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| '''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> That is, the [[rise time]] is selected to be the fastest possible consistent with low [[overshoot (signal)|overshoot]] and [[ringing (signal)|ringing]].</ref><ref name=Toumazou>
| | 49 year old Shoemaker Yeatts from Deloraine, spends time with pursuits including saltwater aquariums, venapro and texting. Has these days finished a journey to Agricultural Landscape of Southern Öland.<br><br>My site ... [http://venaprocritic.com/venapro-hemorrhoid-relief-formula-review/ piles disease] |
| {{cite book
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| |author=C. Toumazu, Moschytz GS & Gilbert B (Editors)
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| |title=Trade-offs in analog circuit design: the designer's companion
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| |year= 2007
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| |pages=272–275
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| |publisher=Springer
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| |location=New York/Berlin/Dordrecht
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| |isbn= 1-4020-7037-3 |url=http://books.google.com/books?id=VoBIOvirkiMC&pg=PA272&lpg=PA272&dq=%22pole+splitting%22&source=web&ots=MC083mOWhv&sig=duZQKaGECaAH80qDj-YNMdRd8nA}}
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| </ref><ref name=Thompson>
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| {{cite book
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| |author=Marc T. Thompson
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| |title=Intuitive analog circuit design: a problem-solving approach using design case studies
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| |year= 2006
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| |pages=200
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| |publisher=Elsevier Newnes
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| |location=Amsterdam
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| |isbn= 0-7506-7786-4 |url=http://books.google.com/books?id=1Tyzjmf0DI8C&pg=PA200&dq=pole+splitting+analog+amplifier&lr=&as_brr=0&sig=gmvG9dtlK48hcqpvf3NwwqcF2Hk}}
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| </ref><ref name=Sansen>
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| {{cite book
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| |author=Wally M. C. Sansen
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| |title=Analog design essentials
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| |year= 2006
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| |pages=§097, p. 266 ''et seq''
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| |publisher=Springer
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| |location=New York; Berlin
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| |isbn=0-387-25746-2
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| |url=http://worldcat.org/isbn/0-387-25746-2}}
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| </ref>
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| == Example of pole splitting ==
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| [[File:Pole Splitting Example.png|thumbnail|250px|Figure 1: 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>''.]]
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| [[File:Pole Splitting Example with Miller Transform.png|thumbnail|250px|Figure 2: Operational amplifier with compensation capacitor transformed using [[Miller's theorem]] to replace the compensation capacitor with a Miller capacitor at the input and a frequency-dependent current source at the output.]]
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| 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>Although this example appears very specific, the associated mathematical analysis is very much used in circuit design.</ref> 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's theorem|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.
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| 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's theorem]], 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,
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| ::<math>
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| \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>
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| which exhibits a [[roll-off]] with frequency beginning at ''f<sub>1</sub>'' where
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| ::<math>
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| \begin{align}
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| f_{1} & = \frac {1} {2 \pi (C_M+C_i)(R_A//R_i) } \\
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| & = \frac {1} {2 \pi \tau_1} \ , \\
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| \end{align}
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| </math>
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| 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, which for ''C<sub>C</sub>'' = 0 F is <math>\frac {1} {2 \pi C_i (R_A//R_i)}</math>.
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| 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
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| ::<math>\ v_o = A_v v_i \ . </math>
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| 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:
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| ::<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>
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| This expression is combined with the gain factor found earlier for the input side of the circuit to obtain the overall gain as
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| ::<math>
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| \frac {v_{\ell}} {v_a} = \frac {v_{\ell}}{v_i} \frac {v_i} {v_a}
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| </math>
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| :::<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>
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| 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's theorem|Miller effect]], the Miller capacitance is given by
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| ::<math>
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| \begin{align}
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| C_M & = C_C \left( 1 - \frac {v_{\ell}} {v_i} \right) \\
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| & = 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 ) \ . \\
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| \end{align}
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| </math>
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| (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:
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| ::<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>
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| with ''D<sub>ω</sub>'' given by a quadratic in ω, namely: | |
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| ::<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>
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| Every quadratic has two factors, and this expression looks simpler if it is rewritten as
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| ::<math>
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| \ D_{ \omega } =(1+j \omega { \tau}_1 )(1+j \omega { \tau}_2 ) </math>
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| :::<math> = 1 + j \omega ( {\tau}_1+{\tau}_2) ) +(j \omega )^2 \tau_1 \tau_2 \ , \ </math>
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| 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>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>.</ref> 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.)
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| 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:
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| ::<math>
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| \begin{align}
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| \ D_{ \omega } & = 1+ j \omega [(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)] \\
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| & = 1+j \omega ( \tau_1 + \tau_2) \approx 1 + j \omega \tau_1 \ , \ \\
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| \end{align}
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| </math>
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| where now ''C<sub>M</sub>'' is redefined using the [[Miller effect|Miller approximation]] as
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| {{anchor|Miller}}
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| ::<math> C_M= C_C \left( 1 - A_v \frac {R_L}{R_L+R_o} \right) \ ,</math>
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| 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> The expression for <math>\tau_1</math> 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>
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| 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
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| ::<math> \tau_2 = \frac {\tau_1 \tau_2} {\tau_1} \approx \frac {\tau_1 \tau_2} {\tau_1 + \tau_2}\ . </math>
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| 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:
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| ::<math>
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| \begin{align}
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| \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)} \\
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| & \approx \frac {C_C C_L +C_L C_i+C_i C_C} {C_M} (R_O//R_L)\ , \\
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| \end{align}
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| </math>
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| 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>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.</ref>
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| 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.
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| === Selection of C<sub>C</sub> ===
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| [[File:Two-pole Bode magnitude plot.png|thumbnail|300px|Figure 3: 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.]]
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| 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>
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| {{cite book
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| |author=A.S. Sedra and K.C. Smith
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| |title=Microelectronic circuits
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| |year= 2004
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| |pages=849 and Example 8.6, p. 853
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| |publisher=Oxford University Press
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| |edition=Fifth Edition
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| |location=New York
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| |isbn= 0-19-514251-9
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| |url=http://worldcat.org/isbn/0-19-514251-9}}
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| </ref> | |
| <ref name=Huijsing> | |
| {{cite book
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| |author=Huijsing, Johan H.
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| |title=Operational amplifiers: theory and design
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| |year= 2001
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| |pages=§6.2, pp.205–206 and Figure 6.2.1
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| |publisher=Kluwer Academic
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| |location=Boston, MA
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| |isbn= 0-7923-7284-0
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| |url=http://books.google.com/books?id=tiuV_agzk_EC&pg=PA102&dq=isbn:0792372840&sig=d-oEw_n992coA6bU0h6gkoJzoUo#PPA206,M1}}
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| </ref> 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> Feucht, Dennis: [http://www.analogzone.com/col_0719.pdf ''Two-pole compensation''] </ref> <ref name=Self>
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| {{cite book
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| |author=Self, Douglas
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| |title=Audio power amplifier design handbook
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| |year= 2006
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| |pages=191–193
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| |publisher=Newnes
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| |location=Oxford
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| |isbn= 0-7506-8072-5
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| |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}}
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| </ref>
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| 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>That is, the frequency is plotted in powers of ten, as 1, 10, 10<sup>2</sup> ''etc''.</ref>) of ( log<sub>10</sub> ''f''<sub>2</sub> − 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:
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| ::Slope per decade of frequency <math>=20 \frac {\mathrm{log_{10}} ( A_v )} {\mathrm{log_{10}} (f_2 / f_1 ) } \ ,</math>
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| 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.
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| 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:
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| ::<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>
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| [[File:Compensation capacitance.png|thumbnail|350px|Figure 4: 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.]]
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| Using the approximations for the time constants developed above,
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| ::<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>
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| or
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| ::<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>
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| 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>''.
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| 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>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.</ref> See Sansen<ref name=Sansen/> or Huijsing<ref name=Huijsing/> and article on [[step response]].
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| ===Slew rate===
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| 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>''.
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| == See also ==
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| * [[Frequency compensation]]
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| * [[Miller effect]]
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| * [[Common source]]
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| * [[Bode plot]]
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| * [[Step_response#Control_of_overshoot|Step response]]
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| == References and notes ==
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| {{reflist}}
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| ==External links==
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| * [http://en.wikibooks.org/wiki/Circuit_Theory/Bode_Plots Bode Plots] in the Circuit Theory [[Wikibooks|Wikibook]]
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| * [http://en.wikibooks.org/wiki/Control_Systems/Bode_Plots Bode Plots] in the Control Systems [[Wikibooks|Wikibook]]
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| [[Category:Analog circuits]]
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| [[Category:Electronic design]]
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