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Richard Clark wrote:
Appeals of authority that are pegged to Cecil are like trying to tread water with a concrete life preserver. Your logic is blighted by a forced conclusion that has nothing to do with the obvious observation that antennas, as transmission lines, are quite evidently non-linear in their characteristic Z. This has been demonstrated and is historic from sources that even Terman's accepts. There exist transmission lines with a changing Z0 along their lengths. Those transmission lines are linear systems. -- 73, Cecil http://www.qsl.net/w5dxp |
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Richard Clark wrote:
"Demanding that "new frequencies" must exist AND then saying that they must be of such-and-such a magnitude to qualify is a hoot." Glad you got a kick out of that. It is not original. In analog microwave systems, often an baseband intermod monitor is used to alarm the operator that nonlinearity has arrived in his system. New frequencies have appeared and have reached a preset arbitrary amplitude sufficient to trigger an alarm. Nothing is perfect so there will always be some intermod. This requires setting a level of these intermod products which will trigger the alarm. This is a standard procedure. Best regards, Richard Harrison, KB5WZI |
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Richard Clark wrote:
"Choose your own numbers, or find an authority to quote a quantitave response." If you can`t detect it, it might as well not exist. If you do detect it, it`s up to you to correct it or not. How many antennas have troubled you with new frequencies? Best regards, Richard Harrison, KB5WZI |
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Richard Harrison wrote:
How many antennas have troubled you with new frequencies? Half a century ago, I installed a ceramic capacitor across the feedpoint of my dipole to change the resonant frequency. I'm sure it generated some new frequencies when it blew up and caught on fire. :-) -- 73, Cecil http://www.qsl.net/w5dxp |
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Roy Lewallen wrote:
I'm sure that somewhere in one of your texts you can find the definition of linear as applied to networks. Once you do, though, a little thought is required to discover that y = mx + b doesn't satisfy the criteria for network linearity. To be linear, a network has to satisfy superposition. This means that: If y1 is the response to excitation x1 and y2 is the response to excitation x2, then the response to x1 + x2 must be y1 + y2. Let's try that with your function. The response to x1 is: y(x1) = mx1 + b The response to x2 is: y(x2) = mx2 + b The sum of y(x1) and y(x2) is: y(x1) + y(x2) = m(x1 + x2) + 2b But response to x1 + x2 is: y(x1 + x2) = m(x1 + x2) + b These are not equal as they must be to satisfy superposition and therefore the requirements for linearity. Roy Lewallen, W7EL Richard Harrison wrote: Roy Lewallen, W7EL wrote: "But of course you realize that the function y = mx + b doesn`t meet the requirements of a linear function when applied to network theory." Works for me. Linear means the graph of the function is a straight line. f(x) = y = mx + b is called linear because its graph is a straight line. A straight line is the shortest distance between two points. In y = mx + b, m is a constant determining the slope of the line. x is is the independent variable. b is the offset or point along the x-axis where the line crosses. y then is a linear function of x because its slope is always mx, but displaced in the x-direction by a constant value, namely b. y is linear the same as IR is linear, or by substitution, E is linear in Ohm`s law where E=IR. For any value of I, voltage = IR and the graph of I versus E is a straight line with a slope equal to R. Resistance is a common factor in network theory. Best regards, Richard Harrison, KB5WZI Not that it means anything, but the linearity requirement is met when b = 0, which, of course, is a subset of the family of equations of the form y = mx + b. 73, Chuck NT3G |
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A good target number for antenna linearity would be one that does not
limit system dynamic range. Our best receivers have a dynamic range of around 120 dB as measured by the minimum discernable signal on the low end, and the point where two-tone third order distortion products are detectable on the high end. 140 dB seems reasonable for an antenna and would theoretically be measured the same way as receiver dynamic range, though setting up a noise-free environment, and coupling large distortion-free signals to a test antenna is a challenge, and is probably one reason we don't see these measurements. The other reason is that there is good evidence that a properly built antenna does not limit system dynamic range. That is, it is very linear in the superposition sense. By the way, generating new frequencies is not necessarily a violation of superposition (though it usually is). Consider a system undergoing a constant Doppler shift. 73, Glenn AC7ZN |
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Richard Clark wrote:
"Your explanation sounds like you are practicing psychaitry, not technoligy." I think quantification is valuable if the measured value is accurate and if the value makes a difference. Antennas are used with transmitters of megawatts of power. These have limitations by regulations on maximum noise and harmonic content. It depends on the jurisdiction, but maximum noise and distortion must be at least 50 dB below the fully modulated level in some locales. I`ve often used the H.P. noise and distortion analyzer to measure off the air to be sure we complied with the regulation. It never occurred to me that our antenna system had a part in noise and distortion production. I expected curvature in a tube`s characteristics or a failed component to cause a rise in noise and distortion. Not once do I recall our antenna system causing distortion anywhere except in the edges of pattern nulls.*This is normal. Receiving antennas on the other hand deliver a satisfactory signal having only microwatts of power. As one responder noted the dynamic range is enormous. This is not really an issue for concern among amateurs. Antennas are in general distortion free. Best regards, RIchard Harrison, KB5WZI |
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Richard Clark wrote:
Cecil would shrug off 59% worth of distortion to define it linear. Richard, seems you suffer from the same affliction as Howard Hughes, repeating the same psychotic nonsense over and over. -- 73, Cecil http://www.qsl.net/w5dxp |
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"Richard Harrison" wrote in message ... snip Antennas are in general distortion free. Yes, but "bad times" can make them non-linear. Consider the bolted joints that get corroded and semi-conductive over time, with rain and temperature changes to help. Nearby RF sources can generate distortion products in my antennas if I haven't kept up my maintenance. |
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wrote in message oups.com... ... By the way, generating new frequencies is not necessarily a violation of superposition (though it usually is). Consider a system undergoing a constant Doppler shift. 73, Glenn AC7ZN Glenn, boy! I can't read all these posts, so I was trying to see where you were going and asking by just skimming, primarily your posts. The above caught my eye. Doppler is not "generating new frequencies" as a non linearity does. A non-linear system will produce harmonics with one exitation frequency and produce the common mixing / IM with multiple exitation (superimposed) frequencies. I think trying to mix relativistic effects in with the stationary world is an unnecessary complication of a linearity discussion. If I have time I'll try to follow the thread to see what you're really after...but. If it takes mega watts to see some non linearity in an antenna, who cares? and more importantly how will you know whree it is occuring since things like the junction of two connectorc can produce enough IM to mask other, smaller sources. If you tried an experiment looking for IM / Mixing you might try to use a receiver becaue a receiver could be a very sensitive detector...but you'd have to have a pretty good receiver. Something like a kW LO and mixer to have a really good intercept. I'm not sure of the point here... Do antennas cause IM? Sounds like a deadend arena to me. 73, Steve, K9DCI |
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Thanks, Steve,
Good post. You do need to read a little mo by mentioning Doppler shifts, I was illustrating that a linear system can generate new frequencies without violating the law of superposition. No relativistic effects needed he a two-tone train whistle will not distort in the superposition sense under Doppler shift, but it can generate new (or different) frequencies. My point was that generation of new frequencies is not necessarily a valid test for nonlinearity. My position on antennas is the same as yours. Earlier posts were confusing as the term 'linearity' was being applied to antennas in two ways. One in the electrical superposition sense (by me) and the other in current distribution along the antenna element. Electrically, antennas are very linear (I believe) even when their current distribution along the element length is not. I stopped posting when I realized we were discussing two different things. 73, Glenn AC7ZN Steve N. wrote: wrote in message oups.com... ... By the way, generating new frequencies is not necessarily a violation of superposition (though it usually is). Consider a system undergoing a constant Doppler shift. 73, Glenn AC7ZN Glenn, boy! I can't read all these posts, so I was trying to see where you were going and asking by just skimming, primarily your posts. The above caught my eye. Doppler is not "generating new frequencies" as a non linearity does. A non-linear system will produce harmonics with one exitation frequency and produce the common mixing / IM with multiple exitation (superimposed) frequencies. I think trying to mix relativistic effects in with the stationary world is an unnecessary complication of a linearity discussion. If I have time I'll try to follow the thread to see what you're really after...but. If it takes mega watts to see some non linearity in an antenna, who cares? and more importantly how will you know whree it is occuring since things like the junction of two connectorc can produce enough IM to mask other, smaller sources. If you tried an experiment looking for IM / Mixing you might try to use a receiver becaue a receiver could be a very sensitive detector...but you'd have to have a pretty good receiver. Something like a kW LO and mixer to have a really good intercept. I'm not sure of the point here... Do antennas cause IM? Sounds like a deadend arena to me. 73, Steve, K9DCI |
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wrote Electrically, antennas are very linear (I believe) even when their current distribution along the element length is not. I stopped posting when I realized we were discussing two different things. In a linear system, there only needs to be a straight line function between the input and output. The actual signals on the input wouldn't be very useful if only straight line functions were allowed on that input. The current distribution along an antenna element length only ever approximates a straight line. The only requirement for that current to obey the rules for a linear system is that it be a linear function of the source current and it is at every point. -- 73, Cecil http://www.qsl.net/w5dxp |
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Cecil Moore wrote: In a linear system, there only needs to be a straight line function between the input and output. The actual signals on the input wouldn't be very useful if only straight line functions were allowed on that input. The current distribution along an antenna element length only ever approximates a straight line. The only requirement for that current to obey the rules for a linear system is that it be a linear function of the source current and it is at every point. I mostly agree with your definition of linearity. Roy's point that an offset in the straight line violates superposition is an example of a straight line violating superposition. Also consider the throwing of two dice. If the dice act independently they can be considered a linear system with two outputs (the numbers that show on the dice) which obeys the law of superposition. If, however, the dice collide when thrown, they now influence each other, the system becomes nonlinear, and the law of superposition is violated. It's pretty hard for me to attach a straight line function to dice. Mixers are especially interesting beasts when viewed in the light of superposition. They are obviously highly nonlinear, yet we regularly speak of mixer linearity. Here is the trick: when the local oscillator is included in the input signal set, the mixer is highly nonlinear as the LO influences every signal that comes in drastically, in the sense of generating new frequencies. But it is convenient to think of the local oscillator as just an internal parameter of the mixer, and to not include it in the input signal set. Under this assumption all the RF input signals substantially obey the law of superposition (that is, they do not influence each other) and discussing mixer linearity has enough meaning that we can characterize it through standard tests. Note also that under this assumption the output frequencies do not match the input frequencies in general, yet the law of superposition holds over a wide dynamic range. Cecil's point that doubling antenna current will double the current at each point in the antenna (I'm rephrasing) has a dual in mixers: doubling any signal will double all the mixer's ouputs due to that signal alone, and not any other outputs. 73, Glenn Dixon AC7ZN |
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wrote:
Cecil's point that doubling antenna current will double the current at each point in the antenna (I'm rephrasing) has a dual in mixers: doubling any signal will double all the mixer's ouputs due to that signal alone, and not any other outputs. Yep, as long as system linearity is maintained. -- 73, Cecil http://www.qsl.net/w5dxp |
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wrote in message oups.com... Thanks, Steve, Good post. Thanks. You do need to read a little mo by mentioning Doppler shifts, I was illustrating that a linear system can generate new frequencies without violating the law of superposition. Hi Glenn, I got that. Perhaps my use of "relativitistic effects" was inappropriate. I still maintain that Doppler shift is not an example of (at least what I would call) "creating new frequencies" because an electronis system did not cause said shift. A system is linear or non-linear, but a system can't "make Doppler happen". It is a frequency change, yes, but an "electronic system" can't cause it. It occurs for reasons other than "system characteristics". Yes, a space craft can be considered part of a system in teh general sense, but not as I believe an "electronic system" should be thought of when discussing such things. Perhaps symmantics in your view, but not mine. I think it is a fundamental, but, perhaps can't explain adequately why. What a thread. usually when a thread gets this long it digresses far outside the original intent. ...Earlier posts were confusing as the term 'linearity' ...electrical superposition sense ... and ... current distribution along the antenna element. So you *were* questioning the linearity of antennas as we understand the term? 73, Steve 73, Glenn AC7ZN Steve N. wrote: wrote in message oups.com... ... By the way, generating new frequencies is not necessarily a violation of superposition (though it usually is). Consider a system undergoing a constant Doppler shift. 73, Glenn AC7ZN Glenn, boy! I can't read all these posts, so I was trying to see where you were going and asking by just skimming, primarily your posts. The above caught my eye. Doppler is not "generating new frequencies" as a non linearity does. A non-linear system will produce harmonics with one exitation frequency and produce the common mixing / IM with multiple exitation (superimposed) frequencies. I think trying to mix relativistic effects in with the stationary world is an unnecessary complication of a linearity discussion. If I have time I'll try to follow the thread to see what you're really after...but. If it takes mega watts to see some non linearity in an antenna, who cares? and more importantly how will you know whree it is occuring since things like the junction of two connectorc can produce enough IM to mask other, smaller sources. If you tried an experiment looking for IM / Mixing you might try to use a receiver becaue a receiver could be a very sensitive detector...but you'd have to have a pretty good receiver. Something like a kW LO and mixer to have a really good intercept. I'm not sure of the point here... Do antennas cause IM? Sounds like a deadend arena to me. 73, Steve, K9DCI |
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I haven't been following this thread lately, but happened on this
particular posting... Seems to me you want to say "Linear TIME INVARIANT system" to get to not generating new frequencies. Certainly I can build a system that is linear but not time-invariant and generate new frequencies with that system. A simple one is a signal going into a potentiometer, coming out the wiper, in which the wiper is rotated continuously. It's linear, but not time invariant, and obviously any input will be amplitued modulated at the rate of the time variation. Is a double-balanced mixer with LO a linear system (for input signals in the intended amplitued range)? Increasing the input amplitude by 1dB causes the output amplitude to increase by 1dB, though the output is not at the same frequency as the input. If the response of the DBM/LO system to input x1 is y1, and to x2 is y2, then is (y1+y2) the response to input (x1+x2)? Is a DBM/LO system time-invariant: if I apply stimulus x1 at time t1 do I get the same response as if I apply it at time t2 (where the response is also shifted by t2-t1)? Perhaps this will be useful food for thought... Cheers, Tom Steve N. wrote: wrote in message oups.com... Thanks, Steve, Good post. Thanks. You do need to read a little mo by mentioning Doppler shifts, I was illustrating that a linear system can generate new frequencies without violating the law of superposition. Hi Glenn, I got that. Perhaps my use of "relativitistic effects" was inappropriate. I still maintain that Doppler shift is not an example of (at least what I would call) "creating new frequencies" because an electronis system did not cause said shift. A system is linear or non-linear, but a system can't "make Doppler happen". It is a frequency change, yes, but an "electronic system" can't cause it. It occurs for reasons other than "system characteristics". Yes, a space craft can be considered part of a system in teh general sense, but not as I believe an "electronic system" should be thought of when discussing such things. Perhaps symmantics in your view, but not mine. I think it is a fundamental, but, perhaps can't explain adequately why. What a thread. usually when a thread gets this long it digresses far outside the original intent. ...Earlier posts were confusing as the term 'linearity' ...electrical superposition sense ... and ... current distribution along the antenna element. So you *were* questioning the linearity of antennas as we understand the term? 73, Steve 73, Glenn AC7ZN Steve N. wrote: wrote in message oups.com... ... By the way, generating new frequencies is not necessarily a violation of superposition (though it usually is). Consider a system undergoing a constant Doppler shift. 73, Glenn AC7ZN Glenn, boy! I can't read all these posts, so I was trying to see where you were going and asking by just skimming, primarily your posts. The above caught my eye. Doppler is not "generating new frequencies" as a non linearity does. A non-linear system will produce harmonics with one exitation frequency and produce the common mixing / IM with multiple exitation (superimposed) frequencies. I think trying to mix relativistic effects in with the stationary world is an unnecessary complication of a linearity discussion. If I have time I'll try to follow the thread to see what you're really after...but. If it takes mega watts to see some non linearity in an antenna, who cares? and more importantly how will you know whree it is occuring since things like the junction of two connectorc can produce enough IM to mask other, smaller sources. If you tried an experiment looking for IM / Mixing you might try to use a receiver becaue a receiver could be a very sensitive detector...but you'd have to have a pretty good receiver. Something like a kW LO and mixer to have a really good intercept. I'm not sure of the point here... Do antennas cause IM? Sounds like a deadend arena to me. 73, Steve, K9DCI |
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I recall a prof or two arm-waving over that one. However, I think if
you formulate your definition of linearity properly, the transfer function y=mx+b will still satisfy linearity. Specifically, if the _response_ is the _change_ that occurs in the output going from x=0 to x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which is exactly the sum of the responses for x1 and x2. Similarly, for a mixer/LO system with RF input and IF output, if the mixer is unbalanced and lets LO get through, it is still a linear system if the change in output when go from zero input to input x1(t) plus the change in output when you go from zero input to input x2(t) is equal to the change in output when you go from zero input to input (x1(t)+x2(t)). But note that a mixer/LO system is NOT time invariant, because the output for x1(t+delta) is in general NOT the same as the output shifted in time by delta for input x1(t). You can most certainly find text books that define linearity differently than I did above. I find the definition above to be a more useful one, however, and it seems to be the one generally accepted in practice, even if it's not stated accurately in words. Cheers, Tom Roy Lewallen wrote: Richard Harrison wrote: Richard Clark, KB7QHC wrote: "Who. in your estimation, does qualify to discuss it?" If it`s about antennas, I nominate Kraus. If it`s about mathematics, many marhematicians qualify. In algebra, y = mx + b, (the point slope formula), is called linear because it is the graph of a straight line. . . . But of course you realize that the function y = mx + b doesn't meet the requirements of a linear function when applied to network theory. Roy Lewallen, W7EL |
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It's not clear to me whether you're proposing an alternative definition
for linearity or for superposition. I've never seen superposition defined as other than that the sum of responses to individual excitations be equal to the response to the sum of the excitations -- that's the definition in Pearson & Maler's _Introductory Circuit Analysis_, Van Valkenburg's _Network Analysis_, and the rather old edition of the _IEEE Standard Dictonary of Electrical and Electronic Terms_ I have. Do you have a reference that gives the definition you propose for superposition? If on the other hand the alternative definition is only for linearity, we'd then be faced with the possibility of having a linear (and time-invariant) circuit which doesn't satisfy superposition. That's not a pleasant circumstance to ponder. Roy Lewallen, W7EL K7ITM wrote: I recall a prof or two arm-waving over that one. However, I think if you formulate your definition of linearity properly, the transfer function y=mx+b will still satisfy linearity. Specifically, if the _response_ is the _change_ that occurs in the output going from x=0 to x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which is exactly the sum of the responses for x1 and x2. Similarly, for a mixer/LO system with RF input and IF output, if the mixer is unbalanced and lets LO get through, it is still a linear system if the change in output when go from zero input to input x1(t) plus the change in output when you go from zero input to input x2(t) is equal to the change in output when you go from zero input to input (x1(t)+x2(t)). But note that a mixer/LO system is NOT time invariant, because the output for x1(t+delta) is in general NOT the same as the output shifted in time by delta for input x1(t). You can most certainly find text books that define linearity differently than I did above. I find the definition above to be a more useful one, however, and it seems to be the one generally accepted in practice, even if it's not stated accurately in words. Cheers, Tom |
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Steve N. wrote: So you *were* questioning the linearity of antennas as we understand the term? No. I was claiming antennas were very linear in the electrical superposition sense. 73, Glenn |
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Roy Lewallen wrote: It's not clear to me whether you're proposing an alternative definition for linearity or for superposition. Lest anyone think that the addition of an offset term (the b in y=mx+b) violating superposition is merely of academic interest, analyze a direct conversion receiver. At my shack I have a Softrock 40, which tunes about thirty KHz of the 40 meter band. At the center of the spectrum (which corresponds to the mixer converting to very low frequencies and DC) is a 1/f bump that the designers have attributed to mixer 1/f noise. But it is not noise at all. When tuned in, it is full of strong signals and strange sounds. These are distortion products, and are caused mainly by the b term in the detector used on the receiver. They wipe out about 3% of the receivable band. Sometimes the b term is benign, but not in this case. 73, Glenn Dixon AC7ZN |
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I would not propose to change the definition for either linearity or
superposition, at least as I saw them a moment ago in one text, the "Linear Circuit Analysis" chapter of "The Electrical Engineering Handbook," Richard C. Dorf, editor. Rather I propose we think more carefully about just what "response" to a stimulus means. If you say that in the system y=mx+b that b is the response to zero input (x=0) then you will conclude that the system is nonlinear. On the other hand, if the "response" to a "stimulus" only has meaning as the _change_ (or difference) in output for a given _change_ (or difference) in input, then it is a linear system. Roy Lewallen wrote: It's not clear to me whether you're proposing an alternative definition for linearity or for superposition. I've never seen superposition defined as other than that the sum of responses to individual excitations be equal to the response to the sum of the excitations -- that's the definition in Pearson & Maler's _Introductory Circuit Analysis_, Van Valkenburg's _Network Analysis_, and the rather old edition of the _IEEE Standard Dictonary of Electrical and Electronic Terms_ I have. Do you have a reference that gives the definition you propose for superposition? If on the other hand the alternative definition is only for linearity, we'd then be faced with the possibility of having a linear (and time-invariant) circuit which doesn't satisfy superposition. That's not a pleasant circumstance to ponder. Roy Lewallen, W7EL K7ITM wrote: I recall a prof or two arm-waving over that one. However, I think if you formulate your definition of linearity properly, the transfer function y=mx+b will still satisfy linearity. Specifically, if the _response_ is the _change_ that occurs in the output going from x=0 to x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which is exactly the sum of the responses for x1 and x2. Similarly, for a mixer/LO system with RF input and IF output, if the mixer is unbalanced and lets LO get through, it is still a linear system if the change in output when go from zero input to input x1(t) plus the change in output when you go from zero input to input x2(t) is equal to the change in output when you go from zero input to input (x1(t)+x2(t)). But note that a mixer/LO system is NOT time invariant, because the output for x1(t+delta) is in general NOT the same as the output shifted in time by delta for input x1(t). You can most certainly find text books that define linearity differently than I did above. I find the definition above to be a more useful one, however, and it seems to be the one generally accepted in practice, even if it's not stated accurately in words. Cheers, Tom |
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K7ITM wrote:
"If you say that in the system (y = mx + b) that b is the response to zero input (x=0) then you will conclude the system is nonlinear." Why? the factor (b) is a constant, a value to be added to (mx) to total a value for (y). (mx) is a straight line. Every value of (b) produces a straight lline parallel with lhe line y=mx when b=0. Factor (b) is merely the offset value of the sloped line in the x direction. y=mx+b is listed in math books as a defining example of a linear equation. (When plotted, a linear equation produces a straight line.) y=mx+b has a special name: "The point slope formula". Perfectly descriptive, too. To clarify everything, graph a few values for yourself. Best regards, Richard Harrison, KB5WZI |
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Richard Harrison wrote:
(mx) is a straight line. Every value of (b) produces a straight lline parallel with lhe line y=mx when b=0. Factor (b) is merely the offset value of the sloped line in the x direction. Why must superposition preserve 'b' (as someone has asserted?) -- 73, Cecil http://www.qsl.net/w5dxp |
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Richard Harrison wrote: K7ITM wrote: "If you say that in the system (y = mx + b) that b is the response to zero input (x=0) then you will conclude the system is nonlinear." Why? the factor (b) is a constant, a value to be added to (mx) to total a value for (y). Because, from 3.3, 'Network Theorems,' in "The Electrical Engineering Handbook," Ed. Richard C. Dorf, Boca Raton: CRC Press LLC, 2000 (you can find it online), "The superposition condition: If the input to the system, e1, causes a response, r1, and if an input to the system, e2, causes a response, r2, then a response, r1 + r2, will occur when the input is e1 + e2." If you take the response to x0=0 to be b, then the response to, say, x1=1 must be m+b, and to x2=2 must be 2*m+b. Then for superposition, and therefore for linearity, the response to (x1+x2)=3 must be m+b+2*m+b = 3*m+2*b, which it is not: it is 3*m+b. I do not suggest a change in the definition of superposition or of homogeneity (which seems to be simply a subset of superpostion anyway) or of linearity. I only suggest that "response" be interpreted as the change in output which occurs when you go from input a to input b. I've generally not seen "response" to be very well defined in those texts which define linearity, though it's quite possible I've missed it. In the PDF file for all of Chapter 3 of the book quoted above, it is certainly not. I guess I'd prefer, basically, to say that the following relationship defines a linear system (MIMO even) -- but it is NOT sufficient to describe all linear systems: dx/dt = Ax + Bu y = Cx + Du where u is a vector of independent input variables, y is a vector of dependent output variables, x represents state variables for the system, and A, B, C and C are matrices of coefficients which are constant in a time-invariant system, but which may be variable with time in a system which is time variant. Cheers, Tom (mx) is a straight line. Every value of (b) produces a straight lline parallel with lhe line y=mx when b=0. Factor (b) is merely the offset value of the sloped line in the x direction. y=mx+b is listed in math books as a defining example of a linear equation. (When plotted, a linear equation produces a straight line.) y=mx+b has a special name: "The point slope formula". Perfectly descriptive, too. To clarify everything, graph a few values for yourself. Best regards, Richard Harrison, KB5WZI |
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