Editing Bode plot (section)

==Gain margin and phase margin==
{{See also|Phase margin}}
Bode plots are used to assess the stability of [[negative-feedback amplifier]]s by finding the gain and [[phase margin]]s of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by

:<math>A_\text{FB} = \frac{A_\text{OL}}{1 + \beta A_\text{OL}},</math>

where ''A''<sub>FB</sub> is the gain of the amplifier with feedback (the ''closed-loop gain''), ''β'' is the ''feedback factor'', and ''A''<sub>OL</sub> is the gain without feedback (the ''open-loop gain''). The gain ''A''<sub>OL</sub> is a complex function of frequency, with both magnitude and phase.<ref group="note">Ordinarily, as frequency increases, the magnitude of the gain drops, and the phase becomes more negative, although these are only trends and may be reversed in particular frequency ranges. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the [[Nyquist plot]] have to be used to assess stability.</ref> Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β''A''<sub>OL</sub> = −1 (that is, the magnitude of β''A''<sub>OL</sub> is unity and its phase is −180°, the so-called [[Barkhausen stability criterion]]). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.

Key to this determination are two frequencies. The first, labeled here as ''f''<sub>180</sub>, is the frequency where the open-loop gain flips sign. The second, labeled here ''f''<sub>0&nbsp;dB</sub>, is the frequency where the magnitude of the product |β''A''<sub>OL</sub>| = 1 = 0&nbsp;dB. That is, frequency ''f''<sub>180</sub> is determined by the condition

:<math>\beta A_\text{OL}(f_{180}) = -|\beta A_\text{OL}(f_{180})| = -|\beta A_\text{OL}|_{180},</math>

where vertical bars denote the [[Absolute value#Complex numbers|magnitude of a complex number]], and frequency ''f''<sub>0&nbsp;dB</sub> is determined by the condition

:<math>|\beta A_\text{OL}(f_\text{0 dB})| = 1.</math>

One measure of proximity to instability is the '''gain margin'''. The Bode phase plot locates the frequency where the phase of β''A''<sub>OL</sub> reaches &minus;180°, denoted here as frequency ''f''<sub>180</sub>. Using this frequency, the Bode magnitude plot finds the magnitude of β''A''<sub>OL</sub>. If |β''A''<sub>OL</sub>|<sub>180</sub> ≥ 1, the amplifier is unstable, as mentioned. If |β''A''<sub>OL</sub>|<sub>180</sub> < 1, instability does not occur, and the separation in dB of the magnitude of |β''A''<sub>OL</sub>|<sub>180</sub> from |β''A''<sub>OL</sub>| = 1 is called the ''gain margin''. Because a magnitude of 1 is 0&nbsp;dB, the gain margin is simply one of the equivalent forms: <math>20 \log_{10} |\beta A_\text{OL}|_{180} = 20 \log_{10} |A_\text{OL}| - 20 \log_{10} \beta^{-1}</math>.

Another equivalent measure of proximity to instability is the ''[[phase margin]]''. The Bode magnitude plot locates the frequency where the magnitude of |β''A''<sub>OL</sub>| reaches unity, denoted here as frequency ''f''<sub>0&nbsp;dB</sub>. Using this frequency, the Bode phase plot finds the phase of β''A''<sub>OL</sub>. If the phase of β''A''<sub>OL</sub>(''f''<sub>0&nbsp;dB</sub>) > &minus;180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when ''f'' = ''f''<sub>180</sub>), and the distance of the phase at ''f''<sub>0&nbsp;dB</sub> in degrees above &minus;180° is called the ''phase margin''.

If a simple ''yes'' or ''no'' on the stability issue is all that is needed, the amplifier is stable if ''f''<sub>0&nbsp;dB</sub> < ''f''<sub>180</sub>. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions ([[minimum phase]] systems). Although these restrictions usually are met, if they are not, then another method must be used, such as the [[Nyquist plot]].<ref name=Lee>
{{cite book 
|author=Thomas H. Lee
|title=The design of CMOS radio-frequency integrated circuits
|section=§14.6. Gain and Phase Margins as Stability Measures
|pages=451–453
|year=2004
|edition=2nd
|publisher=Cambridge University Press
|location=Cambridge UK
|isbn=0-521-83539-9
|url=http://worldcat.org/isbn/0-521-83539-9}}
</ref><ref name=Levine>
{{cite book 
|author=William S. Levine
|title=The control handbook: the electrical engineering handbook series
|section=§10.1. Specifications of Control System
|page=163
|year=1996
|edition=2nd
|publisher=CRC Press/IEEE Press
|location=Boca Raton FL
|isbn=0-8493-8570-9
|url=https://books.google.com/books?id=2WQP5JGaJOgC&q=stability+%22minimum+phase%22&pg=RA1-PA163}}
</ref>
Optimal gain and phase margins may be computed using [[Nevanlinna–Pick interpolation]] theory.<ref name=Tannenbaum>
{{cite book
|author=Allen Tannenbaum | author-link = Allen Tannenbaum
|title=Invariance and Systems Theory: Algebraic and Geometric Aspects
|date = February 1981
|publisher=Springer-Verlag
|location=New York, NY
|isbn=9783540105657}}
</ref>

===Examples using Bode plots===
Figures 6 and 7 illustrate the gain behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode plot for the gain without feedback (the ''open-loop'' gain) ''A''<sub>OL</sub> with the gain with feedback ''A''<sub>FB</sub> (the ''closed-loop'' gain). See [[negative feedback amplifier]] for more detail.

In this example, ''A''<sub>OL</sub> = 100&nbsp;dB at low frequencies, and 1 / β = 58&nbsp;dB. At low frequencies, ''A''<sub>FB</sub> ≈ 58&nbsp;dB as well.

Because the open-loop gain ''A''<sub>OL</sub> is plotted and not the product β ''A''<sub>OL</sub>, the condition ''A''<sub>OL</sub> = 1 / β decides ''f''<sub>0&nbsp;dB</sub>. The feedback gain at low frequencies and for large ''A''<sub>OL</sub> is ''A''<sub>FB</sub> ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain ''A''<sub>OL</sub>), so an equivalent way to find ''f''<sub>0&nbsp;dB</sub> is to look where the feedback gain intersects the open-loop gain. (Frequency ''f''<sub>0&nbsp;dB</sub> is needed later to find the phase margin.)

Near this crossover of the two gains at ''f''<sub>0&nbsp;dB</sub>, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β ''A''<sub>OL</sub> = −1). Beyond the unity gain frequency ''f''<sub>0&nbsp;dB</sub>, the open-loop gain is sufficiently small that ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> (examine the formula at the beginning of this section for the case of small ''A''<sub>OL</sub>).

Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency ''f''<sub>180</sub> where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, ''A''<sub>FB</sub> ≈ ''A''<sub>OL</sub> for small ''A''<sub>OL</sub>.)

Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency ''f''<sub>0&nbsp;dB</sub> and the phase-flip frequency ''f''<sub>180</sub> are very nearly equal in this amplifier, ''f''<sub>180</sub> ≈ ''f''<sub>0&nbsp;dB</sub> ≈ 3.332&nbsp;kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.

Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the condition | β ''A''<sub>OL</sub> | = 1 to lower frequency. In this example, 1 / β = 77&nbsp;dB, and at low frequencies ''A''<sub>FB</sub> ≈ 77&nbsp;dB as well.

Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and ''A''<sub>OL</sub> occurs at  ''f''<sub>0&nbsp;dB</sub> = 1&nbsp;kHz. Notice that the peak in the gain ''A''<sub>FB</sub> near ''f''<sub>0&nbsp;dB</sub> is almost gone.<ref group="note">The critical amount of feedback where the peak in the gain ''just'' disappears altogether is the ''maximally flat'' or [[Butterworth filter#Maximal flatness|Butterworth]] design.</ref><ref name=Sansen>
{{cite book 
|author=Willy M C Sansen
|title=Analog design essentials
|pages=157–163
|year= 2006
|publisher=Springer
|location=Dordrecht, The Netherlands
|isbn=0-387-25746-2
|url=http://worldcat.org/isbn/0-387-25746-2}}
</ref>

Figure 9 is the phase plot. Using the value of ''f''<sub>0&nbsp;dB</sub> = 1&nbsp;kHz found above from the magnitude plot of Figure 8, the open-loop phase at ''f''<sub>0&nbsp;dB</sub> is −135°, which is a phase margin of 45° above −180°.

Using Figure 9, for a phase of −180° the value of ''f''<sub>180</sub> = 3.332&nbsp;kHz (the same result as found earlier, of course<ref group="note">The frequency where the open-loop gain flips sign ''f''<sub>180</sub> does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at ''f''<sub>180</sub> also does not change with a change in β. Therefore, we could use the previous values from Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9.</ref>). The open-loop gain from Figure 8 at ''f''<sub>180</sub> is 58&nbsp;dB, and 1 / β = 77&nbsp;dB, so the gain margin is 19&nbsp;dB.

Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good [[Step response#Step response of feedback amplifiers|step response]]. As a [[rule of thumb]], good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue.<ref name=Sansen/> See also the discussion of phase margin in the [[Step response#Phase margin|step response]] article.

<gallery caption="Examples" widths="300px" perrow="2" class="skin-invert-image">
Image:Magnitude of feedback amplifier.PNG|Figure 6: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. Parameter 1/β = 58&nbsp;dB, and at low frequencies ''A''<sub>FB</sub> ≈ 58&nbsp;dB as well. The gain margin in this amplifier is nearly zero because &#124; β''A''<sub>OL</sub>&#124; = 1 occurs at almost ''f'' = ''f''<sub>180°</sub>.
Image:Phase of feedback amplifier.PNG|Figure 7: Phase of feedback amplifier ''°A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''°A''<sub>OL</sub>. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency ''f'' = ''f''<sub>0&nbsp;dB</sub> where &#124; β''A''<sub>OL</sub>&#124; = 1.
Image:Gain Margin.PNG|Figure 8: Gain of feedback amplifier ''A''<sub>FB</sub> in dB and corresponding open-loop amplifier ''A''<sub>OL</sub>. In this example, 1 / β = 77&nbsp;dB. The gain margin in this amplifier is 19&nbsp;dB.
Image:Phase Margin.PNG|Figure 9: Phase of feedback amplifier ''A''<sub>FB</sub> in degrees and corresponding open-loop amplifier ''A''<sub>OL</sub>. The phase margin in this amplifier is 45°.
</gallery>