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In [[philosophy]] the term '''free will''' refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so.  It is an age-old concern to separate what we can do something about, choose to do, from what we cannot.  The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of ''determinism'', sometimes called ''physical reductionism'',<ref name=Ney/>  and the view that determinism precludes free will is called ''incompatibilism''.<ref name=Vihvelin/> 


'''Pole splitting''' is a phenomenon exploited in some forms of [[frequency compensation]] used in an [[electronic amplifier]]. When a [[capacitor]] is introduced between the input and output sides of the amplifier with the intention of moving the [[Pole (complex analysis)|pole]] lowest in frequency (usually an input pole) to lower frequencies, pole splitting causes the pole next in frequency (usually an output pole) to move to a higher frequency. This pole movement increases the stability of the amplifier and improves its [[step response]] at the cost of decreased speed.<ref name=note1 group=Note/><ref name=Toumazou/><ref name=Thompson/><ref name=Sansen/>
There are several ways to avoid the incompatibilst position, resulting in various ''compatibilist'' positions.<ref name=Timpe/>  One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.  


== Example of pole splitting ==
There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?
{{Image|Pole splitting example.PNG|right|250px| Operational amplifier with compensation capacitor ''C<sub>C</sub>'' between input and output; notice the amplifier has both input impedance ''R<sub>i</sub>'' and output impedance ''R<sub>o</sub>''.}}
{{Image|Pole splitting with Miller transformation.PNG|right|250px|Operational amplifier with compensation capacitor transformed using [[Miller effect|Miller's theorem]] to replace the compensation capacitor with a Miller capacitor at the input and a frequency-dependent current source at the output.}}


This example shows that introduction of the capacitor referred to as C<sub>C</sub> in the amplifier of Figure 1 has two results: first it causes the lowest frequency pole of the amplifier to move still lower in frequency and second, it causes the higher pole to move higher in frequency.<ref name=note2 group=Note/> The amplifier of Figure 1 has a low frequency pole due to the added input resistance ''R<sub>i</sub>'' and capacitance ''C<sub>i</sub>'', with the time constant ''C<sub>i</sub>'' ( ''R<sub>A</sub> // R<sub>i</sub>'' ). This pole is moved down in frequency by the [[Miller effect]]. The amplifier is given a high frequency output pole by addition of the load resistance ''R<sub>L</sub>'' and capacitance ''C<sub>L</sub>'', with the time constant ''C<sub>L</sub>'' ('' R<sub>o</sub> // R<sub>L</sub>'' ). The upward movement of the high-frequency pole occurs because the Miller-amplified compensation capacitor ''C<sub>C</sub>'' alters the frequency dependence of the output voltage divider.
==Science does not apply==


The first objective, to show the lowest pole moves down in frequency, is established using the same approach as the [[Miller effect|Miller's theorem]] article. Following the procedure described in the article on [[Miller effect]], the circuit of Figure 1 is transformed to that of Figure 2, which is electrically equivalent to Figure 1. Application of [[Kirchhoff's current law]] to the input side of Figure 2 determines the input voltage <math>\ v_i</math> to the ideal op amp as a function of the applied signal voltage <math>\ v_a</math>, namely,
One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, [[David Hume]] suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. [[Immanuel Kant]] suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains.  Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.


::<math>
A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise.  It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience.  For example, we choose to explain phenomena like the [[Higgs boson]] found by elaborate means like a [[hadron collider]], but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention.
As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.


  \frac {v_i} {v_a}  = \frac {R_i} {R_i+R_A} \frac {1} {1+j \omega (C_M+C_i) (R_A//R_i)} \ ,</math>
Although not explicitly addressing the issue of free will, it may be noted that [[Ludwig Wittgenstein]] argued that the specialized theories of science, as discussed by [[Rudolf Carnap]] for example, inevitably cover only a limited range of experience. [[Stephen Hawking|Hawking/Mlodinow]] also noted this fact in in their [[model-dependent realism]],<ref name=Hawking/> the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.
{{quote|“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”<ref name=Davies/>}}
::: —— E.B. Davies <span style="font-size:88%">''Epistemological pluralism'', p. 4</span>


which exhibits a [[roll-off]] with frequency beginning at ''f<sub>1</sub>'' where
Still another approach to this matter is analysis of the mind-brain connection (more generally, the [[mind-body problem]]). As suggested by Northoff,<ref name=Northoff/>  there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks.  But observing our own mental activity is not possible in this way – it is a matter of subjective experiences.  The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:
{{quote|“...for each individual there is ''one'' 'mental life' but ''two'' ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the ''observational arrangements'' (the means by which a subject and an external observer access the subject's mental processes).”<ref name=Velmans/> |Max Velmans: |How could conscious experiences affect brains?,  p. 11''
}}


::<math>
A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by [[Niels Bohr]] and by [[Erwin Schrödinger]] in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)
{{quote|“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”<ref name=Pockett/> |Susan Pockett |The neuroscience of movement}}


\begin{align}
f_{1} & =  \frac {1} {2 \pi (C_M+C_i)(R_A//R_i) } \\
      & =  \frac {1} {2 \pi \tau_1} \ , \\
\end{align}


</math>
In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.


which introduces notation <math>\tau_1</math> for the time constant of the lowest pole. This frequency is lower than the initial low frequency of the amplifier, ''f<sub>0</sub>'' say, which for ''C<sub>C</sub>'' = 0 F is:
==Science can be accommodated==
A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do.  Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.


::<math>f_0 =\frac {1} {2 \pi C_i (R_A//R_i)} \ .</math>
==’Will’ ''versus'' ‘action’==
There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,<ref name=Norretranders/> which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, [[addiction|drug addiction]] has been related to alteration of the mechanisms in the brain for [[dopamine]] production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.
{{quote|“Philosophers who distinguish ''freedom of action'' and ''freedom of will'' do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.]<ref name=OConnor/>| Timothy O'Connor |Free Will}}
In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?


Turning to the second objective, showing the higher pole moves still higher in frequency, it is necessary to look at the output side of the circuit, which contributes a second factor to the overall gain, and additional frequency dependence. The voltage <math>\ v_o</math> is determined by the gain of the ideal op amp inside the amplifier as
==Theology==
The ancient Greeks held the view that the gods ''could'' intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:
{{quote| “The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”|spoken by Cassius| Julius Caesar (I, ii, 140-141)}}
The [[Stoicism|Stoics]] wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a ''responsibility''), a sequence demanded by our natures as rational beings.<ref name=Bobzien/>  


::<math>\  v_o = A_v v_i \ . </math>
In [http://www.iep.utm.edu/chrysipp/ Chrysippus of Soli's] view (an apologist for Stoicism), ''fate'' precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.<ref name=Bobzien2/> The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.<ref name=Bobzien3/>


Using this relation and applying Kirchhoff's current law to the output side of the circuit determines the load voltage <math>v_{\ell}</math> as a function of the voltage <math>\ v_{i}</math> at the input to the ideal op amp  as:
==References==
{{reflist|refs=
<ref name=Bobzien>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See §6.3.3 ''The cylinder and cone analogy'',  pp. 258 ''ff''.
</ref>


::<math> \frac {v_{\ell}} {v_i} = A_v \frac {R_L} {R_L+R_o}\,\!</math><math>\sdot \frac {1+j \omega C_C R_o/A_v } {1+j \omega (C_L + C_C ) (R_o//R_L) } \ . </math>
<ref name=Bobzien2>
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular pp. 386 ''ff''.  
</ref>


This expression is combined with the gain factor found earlier for the input side of the circuit to obtain the overall gain as
<ref name=Bobzien3>
 
{{cite book |author=Susanne Bobzien |title=Determinism and Freedom in Stoic Philosophy |url=http://books.google.com/books?id=7kmTeOjHIqkC&printsec=frontcover |year=1998 |publisher=Oxford University Press |isbn=0198237944}} See in particular p. 255.  
::<math>
</ref>
 
\frac {v_{\ell}} {v_a}  = \frac {v_{\ell}}{v_i} \frac {v_i} {v_a}
</math>
 
:::<math>= A_v  \frac {R_i} {R_i+R_A}\sdot \frac {R_L} {R_L+R_o}\,\! </math><math> \sdot \frac {1} {1+j \omega (C_M+C_i) (R_A//R_i)} \,\! </math><math> \sdot \frac {1+j \omega C_C R_o/A_v } {1+j \omega (C_L + C_C ) (R_o//R_L) } \ . </math>
 
This gain formula appears to show a simple two-pole response with two time constants. (It also exhibits a zero in the numerator but, assuming the amplifier gain ''A<sub>v</sub>'' is large, this zero is important only at frequencies too high to matter in this discussion , so the numerator can be approximated as unity.) However, although the amplifier does have a two-pole behavior, the two time-constants are more complicated than the above expression suggests because the Miller capacitance contains a buried frequency dependence that has no importance at low frequencies, but has considerable effect at high frequencies. That is, assuming the output ''R-C'' product,  ''C<sub>L</sub>'' ( ''R<sub>o</sub> // R<sub>L</sub>'' ), corresponds to a frequency well above the low frequency pole, the accurate form of the Miller capacitance must be used, rather than the [[Miller's theorem|Miller approximation]]. According to the article on [[Miller effect]], the Miller capacitance is given by
 
::<math>
\begin{align}
C_M & = C_C \left( 1 - \frac {v_{\ell}} {v_i} \right) \\
    & = C_C \left( 1 - A_v \frac {R_L} {R_L+R_o} \frac {1+j \omega C_C R_o/A_v } {1+j \omega (C_L + C_C ) (R_o//R_L) } \right ) \ . \\
\end{align}
</math>
 
(For a positive Miller capacitance, ''A<sub>v</sub>'' is negative.) Upon substitution of this result into the gain expression and collecting terms, the gain is rewritten as:
 
::<math> \frac {v_{\ell}} {v_a} = A_v  \frac {R_i} {R_i+R_A} \frac {R_L} {R_L+R_o}  \frac {1+j \omega C_C R_o/A_v } {D_{ \omega }} \ , </math>
 
with ''D<sub>ω</sub>'' given by a quadratic in ω, namely:
 
::<math>D_{ \omega }\,\!</math> <math> = [1+j \omega (C_L+C_C) (R_o//R_L)] \,\!</math> <math> \sdot \ [ 1+j \omega C_i (R_A//R_i)] \,\!</math> <math> \ +j \omega C_C (R_A//R_i)\,\! </math> <math>\sdot \left(  1-A_v \frac {R_L} {R_L+R_O} \right) \,\!</math> <math>\ +(j \omega) ^2 C_C C_L (R_A//R_i)  (R_O//R_L) \ . </math>
 
Every quadratic has two factors, and this expression looks simpler if it is rewritten as
 
::<math>
\ D_{ \omega } =(1+j \omega { \tau}_1 )(1+j \omega { \tau}_2 ) </math>
 
:::<math> = 1 + j \omega ( {\tau}_1+{\tau}_2) ) +(j \omega )^2 \tau_1 \tau_2 \ , \ </math>
 
where <math>\tau_1</math> and <math>\tau_2</math> are combinations of the capacitances and resistances in the formula for ''D<sub>ω</sub>''.<ref name=note3 group=Note/> They correspond to the time constants of the two poles of the amplifier. One or the other time constant is the longest; suppose <math>\tau_1</math> is the longest time constant, corresponding to the lowest pole, and suppose <math>\tau_1</math> >> <math>\tau_2</math>. (Good step response requires <math>\tau_1</math> >> <math>\tau_2</math>. See [[Pole splitting#Selection of CC|Selection of C<sub>C</sub>]] below.)
 
At low frequencies near the lowest pole of this amplifier, ordinarily the linear term in ω is more important than the quadratic term, so the low frequency behavior  of ''D<sub>ω</sub>'' is:
 
::<math>
\begin{align}
\ D_{ \omega } & = 1+ j \omega [(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)] \\
              & = 1+j \omega ( \tau_1 + \tau_2) \approx 1 + j \omega \tau_1 \ , \ \\
\end{align}
</math>
 
where now ''C<sub>M</sub>'' is redefined using the [[Miller effect|Miller approximation]] as
{{anchor|Miller}}
::<math> C_M= C_C \left( 1 - A_v \frac {R_L}{R_L+R_o} \right) \ ,</math>
 
which is simply the previous Miller capacitance evaluated at low frequencies. On this basis <math>\tau_1</math> is determined, provided <math>\tau_1</math> >> <math>\tau_2</math>. Because ''C<sub>M</sub>'' is large, the time constant <math>{\tau}_1</math> is much larger than its original value of ''C<sub>i</sub>'' ( ''R<sub>A</sub> // R<sub>i</sub>'' ).<ref name=note4 group=Note/>
 
At high frequencies the quadratic term becomes important. Assuming the above result for <math>\tau_1</math> is valid, the second time constant, the position of the high frequency pole, is found from the quadratic term in ''D<sub>ω</sub>'' as
 
::<math> \tau_2 = \frac {\tau_1 \tau_2} {\tau_1} \approx \frac {\tau_1 \tau_2} {\tau_1 + \tau_2}\ . </math>
 
Substituting in this expression the quadratic coefficient corresponding to the product <math>\tau_1 \tau_2 </math> along with the estimate for <math>\tau_1</math>, an estimate for the position of the second pole is found:
 
::<math>
\begin{align}
\tau_2 & = \frac {(C_C C_L +C_L C_i+C_i C_C)(R_A//R_i)  (R_O//R_L) } {(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)}  \\
        & \approx  \frac {C_C C_L +C_L C_i+C_i C_C} {C_M} (R_O//R_L)\ ,  \\
\end{align}
</math>
 
and because ''C<sub>M</sub>'' is large, it seems <math>\tau_2</math> is reduced in size from its original value ''C<sub>L</sub>'' ( ''R<sub>o</sub>'' // ''R<sub>L</sub>'' ); that is, the higher pole has moved still higher in frequency because of ''C<sub>C</sub>''.<ref name=note5 group=Note/>
 
In short, introduction of capacitor ''C<sub>C</sub>'' moved the low pole lower and the high pole higher, so the term '''pole splitting''' seems a good description.
 
=== Selection of C<sub>C</sub> ===
{{Image|Two-pole Bode magnitude plot.PNG|right|200px|Idealized [[Bode plot]] for a two pole amplifier design. Gain drops from first pole at ''f<sub>1</sub>'' at 20 dB / decade down to second pole at ''f<sub>2</sub>'' where the slope increases to 40 dB / decade.}}
What value is a good choice for ''C<sub>C</sub>''?  For general purpose use, traditional design (often called ''dominant-pole'' or ''single-pole compensation'') requires the amplifier gain to drop at 20 dB/decade from the corner frequency down to 0 dB gain, or even lower.<ref name=Sedra/><ref name=Huijsing/> With this design the amplifier is stable and has near-optimal [[Step_response#Control_of_overshoot|step response]] even as a unity gain voltage buffer. A more aggressive technique is two-pole compensation.<ref name=Feucht/><ref name=Self/>
 
The way to position ''f''<sub>2</sub> to obtain the design is shown in Figure 3. At the lowest pole ''f''<sub>1</sub>, the Bode gain plot breaks slope to fall at 20 dB/decade. The aim is to maintain the 20 dB/decade slope all the way down to zero dB, and taking the ratio of the desired drop in gain (in dB) of 20 log<sub>10</sub> ''A<sub>v</sub>'' to the required change in frequency (on a log frequency scale<ref name=note6 group=Note/>) of ( log<sub>10</sub> ''f''<sub>2</sub> &nbsp;&minus;&nbsp;log<sub>10</sub> ''f''<sub>1</sub> ) = log<sub>10</sub> ( ''f''<sub>2</sub> / ''f''<sub>1</sub> ) the slope of the segment between ''f''<sub>1</sub> and ''f''<sub>2</sub> is:
 
::Slope per decade of frequency  <math>=20  \frac {\mathrm{log_{10}} ( A_v )}  {\mathrm{log_{10}}  (f_2 / f_1 ) } \ ,</math>
 
which is 20 dB/decade provided ''f<sub>2</sub> = A<sub>v</sub> f<sub>1</sub>'' . If ''f<sub>2</sub>'' is not this large, the second break in the Bode plot that occurs at the second pole interrupts the plot before the gain drops to 0 dB with consequent lower stability and degraded step response.
 
Figure 3 shows that to obtain the correct gain dependence on frequency, the second pole is at least a factor ''A<sub>v</sub>'' higher in frequency than the first pole. The gain is reduced a bit by the [[voltage division#Loading effect|voltage dividers]] at the input and output of the amplifier, so with corrections to ''A<sub>v</sub>'' for the voltage dividers at input and output the '''pole-ratio condition''' for good step response becomes:
 
::<math> \frac {\tau_1} {\tau_2} \approx A_v  \frac {R_i} {R_i+R_A}\sdot \frac {R_L} {R_L+R_o} \ , </math>


{{Image|Compensation capacitance.PNG|right|300px|Miller capacitance at low frequencies ''C<sub>M</sub>'' (top) and compensation capacitor ''C<sub>C</sub>'' (bottom) as a function of gain using [[Microsoft Excel|Excel]]. Capacitance units are pF.}}


Using the approximations for the time constants developed above,
<ref name=Davies>
 
{{cite web |title=Epistemological pluralism |author=E Brian Davies |url=http://philsci-archive.pitt.edu/3083/1/EP3single.doc |work=PhilSci Archive |year=2006 }}
::<math> \frac {\tau_1} {\tau_2} \approx \frac {(\tau_1 +\tau_2 ) ^2} {\tau_1 \tau_2} \approx A_v  \frac {R_i} {R_i+R_A}\sdot \frac {R_L} {R_L+R_o} \ ,</math>
 
or
 
::<math> \frac  {[(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)]^2} {(C_C C_L +C_L C_i+C_i C_C)(R_A//R_i)  (R_O//R_L) }  \,\! </math>  <math>{\color{White}\sdot}  = A_v  \frac {R_i} {R_i+R_A}\sdot \frac {R_L} {R_L+R_o} \ ,</math>
 
which provides a quadratic equation to determine an appropriate value for ''C<sub>C</sub>''. Figure 4 shows an example using this equation. At low values of gain this example amplifier satisfies the pole-ratio condition without compensation (that is, in Figure 4 the compensation capacitor ''C<sub>C</sub>'' is small at low gain), but as gain increases, a compensation capacitance rapidly becomes necessary (that is, in Figure 4 the compensation capacitor ''C<sub>C</sub>'' increases rapidly with gain) because the necessary pole ratio increases with gain. For still larger gain, the necessary ''C<sub>C</sub>'' drops with increasing gain because the Miller amplification of ''C<sub>C</sub>'', which increases with gain (see the [[#Miller|Miller equation]] ),  allows a smaller value for ''C<sub>C</sub>''.  
 
To provide more safety margin for design uncertainties, often ''A<sub>v</sub>'' is increased to two or three times ''A<sub>v</sub>'' on the right side of this equation.<ref name=note7 group=Note/> See Sansen<ref name=Sansen/> or Huijsing<ref name=Huijsing/> and article on [[step response]].
 
===Slew rate===
The above is a small-signal analysis. However, when large signals are used, the need to charge and discharge the compensation capacitor adversely affects the amplifier [[slew rate]]; in particular, the response to an input ramp signal is limited by the need to charge ''C<sub>C</sub>''.
 
==Notes==
{{reflist|group=Note|refs=
 
<ref name=note1 group=Note>
That is, the [[Step response|rise time]] is selected to be the fastest possible consistent with low [[Step response|overshoot]] and [[Step response|ringing]].
</ref>
</ref>


<ref name=note2 group=Note>
Although this example appears very specific, the associated mathematical analysis is very much used in circuit design.</ref>


<ref name=note3 group=Note>
<ref name=Hawking>
The sum of the time constants is the coefficient of the term linear in jω and the product of the time constants is the coefficient of the quadratic term in (jω)<sup>2</sup>.
{{cite book |author=Hawking SW, Mlodinow L. |title=The Grand Design |isbn=0553805371 |url= http://www.amazon.com/Grand-Design-Stephen-Hawking/dp/0553805371#reader_0553805371 |pages=pp. 42-43 |chapter=Chapter 3: What is reality?|year=2010|publisher=Bantam Books}}
</ref>
 
<ref name=note4 group=Note>
The expression for ''&tau;<sub>1</sub>'' differs a little from (  ''C<sub>M</sub>+C<sub>i</sub>'' ) ( ''R<sub>A</sub>'' // ''R<sub>i</sub>'' ) as found initially for ''f<sub>1</sub>'', but the difference is minor assuming the load capacitance is not so large that it controls the low frequency response instead of the Miller capacitance.
</ref>
</ref>


<ref name=note5 group=Note>
<ref name=Norretranders>
As an aside, the higher the high-frequency pole is made in frequency, the more likely it becomes for a real amplifier that other poles (not considered in this analysis) play a part.
{{cite book |url=http://www.google.com/search?tbo=p&tbm=bks&q=consciousness%2Bplays%2Ba%2Bsmaller%2Brole%2Bin%2Bhuman%2Blife+intitle:User+intitle:illusion&num=10 |title=The user illusion: Cutting consciousness down to size |quote=Consciousness plays a far smaller role in human life than Western culture has tended to believe |author=Tor Nørretranders |isbn=0140230122 |chapter=Preface |pages=p. ''ix'' |publisher=Penguin Books |year=1998 |edition=Jonathan Sydenham translation of ''Maerk verden'' 1991 ed }}
</ref>
</ref>


<ref name=note6 group=Note>
<ref name=Ney>
That is, the frequency is plotted in powers of ten, as 1, 10, 10<sup>2</sup> ''etc''.
{{cite web |author=Alyssa Ney |title=Reductionism |work=Internet Encyclopedia of Philosophy |date= November 10, 2008 |url=http://www.iep.utm.edu/red-ism/}}
</ref>
</ref>


<ref name=note7 group=Note>
<ref name=Northoff>
A factor of two results in the ''maximally flat'' or [[Butterworth filter|Butterworth]] design for a two-pole amplifier. However, real amplifiers have more than two poles, and a factor greater than two often is necessary.
A rather extended discussion is provided in {{cite book |title=Philosophy of the Brain: The Brain Problem |author=Georg Northoff |url=http://books.google.com/books?id=r0Bf3lLys6AC&printsec=frontcover |publisher=John Benjamins Publishing |isbn=1588114171 |year=2004 |edition=Volume 52 of Advances in Consciousness Research}}
</ref>
</ref>


}}
<ref name=OConnor>
 
{{cite web |title=&thinsp;Free Will |date=Oct 29, 2010 |author=O'Connor, Timothy |url=http://plato.stanford.edu/archives/sum2011/entries/freewill |work=The Stanford Encyclopedia of Philosophy (Summer 2011 Edition) |editor=Edward N. Zalta, ed.}}
== References ==
{{reflist2|refs=
<ref name=Feucht>  
{{cite web
|author=Dennis Feucht
|url=http://www.analogzone.com/col_0719.pdf 
|title=Two-pole compensation}}
</ref>
</ref>


<ref name=Huijsing>
<ref name=Pockett>
{{cite book  
{{cite book|title=&thinsp;Does Consciousness Cause Behavior? |chapter=The neuroscience of movement |author=Susan Pockett |url=http://books.google.com/books?id=G5CaTnNksgkC&pg=PA19&lpg=PA19 |pages= p. 19 |editor=Susan Pockett, WP Banks, Shaun Gallagher, eds.  |publisher=MIT Press |date =2009 |isbn=0262512572}}
|author=Huijsing, Johan H.
|title=Operational amplifiers: theory and design
|year= 2001
|pages=§6.2, pp.205–206 and Figure 6.2.1
|publisher=Kluwer Academic
|location=Boston, MA
|isbn= 0-7923-7284-0
|url=http://books.google.com/books?id=tiuV_agzk_EC&pg=PA102&dq=isbn:0792372840&sig=d-oEw_n992coA6bU0h6gkoJzoUo#PPA206,M1}}
</ref>
</ref>


<ref name=Sansen>
<ref name=Timpe>
{{cite book
{{cite web |author=Kevin Timpe |title=Free will |work=Internet Encyclopedia of Philosophy |date= March 31, 2006 |url=http://www.iep.utm.edu/freewill/#H5}}
|author=Wally M. C. Sansen
|title=Analog design essentials
|year= 2006
|pages=§097, p. 266 ''ff''
|publisher=Springer
|location=New York; Berlin
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>
</ref>


<ref name=Sedra> 
{{cite book
|author=A.S. Sedra and K.C. Smith
|title=Microelectronic circuits
|year= 2004
|pages=p. 849 and Example 8.6, p. 853
|publisher=Oxford University Press
|edition=Fifth Edition
|location=New York
|isbn= 0-19-514251-9
|url=http://worldcat.org/isbn/0-19-514251-9}}
</ref>


<ref name=Self>
<ref name=Velmans>
{{cite book
{{cite journal |journal=Journal of Consciousness Studies |volume=9 |issue=11 |year=2002 |pages=pp. 2-29 |author=Max Velmans  |title=How Could Conscious Experiences Affect Brains? |url=http://cogprints.org/2750/ |year=2002}}
|author=Douglas Self
|title=Audio power amplifier design handbook
|year= 2006
|pages=pp. 191–193
|publisher=Newnes
|location=Oxford
|isbn= 0750680725
|url=http://books.google.com/books?id=BRQZppvawWwC&pg=PA191&lpg=PA191&dq=%22two+pole+compensation%22&source=web&ots=qsxRG-z1Xl&sig=41uVzeYZW3vi3BndJORUNHNZqPY#PPA191,M1}}
</ref>
</ref>


<ref name=Thompson>
<ref name=Vihvelin>
{{cite book
{{cite web |author=Kadri Vihvelin |title=&thinsp;Arguments for Incompatibilism |work=The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) |editor=Edward N. Zalta, ed. |url= http://plato.stanford.edu/archives/spr2011/entries/incompatibilism-arguments/ |date=Mar 1, 2011}}
|author=Marc T. Thompson
|title=Intuitive analog circuit design: a problem-solving approach using design case studies
|year= 2006
|pages=p. 200
|publisher=Elsevier Newnes
|location=Amsterdam
|isbn= 0750677864
|url=http://books.google.com/books?id=1Tyzjmf0DI8C&pg=PA200&dq=pole+splitting+analog+amplifier&lr=&as_brr=0&sig=gmvG9dtlK48hcqpvf3NwwqcF2Hk}}
</ref>
</ref>


<ref name=Toumazou>
{{cite book
|editor=C. Toumazu, Moschytz GS & Gilbert B eds
|title=Trade-offs in analog circuit design: the designer's companion
|year= 2007
|pages=pp. 272–275
|publisher=Springer
|location=New York/Berlin/Dordrecht
|isbn= 1402070373
|url=http://books.google.com/books?id=VoBIOvirkiMC&pg=PA272&lpg=PA272&dq=%22pole+splitting%22&source=web&ots=MC083mOWhv&sig=duZQKaGECaAH80qDj-YNMdRd8nA}}
</ref>
}}
}}

Latest revision as of 03:07, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


In philosophy the term free will refers to consideration of whether an individual has the ability to make decisions or, alternatively, has only the illusion of doing so. It is an age-old concern to separate what we can do something about, choose to do, from what we cannot. The underlying quandary is the idea that science suggests future events are dictated to a great extent, and perhaps entirely, by past events and, inasmuch as the human body is part of the world science describes, its actions also are determined by physical laws and are not affected by human decisions. This view of events is a particular form of determinism, sometimes called physical reductionism,[1] and the view that determinism precludes free will is called incompatibilism.[2]

There are several ways to avoid the incompatibilst position, resulting in various compatibilist positions.[3] One is to limit the scope of scientific description in a manner that excludes human decisions. Another is to argue that even if our actions are strictly determined by the past, it doesn’t seem that way to us, and so we have to find an approach to this issue that somehow marries our intuition of independence with the reality of its fictional nature. A third, somewhat legalistic approach, is to suggest that the ‘will’ to do something is quite different from actually doing it, so ‘free will’ can exist even though there may be no freedom of action.

There is also a theological version of the dilemma. roughly, if a deity or deities, or 'fate', controls our destiny, what place is left for free will?

Science does not apply

One approach to limiting the applicability of science to our decisions is the examination of the notion of cause and effect. For example, David Hume suggested that science did not really deal with causality, but with the correlation of events. So, for example, lighting a match in a certain environment does not ‘’cause’’ an explosion, but is ‘’associated’’ with an explosion. Immanuel Kant suggested that the idea of cause and effect is not a fact of nature but an interpretation put on events by the human mind, a ‘programming’ built into our brains. Assuming this criticism to be true, there may exist classes of events that escape any attempt at cause and effect explanations.

A different way to exempt human decision from the scientific viewpoint is to note that science is a human enterprise. It involves the human creation of theories to explain certain observations, and moreover, the observations it chooses to attempt to explain are selected, and do not encompass all experience. For example, we choose to explain phenomena like the Higgs boson found by elaborate means like a hadron collider, but don’t attempt to explain other phenomena that do not appear amenable to science at this time, often suggesting that they are beneath attention. As time progresses, one may choose to believe that science will explain all experience, but that view must be regarded as speculation analogous to predicting the stock market on the basis of past performance.

Although not explicitly addressing the issue of free will, it may be noted that Ludwig Wittgenstein argued that the specialized theories of science, as discussed by Rudolf Carnap for example, inevitably cover only a limited range of experience. Hawking/Mlodinow also noted this fact in in their model-dependent realism,[4] the observation that, from the scientific viewpoint, reality is covered by a patchwork of theories that are sometimes disjoint and sometimes overlap.

“Whatever might be the ultimate goals of some scientists, science, as it is currently practiced, depends on multiple overlapping descriptions of the world, each of which has a domain of applicability. In some cases this domain is very large, but in others quite small.”[5]
—— E.B. Davies Epistemological pluralism, p. 4

Still another approach to this matter is analysis of the mind-brain connection (more generally, the mind-body problem). As suggested by Northoff,[6] there is an observer-observation issue involved here. Observing a third-person’s mental activity is a matter for neuroscience, perhaps strictly a question of neurons and their interactions through complex networks. But observing our own mental activity is not possible in this way – it is a matter of subjective experiences. The suggestion has been made that ‘’complementary’’ descriptions of nature are involved, that may be simply different perspectives upon the same reality:

“...for each individual there is one 'mental life' but two ways of knowing it: first-person knowledge and third-person knowledge. From a first-person perspective conscious experiences appear causally effective. From a third person perspective the same causal sequence can be explained in neural terms. It is not the case that the view from one perspective is right and the other wrong. These perspectives are complementary. The differences between how things appear from a first-person versus a third-person perspective has to do with differences in the observational arrangements (the means by which a subject and an external observer access the subject's mental processes).”[7]

—Max Velmans: , How could conscious experiences affect brains?, p. 11

A related view is that the two descriptions may be mutually exclusive. That is, the connection between subjective experience and neuronal activity may run into a version of the measurement-observation interference noticed by Niels Bohr and by Erwin Schrödinger in the early days of quantum mechanics. (The measurement of the position of a particle caused the particle to change position in an unknown way.)

“...it is important to be clear about exactly what experience one wants one's subjects to introspect. Of course, explaining to subjects exactly what the experimenter wants them to experience can bring its own problems–...instructions to attend to a particular internally generated experience can easily alter both the timing and he content of that experience and even whether or not it is consciously experienced at all.”[8]

—Susan Pockett , The neuroscience of movement


In any case, so far as free will is concerned, the implication of 'complementarity' is that 'free will' may be a description that is either an alternative to the scientific view, or possibly a view that can be entertained only if the scientific view is abandoned.

Science can be accommodated

A second approach is to argue that we can accommodate our subjective notions of free will with a deterministic reality. One way to do this is to argue that although we cannot do differently, in fact we really don’t want to do differently, and so what we ‘decide’ to do always agrees with what we (in fact) have to do. Our subjective vision of the decision process as ‘voluntary’ is just a conscious concomitant of the unconscious and predetermined move to action.

’Will’ versus ‘action’

There is growing evidence of the pervasive nature of subconscious thought upon our actions, and the capriciousness of consciousness,[9] which may switch focus from a sip of coffee to the writing of a philosophical exposition without warning. There also is mounting evidence that our consciousness is greatly affected by events in the brain beyond our control. For example, drug addiction has been related to alteration of the mechanisms in the brain for dopamine production, and withdrawal from addiction requires a reprogramming of this mechanism that is more than a simple act of will. The ‘will’ to overcome addiction can become separated from the ability to execute that will.

“Philosophers who distinguish freedom of action and freedom of will do so because our success in carrying out our ends depends in part on factors wholly beyond our control. Furthermore, there are always external constraints on the range of options we can meaningfully try to undertake. As the presence or absence of these conditions and constraints are not (usually) our responsibility, it is plausible that the central loci of our responsibility are our choices, or ‘willings’.”[Italics not in original.][10]

— Timothy O'Connor , Free Will

In effect, could the 'will' be a subjective perception which might operate outside the realm of scientific principle, while its execution is not?

Theology

The ancient Greeks held the view that the gods could intervene in the course of events, and it was possible on occasion to divine their intentions or even to change them. That view leaves a role for free will, although it can be limited in scope by the gods. A more complete restriction is the belief that the gods are omniscient and have perfect foreknowledge of events, which obviously includes human decisions. This view leads to the belief that, while the gods know what we will choose, humans do not, and are faced therefore with playing the role of deciding our actions, even though they are scripted, a view contradicted by Cassius in arguing with Brutus, a Stoic:

“The fault, dear Brutus, is not in our stars, But in ourselves, that we are underlings.”

—spoken by Cassius, Julius Caesar (I, ii, 140-141)

The Stoics wrestled with this problem, and one argument for compatibility took the view that although the gods controlled matters, what they did was understandable using human intellect. Hence, when fate presented us with an issue, there was a duty to sort through a decision, and assent to it (a responsibility), a sequence demanded by our natures as rational beings.[11]

In Chrysippus of Soli's view (an apologist for Stoicism), fate precipitates an event, but our nature determines its course, in the same way that bumping a cylinder or a cone causes it to move, but it rolls or it spins according to its nature.[12] The actual course of events depends upon the nature of the individual, who therefore bears a personal responsibility for the resulting events. It is not clear whether the individual is thought to have any control over their nature, or even whether this question has any bearing upon their responsibility.[13]

References

  1. Alyssa Ney (November 10, 2008). Reductionism. Internet Encyclopedia of Philosophy.
  2. Kadri Vihvelin (Mar 1, 2011). Edward N. Zalta, ed.: Arguments for Incompatibilism. The Stanford Encyclopedia of Philosophy (Spring 2011 Edition).
  3. Kevin Timpe (March 31, 2006). Free will. Internet Encyclopedia of Philosophy.
  4. Hawking SW, Mlodinow L. (2010). “Chapter 3: What is reality?”, The Grand Design. Bantam Books, pp. 42-43. ISBN 0553805371. 
  5. E Brian Davies (2006). Epistemological pluralism. PhilSci Archive.
  6. A rather extended discussion is provided in Georg Northoff (2004). Philosophy of the Brain: The Brain Problem, Volume 52 of Advances in Consciousness Research. John Benjamins Publishing. ISBN 1588114171. 
  7. Max Velmans (2002). "How Could Conscious Experiences Affect Brains?". Journal of Consciousness Studies 9 (11): pp. 2-29.
  8. Susan Pockett (2009). “The neuroscience of movement”, Susan Pockett, WP Banks, Shaun Gallagher, eds.:  Does Consciousness Cause Behavior?. MIT Press, p. 19. ISBN 0262512572. 
  9. Tor Nørretranders (1998). “Preface”, The user illusion: Cutting consciousness down to size, Jonathan Sydenham translation of Maerk verden 1991 ed. Penguin Books, p. ix. ISBN 0140230122. “Consciousness plays a far smaller role in human life than Western culture has tended to believe” 
  10. O'Connor, Timothy (Oct 29, 2010). Edward N. Zalta, ed.: Free Will. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
  11. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See §6.3.3 The cylinder and cone analogy, pp. 258 ff.
  12. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular pp. 386 ff.
  13. Susanne Bobzien (1998). Determinism and Freedom in Stoic Philosophy. Oxford University Press. ISBN 0198237944.  See in particular p. 255.