'cept its not on that line anymore... its sticking up
in the air.
eL
(looks like someone spilled the beans already... hopefully 'They' won't be out to get him, too!)
Re: Re: Re: What is the fault in my Calculator?
Yeah, you're right, but you are using j in that, which was what we were trying to get the root of in the first place. Not fair!
Jan Didden
dkemppai said:
I don't belive you.
There is a number you can square to get -1, it is (0,j1) in rectangular notation (or 1 at an angle of 90 in polar notation).
Not a lot of people need to use it
-Dan
Yeah, you're right, but you are using j in that, which was what we were trying to get the root of in the first place. Not fair!
Jan Didden
Re: Re: Re: What is the fault in my Calculator?
And -j of course which squared gives -1. Or i and -i as in maths as I think j is used for physics/electronics mainly. Confusing to say the least. It does exist however.
The solution to SQRT(-1) is thus j and -j.
/UrSv
dkemppai said:
I don't belive you.
There is a number you can square to get -1, it is (0,j1) in rectangular notation (or 1 at an angle of 90 in polar notation).
Not a lot of people need to use it
-Dan
And -j of course which squared gives -1. Or i and -i as in maths as I think j is used for physics/electronics mainly. Confusing to say the least. It does exist however.
The solution to SQRT(-1) is thus j and -j.
/UrSv
reactance is futile!
Halojoy,
Impedance is not the same as reactance. Impedance is the combination of the resistance and reactance of a circuit. If a circuit is a pure capacitor or a pure inductor then its impedance is the same as its reactance. A pure resistor's impedance is the same as it's resistance because it has no reactance.
Reactance is different from resistance because it causes a phase difference between voltage and current. A resistance does not.
Now, impedance is the ratio of voltage to current. Because the current may not be in phase with the voltage the impedance is "complex" - it's value is best expressed using complex numbers to represent the different contributions of resistance and reactance.
Impedance Z = R + j.X
Where R is a resistance and X is a reactance. The j just says that the reactance is pi/2 radians out of phase. Note that X can be positive or negative depending upon how much of the reactance is capacitive and how much is inductive. This is the universal form for expressing impedance and from it you can calculate a magnitude and a phase angle.
Impedance magnitude = sqrt( R^2 + X^2)
Impedance phase = arctan(X/R) in radians
When people talk about impedance as a specific number they are really talking about impedance magnitude. However, impedance is actually both a magnitude and a phase. You can see that resistance and reactance must be combined geometrically to get the magnitude, unlike two resistances.
You can also think about this geometrically in the form of a right-angled triangle. You make the adjacent side length equal the resistance, the opposite side length equal the reactance and the the impedance magnitude is the length of the hypoteneuse and the impedance phase angle is the angle between adjacent and hypotenuse.
The impedance of a pure resistor is R.
The impedance of a pure capacitor is -j/(wC)
The impedance of a pure inductor is jwL
where w is the frequency in radians per second
You can calculate the combined impedance of several components just as you do with resistors.
!!!Note that: j.j = -1 and 1/j=-j.
Eg:
Capacitor in series with a resistor:
Z = R - j/(wC)
Z(mag) = sqrt (R^2 + (1/(wC))^2)
Z(phase) = arctan(-1/(wCR)) ....the phase angle is negative
Capacitor in parallel with inductor:
Z = 1/( 1/Zc + 1/Zl )
= 1/( jwC - j/(wL) )
= 1/ ( j.(wC - 1/(wL)))
= 0 - j/[wC - 1/(wL)]
Z(mag) = sqrt( 0^2 + {1/[wC-1/(wL)]}^2 ) = 1/[wC-1/(wL)]
Z(phase) = arctan( -[wC-1/(wL)] )
Note the subtraction of capacitive and inductive reactance. Capacitors are essentially inverse inductors. Note also that when wC = 1/(wL) the Z(mag) becomes zero and the Z(phase) becomes zero too. The value of w when this happens is called the "resonant frequency" of the circuit. At resonance this series circuit becomes a "pure" short circuit. A similar thing happens when you have a capacitor and inductor in parallel, except at resonance the circuit becomes a "pure" open-circuit.
You can make up any combinations of these components and if you get the formula for the impedance in the form R+jX you can easily calculate the phase and magnitude.
Halojoy,
Impedance is not the same as reactance. Impedance is the combination of the resistance and reactance of a circuit. If a circuit is a pure capacitor or a pure inductor then its impedance is the same as its reactance. A pure resistor's impedance is the same as it's resistance because it has no reactance.
Reactance is different from resistance because it causes a phase difference between voltage and current. A resistance does not.
Now, impedance is the ratio of voltage to current. Because the current may not be in phase with the voltage the impedance is "complex" - it's value is best expressed using complex numbers to represent the different contributions of resistance and reactance.
Impedance Z = R + j.X
Where R is a resistance and X is a reactance. The j just says that the reactance is pi/2 radians out of phase. Note that X can be positive or negative depending upon how much of the reactance is capacitive and how much is inductive. This is the universal form for expressing impedance and from it you can calculate a magnitude and a phase angle.
Impedance magnitude = sqrt( R^2 + X^2)
Impedance phase = arctan(X/R) in radians
When people talk about impedance as a specific number they are really talking about impedance magnitude. However, impedance is actually both a magnitude and a phase. You can see that resistance and reactance must be combined geometrically to get the magnitude, unlike two resistances.
You can also think about this geometrically in the form of a right-angled triangle. You make the adjacent side length equal the resistance, the opposite side length equal the reactance and the the impedance magnitude is the length of the hypoteneuse and the impedance phase angle is the angle between adjacent and hypotenuse.
The impedance of a pure resistor is R.
The impedance of a pure capacitor is -j/(wC)
The impedance of a pure inductor is jwL
where w is the frequency in radians per second
You can calculate the combined impedance of several components just as you do with resistors.
!!!Note that: j.j = -1 and 1/j=-j.
Eg:
Capacitor in series with a resistor:
Z = R - j/(wC)
Z(mag) = sqrt (R^2 + (1/(wC))^2)
Z(phase) = arctan(-1/(wCR)) ....the phase angle is negative
Capacitor in parallel with inductor:
Z = 1/( 1/Zc + 1/Zl )
= 1/( jwC - j/(wL) )
= 1/ ( j.(wC - 1/(wL)))
= 0 - j/[wC - 1/(wL)]
Z(mag) = sqrt( 0^2 + {1/[wC-1/(wL)]}^2 ) = 1/[wC-1/(wL)]
Z(phase) = arctan( -[wC-1/(wL)] )
Note the subtraction of capacitive and inductive reactance. Capacitors are essentially inverse inductors. Note also that when wC = 1/(wL) the Z(mag) becomes zero and the Z(phase) becomes zero too. The value of w when this happens is called the "resonant frequency" of the circuit. At resonance this series circuit becomes a "pure" short circuit. A similar thing happens when you have a capacitor and inductor in parallel, except at resonance the circuit becomes a "pure" open-circuit.
You can make up any combinations of these components and if you get the formula for the impedance in the form R+jX you can easily calculate the phase and magnitude.
It's all in the mind
UrSv,
I agree with what you've said. Yes, in pure mathematics i is most often used and I have also seen j,k,l... used for multidimentional vectoring. I believe j is invariably used in electronics to avoid the mix up with current.
I'm not sure i or j literally exist, however. In fact, I'm not convinced that any mathematics literally exists - you know, as if it were a feature of the physical world. What exactly is the number 3? I think mathematics, like language, is all made-up by adults to torture school kids. Err...I mean to help our ape-brains model the real world.
UrSv,
I agree with what you've said. Yes, in pure mathematics i is most often used and I have also seen j,k,l... used for multidimentional vectoring. I believe j is invariably used in electronics to avoid the mix up with current.
I'm not sure i or j literally exist, however. In fact, I'm not convinced that any mathematics literally exists - you know, as if it were a feature of the physical world. What exactly is the number 3? I think mathematics, like language, is all made-up by adults to torture school kids. Err...I mean to help our ape-brains model the real world.
Some notes on s, jw and other things
This turned out much, much longer than intended, so please bear
with me and don't bother to read if you already know these
things (or, rather, do read it anyway, so yoiu can point out the
errrors that have most surely sneaked in).
Things have been a bit scattered in this thread, so I'll make an
attempt att collect a few things that have been said, and a few
that haven't, for those of you who did not learn about these
things in school (or elsewhere). It is a bit lenghty, but yet
hopelessly incomplete and brief.
The j is used in electronics as a replacement for the i in
mathematics, since i is so heavily used for currents. It is more
appropriate to say that j*j = -1 than to say that j = sqrt(-1)
since sqrt(-1) is undefined. The symbol j is never evaluated and
is only used in algebraic manipulations of expressions, so we
never need to know what the value of j is, or if it has one. A
number of the form ja or -ja is called an imaginary number. A
number of the form a+jb or a-jb is called a complex number.
The complex number can be though of a two-dimensional vectors,
where the imaginary numbers are orthogonal vectors to the
real numbers. The purpose of introducing imaginary numbers is
to avoid vector notation, so we can write eg. a+jb instead of
(a, jb). This simplifies certain things in mathematics, like the
Euler formulas.
A complex number
a+jb
can be thought of as a vector with length
sqrt(a^2+b^2)
and angle
arctan(b/a).
In particular, if a+jb is a voltage or a current, we get the
amplitude and phase angle.
From the definition of j we also get that
j^3 = j^2*j = -1*j = -j
j^4 = j^2*j^2 = -1*(-1) = 1
1/j = j/j^2 = j/(-1) = -j
which are useful when manipulating complex numbers.
Since radians, not degrees, is the basic unit for angles in
mathematics, we often use the expression 2*pi*f, where f is
a frequency. This expression is commonly abbreviated as
small omega, which I will write as w for lack of greek letters.
This has given the name to the j-omega method (or the jw
method) which is commonly used in electronics. We can then
write the reactance of an inductor L as jwL and the reactance
of a capacitor as 1/(jwC) = -j/(wC). From this we can see that
reactance is orthogonal (ie. rotated 90 degrees from) resistance.
For instance, an inductor with inductance L and DC resistance R
has the impedance R+jwL which is a vector. The magnitude of
the impedance is sqrt(R^2+(wL)^2) and
the impedance has a phase angle of arctan(wL/R), ie. a perfect
inductor with R=0 has the phase angle 90 deg. Similarily, a
perfect capacitor will have a phase angle of -90 deg.
We can now apply Ohms and Kirchoffs laws to impedances in the
same way as we do with resistances, just that we use complex
numbers instead. Connecting a resistor R in series with a
capacitor C gives the total impedance R + 1/(jwC). Since
this expression contains w, it is obviously not constant but
frequency dependent, as expected.
Example:
A sinewave with frequency f and amplitude (peak value) A is
feeding a resistor R in series with a capacitor C. What is the voltage over the capacitor?
The signal Asin(2*pi*f*t) = Asin(wt). Assuming perfect components,
the resistor has impedance R and the capacitor has impedance
1/(jwC), so these components act as a voltage divider where
the voltage over the capacitor is
1/(jwC)
-------------- * Asin(wt)
R + 1/(jwC)
This can be simplified to
1
------------- * Asin(wt)
jwRC + 1
So the amplitude of the voltage over C is
1
---------------------------- * Asin(wt)
sqrt( (wRC)^2 + 1^2)
and since Zc = 1/(jwC) = -j/(wC) we get the phase angle
-1/(wC) -1
arctan ---------- = arctan----------
R wRC
compared to the phase of the signal Asin(wt), in this case the
phase is lagging behind the signal source.
The jw method (and Fourier transforms) only work for signals
that do not change over time, that is, signals that can be made
up as a sum of sines and cosines. This is usually sufficient for
most common situations in audio. When transient signals need
to be accounted for, we need to use the Laplace transform and
this is where the s enters. The s is just a trick to make the
transforms converge and s = jw + c for some suitable constant
c that gives convergence. If we do not have any transient
behaviour the ordinary Fourier transform will converge so we
can choose c = 0 and, thus, s collapses to jw. That is, in many
cases where we see an s in formulas it is nothing but a shorthand
for jw. It is common practice to write s even when not required
since it makes the formulas simpler to read, write and manipulate.
The reactance of an inductor can thus be written sL and the
reactance of a capacitor is 1/(sL).
Example
What is the impedance of an inductor L in series with a
capacitance C?
1
Z = sL + ----
sC
Replacing s with jw, we get
1 1 1
Z = jwL + -------- = jwL - j------ = j(wL - -----)
jwC wC wC
The magnitude of Z is wL-1/(wC) Ohms and the phase angle
is +90 deg or -90 deg depending on the sign of wL-1/(wC).
This turned out much, much longer than intended, so please bear
with me and don't bother to read if you already know these
things (or, rather, do read it anyway, so yoiu can point out the
errrors that have most surely sneaked in).
Things have been a bit scattered in this thread, so I'll make an
attempt att collect a few things that have been said, and a few
that haven't, for those of you who did not learn about these
things in school (or elsewhere). It is a bit lenghty, but yet
hopelessly incomplete and brief.
The j is used in electronics as a replacement for the i in
mathematics, since i is so heavily used for currents. It is more
appropriate to say that j*j = -1 than to say that j = sqrt(-1)
since sqrt(-1) is undefined. The symbol j is never evaluated and
is only used in algebraic manipulations of expressions, so we
never need to know what the value of j is, or if it has one. A
number of the form ja or -ja is called an imaginary number. A
number of the form a+jb or a-jb is called a complex number.
The complex number can be though of a two-dimensional vectors,
where the imaginary numbers are orthogonal vectors to the
real numbers. The purpose of introducing imaginary numbers is
to avoid vector notation, so we can write eg. a+jb instead of
(a, jb). This simplifies certain things in mathematics, like the
Euler formulas.
A complex number
a+jb
can be thought of as a vector with length
sqrt(a^2+b^2)
and angle
arctan(b/a).
In particular, if a+jb is a voltage or a current, we get the
amplitude and phase angle.
From the definition of j we also get that
j^3 = j^2*j = -1*j = -j
j^4 = j^2*j^2 = -1*(-1) = 1
1/j = j/j^2 = j/(-1) = -j
which are useful when manipulating complex numbers.
Since radians, not degrees, is the basic unit for angles in
mathematics, we often use the expression 2*pi*f, where f is
a frequency. This expression is commonly abbreviated as
small omega, which I will write as w for lack of greek letters.
This has given the name to the j-omega method (or the jw
method) which is commonly used in electronics. We can then
write the reactance of an inductor L as jwL and the reactance
of a capacitor as 1/(jwC) = -j/(wC). From this we can see that
reactance is orthogonal (ie. rotated 90 degrees from) resistance.
For instance, an inductor with inductance L and DC resistance R
has the impedance R+jwL which is a vector. The magnitude of
the impedance is sqrt(R^2+(wL)^2) and
the impedance has a phase angle of arctan(wL/R), ie. a perfect
inductor with R=0 has the phase angle 90 deg. Similarily, a
perfect capacitor will have a phase angle of -90 deg.
We can now apply Ohms and Kirchoffs laws to impedances in the
same way as we do with resistances, just that we use complex
numbers instead. Connecting a resistor R in series with a
capacitor C gives the total impedance R + 1/(jwC). Since
this expression contains w, it is obviously not constant but
frequency dependent, as expected.
Example:
A sinewave with frequency f and amplitude (peak value) A is
feeding a resistor R in series with a capacitor C. What is the voltage over the capacitor?
The signal Asin(2*pi*f*t) = Asin(wt). Assuming perfect components,
the resistor has impedance R and the capacitor has impedance
1/(jwC), so these components act as a voltage divider where
the voltage over the capacitor is
1/(jwC)
-------------- * Asin(wt)
R + 1/(jwC)
This can be simplified to
1
------------- * Asin(wt)
jwRC + 1
So the amplitude of the voltage over C is
1
---------------------------- * Asin(wt)
sqrt( (wRC)^2 + 1^2)
and since Zc = 1/(jwC) = -j/(wC) we get the phase angle
-1/(wC) -1
arctan ---------- = arctan----------
R wRC
compared to the phase of the signal Asin(wt), in this case the
phase is lagging behind the signal source.
The jw method (and Fourier transforms) only work for signals
that do not change over time, that is, signals that can be made
up as a sum of sines and cosines. This is usually sufficient for
most common situations in audio. When transient signals need
to be accounted for, we need to use the Laplace transform and
this is where the s enters. The s is just a trick to make the
transforms converge and s = jw + c for some suitable constant
c that gives convergence. If we do not have any transient
behaviour the ordinary Fourier transform will converge so we
can choose c = 0 and, thus, s collapses to jw. That is, in many
cases where we see an s in formulas it is nothing but a shorthand
for jw. It is common practice to write s even when not required
since it makes the formulas simpler to read, write and manipulate.
The reactance of an inductor can thus be written sL and the
reactance of a capacitor is 1/(sL).
Example
What is the impedance of an inductor L in series with a
capacitance C?
1
Z = sL + ----
sC
Replacing s with jw, we get
1 1 1
Z = jwL + -------- = jwL - j------ = j(wL - -----)
jwC wC wC
The magnitude of Z is wL-1/(wC) Ohms and the phase angle
is +90 deg or -90 deg depending on the sign of wL-1/(wC).
Thanks, those formulas very useful in audio
In my calculator I have 2 programs P1 P2 and 6 constants K1-K6
K4 is always SQR 2 = 1.4142... half of that is 0.7071...
P1- program always puts out:
1/(2pi x K1 x K2 x K3)
K1 K2 K3, I think of as
f R C
but I have always have K1 set as 1.0
so by putting 2 values into K2 K3
I get the third value - the unknown values
I know not much of vector maths, only what I have use for
I know a sinus curve have 360 degrees (could be called radians also)
and at 45 and 135 it crosses 0.7071.. of its maxium.
sinus and cosinus is useful in audio
a sinus wave is to be thought of as points on a circle
transported to the right, as time goes from zero and onwards
If I think of symbol w = 2 pi f
it is not as hard to grasp
In my calculator I have 2 programs P1 P2 and 6 constants K1-K6
K4 is always SQR 2 = 1.4142... half of that is 0.7071...
P1- program always puts out:
1/(2pi x K1 x K2 x K3)
K1 K2 K3, I think of as
f R C
but I have always have K1 set as 1.0
so by putting 2 values into K2 K3
I get the third value - the unknown values
I know not much of vector maths, only what I have use for
I know a sinus curve have 360 degrees (could be called radians also)
and at 45 and 135 it crosses 0.7071.. of its maxium.
sinus and cosinus is useful in audio
a sinus wave is to be thought of as points on a circle
transported to the right, as time goes from zero and onwards
If I think of symbol w = 2 pi f
it is not as hard to grasp
- Status
- This old topic is closed. If you want to reopen this topic, contact a moderator using the "Report Post" button.