User:John R. Brews/Sample2

Figure 1(a): The Bode plot for a first-order (one-pole) highpass filter; the straight-line approximations are labeled "Bode pole"; phase varies from 90° at low frequencies (due to the contribution of the numerator, which is 90° at all frequencies) to 0° at high frequencies (where the phase contribution of the denominator is −90° and cancels the contribution of the numerator).

Figure 1(b): The Bode plot for a first-order (one-pole) lowpass filter; the straight-line approximations are labeled "Bode pole"; phase is 90° lower than for Figure 1(a) because the phase contribution of the numerator is 0° at all frequencies.

A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot:

A Bode magnitude plot is a graph of log magnitude versus frequency, plotted with a log-frequency axis, to show the transfer function or frequency response of a linear, time-invariant system.

The magnitude axis of the Bode plot is usually expressed as decibels, that is, 20 times the common logarithm of the amplitude gain. With the magnitude gain being logarithmic, Bode plots make multiplication of magnitudes a simple matter of adding distances on the graph (in decibels), since

$\log(a\cdot b)=\log(a)+\log(b)\,$.

A Bode phase plot is a graph of phase versus frequency, also plotted on a log-frequency axis, usually used in conjunction with the magnitude plot, to evaluate how much a frequency will be phase-shifted. For example a signal described by: Asin(ωt) may be attenuated but also phase-shifted. If the system attenuates it by a factor x and phase shifts it by −Φ the signal out of the system will be (A/x) sin(ωt − Φ). The phase shift Φ is generally a function of frequency.

Phase can also be added directly from the graphical values, a fact that is mathematically clear when phase is seen as the imaginary part of the complex logarithm of a complex gain.

In Figure 1(a), the Bode plots are shown for the one-pole highpass filter function:

$\mathrm {T_{High}} (f)={\frac {jf/f_{1}}{1+jf/f_{1}}}\ ,$

where f is the frequency in Hz, and f₁ is the pole position in Hz, f₁ = 100 Hz in the figure. Using the rules for complex numbers, the magnitude of this function is

$\mid \mathrm {T_{High}} (f)\mid ={\frac {f/f_{1}}{\sqrt {1+(f/f_{1})^{2}}}}\ ,$

while the phase is:

$\phi _{T_{High}}=90^{\circ }-\mathrm {tan^{-1}} (f/f_{1})\ .$

Care must be taken that the inverse tangent is set up to return degrees, not radians. On the Bode magnitude plot, decibels are used, and the plotted magnitude is:

$20\ \mathrm {log_{10}} \mid \mathrm {T_{High}} (f)\mid \ =20\ \mathrm {log_{10}} \left(f/f_{1}\right)$

  $\ -20\ \mathrm {log_{10}} \left({\sqrt {1+(f/f_{1})^{2}}}\right)\ .$

In Figure 1(b), the Bode plots are shown for the one-pole lowpass filter function:

$\mathrm {T_{Low}} (f)={\frac {1}{1+jf/f_{1}}}\ .$

Also shown in Figure 1(a) and 1(b) are the straight-line approximations to the Bode plots that are used in hand analysis, and described later.

The magnitude and phase Bode plots can seldom be changed independently of each other — changing the amplitude response of the system will most likely change the phase characteristics and vice versa. For minimum-phase systems the phase and amplitude characteristics can be obtained from each other with the use of the Hilbert transform.

If the transfer function is a rational function with real poles and zeros, then the Bode plot can be approximated with straight lines. These asymptotic approximations are called straight line Bode plots or uncorrected Bode plots and are useful because they can be drawn by hand following a few simple rules. Simple plots can even be predicted without drawing them.

The approximation can be taken further by correcting the value at each cutoff frequency. The plot is then called a corrected Bode plot.

Rules for hand-made Bode plot

The main idea about Bode plots is that one can think of the log of a function in the form:

$f(x)=A\prod (x+c_{n})^{a_{n}}$

as a sum of the logs of its poles and zeros:

$\log(f(x))=\log(A)+\sum a_{n}\log(x+c_{n}).$

This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.

Straight-line amplitude plot

Amplitude decibels is usually done using the $20\log _{10}(X)$ version. Given a transfer function in the form

$H(s)=A\prod {\frac {(s+x_{n})^{a_{n}}}{(s+y_{n})^{b_{n}}}}$

where $x_{n}$ and $y_{n}$ are constants, $s=j\omega$, $a_{n}$, $b_{n}$ > 0, and H is the transfer function:

• at every value of s where $\omega =x_{n}$ (a zero), increase the slope of the line by $20\cdot a_{n}dB$ per decade.
• at every value of s where $\omega =y_{n}$ (a pole), decrease the slope of the line by $20\cdot b_{n}dB$ per decade.
• The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency ω into the function and finding |H(jω)|.
• The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and are found using the first two rules.

To handle irreducible 2nd order polynomials, $ax^{2}+bx+c\$ can, in many cases, be approximated as $({\sqrt {a}}x+{\sqrt {c}})^{2}$.

Note that zeros and poles happen when ω is equal to a certain $x_{n}$ or $y_{n}$. This is because the function in question is the magnitude of H(jω), and since it is a complex function, $|H(j\omega )|={\sqrt {H\cdot H^{*}}}$. Thus at any place where there is a zero or pole involving the term $(s+x_{n})$, the magnitude of that term is ${\sqrt {(x_{n}+j\omega )\cdot (x_{n}-j\omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}$.

Corrected amplitude plot

To correct a straight-line amplitude plot:

• at every zero, put a point $3\cdot a_{n}\ \mathrm {dB}$ above the line,
• at every pole, put a point $3\cdot b_{n}\ \mathrm {dB}$ below the line,
• draw a smooth line through those points using the straight lines as asymptotes (lines which the curve approaches).

Note that this correction method does not incorporate how to handle complex values of $x_{n}$ or $y_{n}$. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.

Straight-line phase plot

Given a transfer function in the same form as above:

$H(s)=A\prod {\frac {(s+x_{n})^{a_{n}}}{(s+y_{n})^{b_{n}}}}$

the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by $-\mathrm {arctan} (\mathrm {Im} [H(s)]/\mathrm {Re} [H(s)])$.

To draw the phase plot, for each pole and zero:

• if A is positive, start line (with zero slope) at 0 degrees
• if A is negative, start line (with zero slope) at 180 degrees
• at every $\omega =x_{n}$ (for stable zeros – $Re(z)<0$), increase the slope by $45\cdot a_{n}$ degrees per decade, beginning one decade before $\omega =x_{n}$ (i.e. ${\frac {x_{n}}{10}}$)
• at every $\omega =y_{n}$ (for stable poles – $Re(p)<0$), decrease the slope by $45\cdot b_{n}$ degrees per decade, beginning one decade before $\omega =y_{n}$ (i.e. ${\frac {y_{n}}{10}}$)
• "unstable" (right half plane) poles and zeros ($Re(s)>0$) have opposite behavior
• flatten the slope again when the phase has changed by $90\cdot a_{n}$ degrees (for a zero) or $90\cdot b_{n}$ degrees (for a pole),
• After plotting one line for each pole or zero, add the lines together to obtain the final phase plot; that is, the final phase plot is the superposition of each earlier phase plot.

Example

A passive (unity pass band gain) lowpass RC filter, for instance has the following transfer function expressed in the frequency domain:

$H(jf)={\frac {1}{1+j2\pi fRC}}$

From the transfer function it can be determined that the cutoff frequency point f_c (in hertz) is at the frequency

$f_{\mathrm {c} }={1 \over {2\pi RC}}$

or (equivalently) at

$\omega _{\mathrm {c} }={1 \over {RC}}$ where $\omega _{\mathrm {c} }=2\pi f_{\mathrm {c} }$ is the angular cutoff frequency in radians per second.

The transfer function in terms of the angular frequencies becomes:

$H(j\omega )={1 \over 1+j{\omega \over {\omega _{\mathrm {c} }}}}$

The above equation is the normalized form of the transfer function. The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.

Magnitude plot

The magnitude (in decibels) of the transfer function above, (normalized and converted to angular frequency form), given by the decibel gain expression $A_{\mathrm {vdB} }$:

$A_{\mathrm {vdB} }\ =\ 20\ \log |H(j\omega )|\ =\ 20\ \log {1 \over \left|1+j{\omega \over {\omega _{\mathrm {c} }}}\right|}$

$=\ -20\ \log \left|1+j{\omega \over {\omega _{\mathrm {c} }}}\right|$   $=\ -10\ \log {\left[1+{\frac {\omega ^{2}}{\omega _{\mathrm {c} }^{2}}}\right]}$

when plotted versus input frequency $\omega$ on a logarithmic scale, can be approximated by two lines and it forms the asymptotic (approximate) magnitude Bode plot of the transfer function:

• for angular frequencies below $\omega _{\mathrm {c} }$ it is a horizontal line at 0 dB since at low frequencies the ${\omega \over {\omega _{\mathrm {c} }}}$ term is small and can be neglected, making the decibel gain equation above equal to zero,
• for angular frequencies above $\omega _{\mathrm {c} }$ it is a line with a slope of −20 dB per decade since at high frequencies the ${\omega \over {\omega _{\mathrm {c} }}}$ term dominates and the decibel gain expression above simplifies to $-20\log {\omega \over {\omega _{\mathrm {c} }}}$ which is a straight line with a slope of -20 dB per decade.

These two lines meet at the corner frequency. From the plot, it can be seen that for frequencies well below the corner frequency, the circuit has an attenuation of 0 dB, corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals the amplitude of the input. Frequencies above the corner frequency are attenuated – the higher the frequency, the higher the attenuation.

Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

Figure 4: Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots

Figure 5: Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots

Phase plot

The phase Bode plot is obtained by plotting the phase angle of the transfer function given by: $\phi =-\tan ^{-1}{\omega \over {\omega _{\mathrm {c} }}}$ versus $\omega$, where $\omega$ and $\omega _{\mathrm {c} }$ are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio ${\omega \over {\omega _{\mathrm {c} }}}$ is small and therefore the phase angle is close to zero. As the ratio increases the absolute value of the phase increases and becomes –45 degrees when $\omega =\omega _{\mathrm {c} }$. As the ratio increases for input frequencies much greater than the corner frequency, the phase angle asymptotically approaches -90 degrees. The frequency scale for the phase plot is logarithmic.

Normalized plot

The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio ${\omega \over {\omega _{\mathrm {c} }}}$. In such a case the plot is said to be normalized and units of the frequencies are no longer used since all input frequencies are now expressed as multiples of the cutoff frequency $\omega _{\mathrm {c} }$.

An example with pole and zero

Figures 2-5 further illustrate construction of Bode plots. This example with both a pole and a zero shows how to use superposition. To begin, the components are presented separately.

Figure 2 shows the Bode magnitude plot for a zero and a low-pass pole, and compares the two with the Bode straight line plots. The straight-line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade. The second Figure 3 does the same for the phase. The phase plots are horizontal up to a frequency a factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°.

Figure 4 and Figure 5 show how superposition (simple addition) of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero has been moved to higher frequency than the pole to make a more interesting example. Notice in Figure 4 that the 20 dB/decade drop of the pole is arrested by the 20 dB/decade rise of the zero resulting in a horizontal magnitude plot for frequencies above the zero location. Notice in Figure 5 in the phase plot that the straight-line approximation is pretty approximate in the region where both pole and zero affect the phase. Notice also in Figure 5 that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole (zero) location. Where the phase of the pole and the zero both are present, the straight-line phase plot is horizontal because the 45°/decade drop of the pole is arrested by the overlapping 45°/decade rise of the zero in the limited range of frequencies where both are active contributors to the phase.

Gain margin and phase margin

Bode plots are used to assess the stability of negative feedback amplifiers by finding the gain and phase margins of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by

$A_{FB}={\frac {A_{OL}}{1+\beta A_{OL}}}\ ,$

where A_FB is the gain of the amplifier with feedback (the closed-loop gain), β is the feedback factor and A_OL is the gain without feedback (the open-loop gain). The gain A_OL is a complex function of frequency, with both magnitude and phase. Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product βA_OL = −1. (That is, the magnitude of βA_OL is unity and its phase is −180°, the so-called Barkhausen criteria). 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 f₁₈₀, is the frequency where the open-loop gain flips sign. The second, labeled here f_0dB, is the frequency where the magnitude of the product | β A_OL | = 1 (in dB, magnitude 1 is 0 dB). That is, frequency f₁₈₀ is determined by the condition:

$\beta A_{OL}\left(f_{180}\right)=-|\beta A_{OL}\left(f_{180}\right)|=-|\beta A_{OL}|_{180}\ ,$

where vertical bars denote the magnitude of a complex number (for example, | a + j b | = [ a² + b²]^1/2 ), and frequency f_0dB is determined by the condition:

$|\beta A_{OL}\left(f_{0dB}\right)|=1\ .$

One measure of proximity to instability is the gain margin. The Bode phase plot locates the frequency where the phase of βA_OL reaches −180°, denoted here as frequency f₁₈₀. Using this frequency, the Bode magnitude plot finds the magnitude of βA_OL. If |βA_OL|₁₈₀ = 1, the amplifier is unstable, as mentioned. If |βA_OL|₁₈₀ < 1, instability does not occur, and the separation in dB of the magnitude of |βA_OL|₁₈₀ from |βA_OL| = 1 is called the gain margin. Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log₁₀( |βA_OL|₁₈₀) = 20 log₁₀( |A_OL|₁₈₀) − 20 log₁₀( 1 / β ).

Another equivalent measure of proximity to instability is the phase margin. The Bode magnitude plot locates the frequency where the magnitude of |βA_OL| reaches unity, denoted here as frequency f_0dB. Using this frequency, the Bode phase plot finds the phase of βA_OL. If the phase of βA_OL( f_0dB) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when f = f₁₈₀), and the distance of the phase at f_0dB in degrees above −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_0dB < f₁₈₀. 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 another method must be used, such as the Nyquist plot.

Figure 6: Gain of feedback amplifier A_FB in dB and corresponding open-loop amplifier A_OL. The gain margin in this amplifier is nearly zero because | βA_OL| = 1 occurs at almost f = f_180°.

Figure 7: Phase of feedback amplifier °A_FB in degrees and corresponding open-loop amplifier °A_OL. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency f = f_0dB where | βA_OL| = 1.

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_OL with the gain with feedback A_FB (the closed-loop gain). See negative feedback amplifier for more detail.

Because the open-loop gain A_OL is plotted and not the product β A_OL, the condition A_OL = 1 / β decides f_0dB. The feedback gain at low frequencies and for large A_OL is A_FB ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain A_OL), so an equivalent way to find f_0dB is to look where the feedback gain intersects the open-loop gain. (Frequency f_0dB is needed later to find the phase margin.)

Near this crossover of the two gains at f_0dB, 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_OL = −1). Beyond the unity gain frequency f_0dB, the open-loop gain is sufficiently small that A_FB ≈ A_OL (examine the formula at the beginning of this section for the case of small A_OL).

Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency f₁₈₀ 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_FB ≈ A_OL for small A_OL.)

Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency f_0dB and the phase-flip frequency f₁₈₀ are very nearly equal in this amplifier, f₁₈₀ ≈ f_0dB ≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.

Figure 8: Gain of feedback amplifier A_FB in dB and corresponding open-loop amplifier A_OL. The gain margin in this amplifier is 19 dB.

Figure 9: Phase of feedback amplifier A_FB in degrees and corresponding open-loop amplifier A_OL. The phase margin in this amplifier is 45°.

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 the condition | β A_OL | = 1 to lower frequency.

Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and A_OL occurs at f_0dB = 1 kHz. Notice that the peak in the gain A_FB near f_0dB is almost gone.

Figure 9 is the phase plot. Using the value of f_0dB = 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at f_0dB is −135°, which is a phase margin of 45° above −180°.

Using Figure 9, for a phase of −180° the value of f₁₈₀ = 3.332 kHz (the same result as found earlier, of course). The open-loop gain from Figure 8 at f₁₈₀ is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.

As an aside, it should be noted that stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good 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. See also the discussion of phase margin in the step response article.