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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pole splitting ： ウィキペディア英語版 Pole splitting 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 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.〔That is, the rise time is selected to be the fastest possible consistent with low overshoot and ringing.〕〔 〕〔 〕〔 〕 == Example of pole splitting == This example shows that introduction of the capacitor referred to as CC 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.〔Although this example appears very specific, the associated mathematical analysis is very much used in circuit design.〕 The amplifier of Figure 1 has a low frequency pole due to the added input resistance ''Ri'' and capacitance ''Ci'', with the time constant ''Ci'' ( ''RA // Ri'' ). 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 ''RL'' and capacitance ''CL'', with the time constant ''CL'' ('' Ro // RL'' ). The upward movement of the high-frequency pole occurs because the Miller-amplified compensation capacitor ''CC'' alters the frequency dependence of the output voltage divider. The first objective, to show the lowest pole moves down in frequency, is established using the same approach as the 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 $\backslash \; v\_i$ to the ideal op amp as a function of the applied signal voltage $\backslash \; v\_a$, namely, ::$$ \frac = \frac \frac \ , which exhibits a roll-off with frequency beginning at ''f1'' where ::$$ \begin f_ & = \frac \\ & = \frac \ , \\ \end which introduces notation $\backslash tau\_1$ for the time constant of the lowest pole. This frequency is lower than the initial low frequency of the amplifier, which for ''CC'' = 0 F is $\backslash frac$. 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 $\backslash \; v\_o$ is determined by the gain of the ideal op amp inside the amplifier as ::$\backslash \; v\_o\; =\; A\_v\; v\_i\; \backslash \; .$ Using this relation and applying Kirchhoff's current law to the output side of the circuit determines the load voltage $v\_$ as a function of the voltage $\backslash \; v\_$ at the input to the ideal op amp as: ::$\backslash frac\; =\; A\_v\; \backslash frac\; \backslash ,\backslash !$$\backslash sdot\; \backslash frac\; \backslash \; .$ This expression is combined with the gain factor found earlier for the input side of the circuit to obtain the overall gain as ::$$ \frac = \frac \frac :::$=\; A\_v\; \backslash frac\; \backslash sdot\; \backslash frac\; \backslash ,\backslash !$$\backslash sdot\; \backslash frac\; \backslash ,\backslash !$$\backslash sdot\; \backslash frac\; \backslash \; .$ 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 ''Av'' 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, ''CL'' ( ''Ro // RL'' ), corresponds to a frequency well above the low frequency pole, the accurate form of the Miller capacitance must be used, rather than the Miller approximation. According to the article on Miller effect, the Miller capacitance is given by ::$$ \begin C_M & = C_C \left( 1 - \frac \right) \\ & = C_C \left( 1 - A_v \frac \frac \right ) \ . \\ \end (For a positive Miller capacitance, ''Av'' is negative.) Upon substitution of this result into the gain expression and collecting terms, the gain is rewritten as: ::$\backslash frac\; =\; A\_v\; \backslash frac\; \backslash frac\; \backslash frac\; \backslash ,\backslash !$ $=\; (\backslash omega\; (C\_L+C\_C)\; (R\_o//R\_L))\; \backslash ,\backslash !$ $\backslash sdot\; \backslash \; (1+j\; \backslash omega\; C\_i\; (R\_A//R\_i))\; \backslash ,\backslash !$ $\backslash \; +j\; \backslash omega\; C\_C\; (R\_A//R\_i)\backslash ,\backslash !$ $\backslash sdot\; \backslash left(\; 1-A\_v\; \backslash frac\; \backslash right)\; \backslash ,\backslash !$ $\backslash \; +(j\; \backslash omega)\; ^2\; C\_C\; C\_L\; (R\_A//R\_i)\; (R\_O//R\_L)\; \backslash \; .$ Every quadratic has two factors, and this expression looks simpler if it is rewritten as ::$$ \ D_ =(1+j \omega _1 )(1+j \omega _2 ) :::$=\; 1\; +\; j\; \backslash omega\; (\; \_1+\_2)\; )\; +(j\; \backslash omega\; )^2\; \backslash tau\_1\; \backslash tau\_2\; \backslash \; ,\; \backslash$ where $\backslash tau\_1$ and $\backslash tau\_2$ are combinations of the capacitances and resistances in the formula for ''Dω''.〔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)2.〕 They correspond to the time constants of the two poles of the amplifier. One or the other time constant is the longest; suppose $\backslash tau\_1$ is the longest time constant, corresponding to the lowest pole, and suppose $\backslash tau\_1$ >> $\backslash tau\_2$. (Good step response requires $\backslash tau\_1$ >> $\backslash tau\_2$. See Selection of CC 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 ''Dω'' is: ::$$ \begin \ D_ & = 1+ j \omega ((R_A//R_i) +(C_L+C_C) (R_o//R_L) ) \\ & = 1+j \omega ( \tau_1 + \tau_2) \approx 1 + j \omega \tau_1 \ , \ \\ \end where now ''CM'' is redefined using the Miller approximation as ::$C\_M=\; C\_C\; \backslash left(\; 1\; -\; A\_v\; \backslash frac\; \backslash right)\; \backslash \; ,$ which is simply the previous Miller capacitance evaluated at low frequencies. On this basis $\backslash tau\_1$ is determined, provided $\backslash tau\_1$ >> $\backslash tau\_2$. Because ''CM'' is large, the time constant $\_1$ is much larger than its original value of ''Ci'' ( ''RA // Ri'' ).〔The expression for $\backslash tau\_1$ differs a little from ( ''CM+Ci'' ) ( ''RA'' // ''Ri'' ) as found initially for ''f1'', but the difference is minor assuming the load capacitance is not so large that it controls the low frequency response instead of the Miller capacitance.〕 At high frequencies the quadratic term becomes important. Assuming the above result for $\backslash tau\_1$ is valid, the second time constant, the position of the high frequency pole, is found from the quadratic term in ''Dω'' as ::$\backslash tau\_2\; =\; \backslash frac\; \backslash approx\; \backslash frac\; \backslash \; .$ Substituting in this expression the quadratic coefficient corresponding to the product $\backslash tau\_1\; \backslash tau\_2$ along with the estimate for $\backslash tau\_1$, an estimate for the position of the second pole is found: ::$$ \begin \tau_2 & = \frac \\ & \approx \frac (R_O//R_L)\ , \\ \end and because ''CM'' is large, it seems $\backslash tau\_2$ is reduced in size from its original value ''CL'' ( ''Ro'' // ''RL'' ); that is, the higher pole has moved still higher in frequency because of ''CC''.〔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.〕 In short, introduction of capacitor ''CC'' moved the low pole lower and the high pole higher, so the term pole splitting seems a good description. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pole splitting」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. 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