How To Draw The Bode Plot (Sjsu)

The Asymptotic Bode Diagram: Derivation of Approximations

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Introduction

Given an arbitrary transfer role, such equally

$$H(southward)=\frac{100s + 100}{s^2 + 110s + 1000}$$

if you wanted to make a mode plot y'all could calculate the value of H(s) over a range of frequencies (recall s=j·ω for a Bode plot), and plot them.  This is what a reckoner would naturally practise.  For case if you use MATLAB® and enter the commands

>> mySys=tf(100*[ane 1],[1 110 1000]) mySys =      100 s + 100   ------------------   south^2 + 110 s + thou >> bode(mySys)

yous get a plot like the one shown below.  The asymptotic solution is given elsewhere.

Yet, at that place are reasons to develop a method for sketching Bode diagrams manually.  By cartoon the plots past mitt you lot develop an agreement about how the locations of poles and zeros upshot the shape of the plots.  With this knowledge you can predict how a organization behaves in the frequency domain past merely examining its transfer part.  On the other hand, if you know the shape of transfer part that you want, you lot can use your knowledge of Bode diagrams to generate the transfer function.

The first task when drawing a Bode diagram by hand is to rewrite the transfer function so that all the poles and zeros are written in the form (1+s/ω₀).  The reasons for this will go credible when deriving the rules for a existent pole.  A derivation will be washed using the transfer office from in a higher place, but information technology is too possible to do a more generic derivation.  Let's rewrite the transfer role from higher up.

$$\eqalign{ H(s) &= 100{{south + 1} \over {(s + 10)(s + 100)}} = 100{{1 + s/one} \over {ten \cdot (one + southward/ten) \cdot 100 \cdot (1 + southward/100)}} \cr &= 0.1{{ane + south/ane} \over {(1 + s/10)(1 + s/100)}} \cr} $$

Now permit'south examine how nosotros can easily draw the magnitude and phase of this office when due south=j ω.

First note that this expression is made up of 4 terms, a constant (0.ane), a zero (at s=-1), and two poles (at s=-10 and s=-100). Nosotros tin rewrite the function (with due south=jω) every bit four individual phasors (i.e., magnitude and phase), each phasor is within a set of square brackets to make them more easily distinguished from each other..

$$\eqalign{ H(j\omega ) &= 0.one{{i + j\omega /1} \over {(1 + j\omega /x)(1 + j\omega /100)}} \cr &= \left[ {\left| {0.1} \right|\angle \left( {0.1} \correct)} \right]{{\left[ {\left| {1 + j\omega /1} \right|\angle \left( {i + j\omega /1} \correct)} \right]} \over {\left[ {\left| {1 + j\omega /10} \right|\angle \left( {1 + j\omega /ten} \right)} \right]\left[ {\left| {1 + j\omega /100} \right|\angle \left( {1 + j\omega /100} \right)} \right]}} \cr} $$

Nosotros volition prove (below) that cartoon the magnitude and phase of each private phasor is adequately straightforward.  The difficulty lies in trying to describe the magnitude and phase of the more complicated function, H(jω).  To beginning, we volition write H(jω) as a unmarried phasor:

$$\eqalign{ H(j\omega ) &= \left( {\left| {0.1} \correct|{{\left| {1 + j\omega /1} \correct|} \over {\left| {1 + j\omega /10} \right|\left| {1 + j\omega /100} \right|}}} \correct)\left( {\angle \left( {0.one} \right) + \angle \left( {1 + j\omega /1} \right) - \angle \left( {1 + j\omega /10} \right) - \angle \left( {1 + j\omega /100} \right)} \correct) \cr &= \left| {H(j\omega )} \correct|\bending H(j\omega ) \cr \cr \left| {H(j\omega )} \right| &= \left| {0.1} \correct|{{\left| {1 + j\omega /1} \right|} \over {\left| {one + j\omega /10} \correct|\left| {one + j\omega /100} \right|}} \cr \bending H(j\omega ) &= \angle \left( {0.1} \right) + \angle \left( {one + j\omega /i} \right) - \bending \left( {one + j\omega /10} \correct) - \angle \left( {1 + j\omega /100} \right) \cr} $$

Drawing the stage is fairly elementary.  We can draw each phase term separately, and so simply add (or decrease) them.  The magnitude term is non and then straightforward because the magnitude terms are multiplied, it would exist much easier if they were added - then we could depict each term on a graph and just add them. We can accomplish this by usinga logarthmic scale (and then multiplication and sectionalization get addition and subtraction). Instead of a simple logarithm, we will use a deciBel (or dB) scale.

A Magnitude Plot

Ane way to transform multiplication into add-on is by using the logarithm.  Instead of using a uncomplicated logarithm, we will utilize a deciBel (named for Alexander Graham Bell). The human relationship betwixt a quantity, Q, and its deciBel representation, Ten, is given by:

$$X = 20 \cdot log{_{ten}}\left( Q \correct)$$

And then if Q=100 so Ten=twoscore; Q=0.01 gives X=-40; 10=3 gives Q=ane.41; and then on.

If we represent the magnitude of H(s) in deciBels several things happen.

$$\eqalign{ \left| {H(south)} \right| &= \left| {0.1} \right|{{\left| {ane + j\omega /i} \right|} \over {\left| {1 + j\omega /10} \right|\left| {1 + j\omega /100} \right|}} \cr 20 \cdot {\log _{10}}\left( {\left| {H(s)} \right|} \right) &= 20 \cdot {\log _{10}}\left( {\left| {0.1} \right|{{\left| {1 + j\omega /i} \correct|} \over {\left| {1 + j\omega /x} \correct|\left| {1 + j\omega /100} \right|}}} \correct) \cr &= twenty \cdot {\log _{x}}\left( {\left| {0.one} \correct|} \right) + 20 \cdot {\log _{10}}\left( {\left| {ane + j\omega /i} \right|} \right) + 20 \cdot {\log _{ten}}\left( {{1 \over {\left| {1 + j\omega /ten} \right|}}} \right) + 20 \cdot {\log _{10}}\left( {{1 \over {\left| {i + j\omega /100} \right|}}} \right) \cr &= xx \cdot {\log _{x}}\left( {\left| {0.1} \right|} \right) + 20 \cdot {\log _{10}}\left( {\left| {1 + j\omega /i} \right|} \right) - twenty \cdot {\log _{x}}\left( {\left| {1 + j\omega /x} \correct|} \correct) - xx \cdot {\log _{x}}\left( {\left| {i + j\omega /100} \right|} \right) \cr} $$

The advantages of using deciBels (and of writing poles and zeros in the grade (1+s/ω₀)) are now revealed.  The fact that the deciBel is a logarithmic term transforms the multiplications and divisions of the individual terms to additions and subtsractions.  Another benefit is apparent in the terminal line that reveals just two types of terms, a abiding term and terms of the class 20·log10(|1+jω/ω₀|).  Plotting the constant term is trivial, nevertheless the other terms are not and then straightforward.  These plots will be discussed below.  Withal, in one case these plots are drawn for the private terms, they tin can simply be added together to get a plot for H(s).

A Stage Plot

If we await at the phase of the transfer function, we meet much the aforementioned thing: The phase plot is easy to describe if we take our lead from the magnitude plot.  Get-go note that the transfer function is made up of four terms.  If we want

$$\angle H(s) = \angle \left( {0.i} \right) + \angle \left( {one + j\omega /ane} \right) - \angle \left( {1 + j\omega /10} \right) - \angle \left( {one + j\omega /100} \right)$$

Once more there are just two types of terms, a constant term and terms of the form (i+jω/ω ₀).  Plotting the constant term is niggling;  the other terms are discussed beneath.

A more generic derivation

The discussion to a higher place dealt with simply a single transfer role.  Another derivation that is more general, but a little more than complicated mathematically is hither.

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Making a Bode Diagram

Post-obit the word above, the fashion to brand a Bode Diagram is to split the function up into its constituent parts, plot the magnitude and phase of each part, and then add them up.  The following gives a derivation of the plots for each blazon of constituent part.  Examples, including rules for making the plots follow in the next document, which is more of a "How to" description of Bode diagrams.

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A Constant Term

Consider a constant term:$H(south) = H(j\omega) = M$

Magnitude

Clearly the magnitude is constant as ω varies. $\left| {H(j\omega )} \correct| = |K|$

Stage

The phase is too constant.  If Grand is positive, the phase is 0° (or any even multiple of 180°, i.e., ±360°).  If G is negative the stage is -180°, or any odd multiple of 180°.  We will use -180° considering that is what MATLAB® uses.  Expressed in radians we can say that if K is positive the phase is 0 radians, if K is negative the stage is -π radians.

Example: Bode Plot of Gain Term

\[\eqalign{
H(s)& = H\left( {j\omega } \correct) = xv \\ \left| {H\left( {j\omega } \right)} \right| &= \left| 15 \right| = xv =23.5\,dB \\ \angle H\left( {j\omega } \right) &= \bending 15 = 0^\circ } \]

The magnitude (in dB is calculated as $$twenty \cdot {\log _{10}}\left( {15} \right) = 23.five$$.

Key Concept: Bode Plot of Gain Term

• For a constant term, the magnitude plot is a directly line.
• The phase plot is besides a straight line, either at 0° (for a positive constant) or ±180° (for a negative abiding).

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A Real Pole

Consider a simple real pole : $H\left( south \correct) = {ane \over {1 + {south \over {{\omega _0}}}}},\quad \quad H\left( {j\omega } \right) = {one \over {ane + j{\omega \over {{\omega _0}}}}}$

The frequency ω₀ is chosen the intermission frequency, the corner frequency or the 3 dB frequency (more on this last proper name later on). The analysis given beneath assumes ω₀ is positive. For negative ω₀ here.

Magnitude

The magnitude is given by

$$\eqalign{ & \left| {H\left( {j\omega } \right)} \correct| = \left| {{one \over {1 + j{\omega \over {{\omega _0}}}}}} \right| = {ane \over {\sqrt {{1^2} + {{\left( {{\omega \over {{\omega _0}}}} \right)}^two}} }} \cr & {\left| {H\left( {j\omega } \correct)} \right|_{dB}} = 20 \cdot {\log _{10}}\left( {{one \over {\sqrt {ane + {{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} }}} \right) \cr} $$

Allow's consider three cases for the value of the frequency, and determine the magnitude in each example.:

Case 1) ω<<ω₀.  This is the low frequency case with ω/ω₀→0.  Nosotros can write an approximation for the magnitude of the transfer part:

\[\sqrt {1 + {{\left( {{\omega \over {{\omega _0}}}} \correct)}^2}} \approx \textrm{1, and }{\left| {H\left( {j\omega } \correct)} \correct|_{dB}} \approx twenty \cdot {\log _{10}}\left( {{1 \over 1}} \right) = 0\]

This low frequency approximation is shown in bluish on the diagram below.

Case 2) ω>>ω₀.  This is the loftier frequency case with ω/ω₀→∞. We tin can write an approximation for the magnitude of the transfer office:

$\sqrt {1 + {{\left( {{\omega \over {{\omega _0}}}} \correct)}^2}} \approx \sqrt {{{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} \approx {\omega \over {{\omega _0}}}$, and so
${\left| {H\left( {j\omega } \right)} \correct|_{dB}} \approx 20 \cdot {\log _{10}}\left( {{{{\omega _0}} \over \omega }} \right)$

The high frequency approximation is at shown in green on the diagram beneath.  Information technology is a direct line with a slope of -20 dB/decade going through the break frequency at 0 dB (if ω=ω₀ the approximation simplifies to 0 dB; ω=ten·ω₀ gives an judge gain of 0.1, or -xx dB and so on). That is, the approximation goes through 0 dB at ω=ω₀, and for every factor of 10 increase in frequency, the magnitude drops by 20 dB..

Case 3) ω=ω₀. At the pause frequency

\[{\left| {H\left( {j{\omega _0}} \right)} \correct|_{dB}} = 20 \cdot {\log _{ten}}\left( {{i \over {\sqrt {1 + {{\left( {{{{\omega _0}} \over {{\omega _0}}}} \right)}^2}} }}} \correct) = twenty \cdot {\log _{10}}\left( {{i \over {\sqrt 2 }}} \right) \approx - 3\;dB\]

This point is shown as a red circle on the diagram.

To describe a piecewise linear approximation, utilise the depression frequency asymptote up to the intermission frequency, and the high frequency asymptote thereafter.

The resulting asymptotic approximation is shown highlighted in transparent magenta.  The maximum error betwixt the asymptotic approximation and the exact magnitude office occurs at the suspension frequency and is approximately -iii dB.

Magnitude of a existent pole: The piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the suspension frequency and then drops at xx dB per decade every bit frequency increases (i.e., the slope is -twenty dB/decade).

Stage

The stage of a single real pole is given by is given by

$$\angle H\left( {j\omega } \right) = \bending \left( {{1 \over {1 + j{\omega \over {{\omega _0}}}}}} \right) = - \bending \left( {i + j{\omega \over {{\omega _0}}}} \right) = - \arctan \left( {{\omega \over {{\omega _0}}}} \right)$$

Permit u.s. again consider three cases for the value of the frequency:

Example one) ω<<ω₀.  This is the low frequency case with ω/ω₀→0.  At these frequencies Nosotros can write an approximation for the phase of the transfer role

$$\angle H\left( {j\omega } \right) \approx -\arctan \left( 0 \correct) = 0^\circ = 0\;rad$$

The low frequency approximation is shown in blue on the diagram below.

Instance ii) ω>>ω₀.  This is the high frequency case with ω/ω₀→∞.  We can write an approximation for the phase of the transfer function

$$\bending H\left( {j\omega } \correct) \approx - \arctan \left( \infty \right) = - 90^\circ = - {\pi \over 2}\;rad$$

The high frequency approximation is at shown in green on the diagram below.  It is a horizontal line at -90°.

Example 3) ω=ω₀.  The break frequency.  At this frequency

$$\angle H\left( {j\omega } \right) = - \arctan \left( one \correct) = - 45^\circ = - {\pi \over 4}\;rad$$

This indicate is shown every bit a ruby-red circle on the diagram.

A piecewise linear approximation is not every bit like shooting fish in a barrel in this case because the high and low frequency asymptotes don't intersect.  Instead we apply a rule that follows the exact part fairly closely, just is also somewhat arbitrary.  Its main advantage is that it is easy to recollect.

Stage of a real pole: The piecewise linear asymptotic Bode plot for phase follows the low frequency asymptote at 0° until ane 10th the intermission frequency (0.i·ω₀) then decrease linearly to run across the loftier frequency asymptote at 10 times the break frequency (10·ω₀). This line is shown above.  Note that there is no error at the break frequency and about 5.7° of error at 0.1·ω₀ and x·ω₀ the intermission frequency.

Example: Existent Pole

The first instance is a simple pole at v radians per second.  The asymptotic approximation is magenta, the exact function is a dotted black line.

$$H(s)=\frac{ane}{one+\frac{s}{5}}$$

Example:  Repeated Real Pole

The 2nd example shows a double pole at 30 radians per 2d.  Note that the gradient of the asymptote is -40 dB/decade and the phase goes from 0 to -180°. The effect of repeating a pole is to double the gradient of the magnitude to -xl dB/decade and the gradient of the phase to -ninety°/decade.

$$H(s)=\frac{1}{\left(1+\frac{s}{30}\right)^two}$$

Cardinal Concept: Bode Plot for Real Pole

• For a simple real pole the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the intermission frequency and then drops at 20 dB per decade (i.e., the gradient is -20 dB/decade).   An northward^th social club pole has a slope of -xx·northward dB/decade.
• The stage plot is at 0° until one tenth the interruption frequency and and then drops linearly to -ninety° at 10 times the break frequency.  An due north^th order pole drops to -90°·north.

The analysis given above assumes ω₀ is positive. For negative ω₀ here.

Aside: a different formulation of the phase approximation

There is another approximation for phase that is occasionally used. The approximation is adult by matching the slope of the actual stage term to that of the approximation at ω=ω₀. Using math similar to that given here (for the underdamped case) it can be shown that by cartoon a line starting at 0° at ω=ω₀/e^π/ii=ω₀/iv.81 (or ω₀·e^-π/2) to -90° at ω=ω₀·4.81 nosotros get a line with the same gradient as the actual function at ω=ω₀. The approximation described previously is much more commonly used every bit is easier to remember every bit a line fatigued from 0° at ω₀/five to -ninety° at ω₀·5, and easier to draw on semi-log paper. The latter is shown on the diagram below.

Although this method is more accurate in the region around ω=ω₀ in that location is a larger maximum error (more than x°) about ω₀/five and ω₀·5 when compared to the method described previously.

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A Existent Zip

The piecewise linear approximation for a zero is much like that for a pole  Consider a simple nil: $H(due south)=1+\frac{s}{\omega_0},\quad H(j\omega)=i+j\frac{\omega}{\omega_0}$.

Magnitude

The development of the magnitude plot for a nothing follows that for a pole.  Refer to the previous department for details.  The magnitude of the zero is given by

$$\left| {H\left( {j\omega } \right)} \right| = \left| {1 + j{\omega \over {{\omega _0}}}} \right|$$

Again, every bit with the case of the real pole, in that location are three cases:

1. At  depression frequencies, ω<<ω₀, the gain is approximately 1 (or 0 dB).
2. At high frequencies, ω>>ω₀, the gain increases at 20 dB/decade and goes through the break frequency at 0 dB.
3. At the break frequency, ω=ω₀, the proceeds is well-nigh 3 dB.

Magnitude of a Real Nix: For a simple real zero the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency so increases at xx dB per decade (i.e., the slope is +twenty dB/decade).

Phase

The stage of a simple zero is given by:

$$\angle H\left( {j\omega } \right) = \angle \left( {1 + j{\omega \over {{\omega _0}}}} \right) = \arctan \left( {{\omega \over {{\omega _0}}}} \right)$$

The phase of a single real zero also has three cases (which tin be derived similarly to those for the real pole, given to a higher place):

1. At  low frequencies, ω<<ω₀, the phase is approximately zero.
2. At loftier frequencies, ω>>ω₀, the phase is +90°.
3. At the break frequency, ω=ω₀, the stage is +45°.

Phase of a Real Naught: Follow the low frequency asymptote at 0° until one tenth the pause frequency (0.1 ω₀) and then increase linearly to encounter the high frequency asymptote at x times the break frequency (10 ω₀).

Example: Real Zero

This case shows a simple zero at 30 radians per 2nd.  The asymptotic approximation is magenta, the exact function is the dotted black line.

$$H(s)=1+\frac{southward}{30}$$

Key Concept: Bode Plot of Real Zero:

• The plots for a real zero are like those for the real pole but mirrored near 0dB or 0°.
• For a unproblematic real zero the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the intermission frequency so rises at +twenty dB per decade (i.e., the slope is +twenty dB/decade).   An n ^th society zero has a slope of +20·northward dB/decade.
• The stage plot is at 0° until 1 tenth the pause frequency and and then rises linearly to +90° at 10 times the break frequency.  An northward^thursday order zero rises to +90°·n.

The analysis given above assumes the ω₀ is positive. For negative ω₀ here.

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A Pole at the Origin

A pole at the origin is easily drawn exactly.  Consider

$$H\left( s \correct) = {one \over s},\quad H\left( {j\omega } \correct) = {1 \over {j\omega }} = - {j \over \omega }$$

Magnitude

The magnitude is given by

$$\eqalign{
\left| {H\left( {j\omega } \correct)} \right| &= \left| { - {j \over \omega }} \right| = {1 \over \omega } \cr
{\left| {H\left( {j\omega } \correct)} \correct|_{dB}} &= 20\cdot{\log _{x}}\left( {{i \over \omega }} \right) = - xx\cdot{\log _{10}}\left( \omega \right) \cr} $$

In this case there is no need for approximate functions and asymptotes, we can plot the exact funtion. The function is represented by a straight line on a Bode plot with a slope of -20 dB per decade and going through 0 dB at ane rad/ sec.  Information technology also goes through 20 dB at 0.1 rad/sec, -twenty dB at 10 rad/sec... Since there are no parameters (i.eastward., ω₀) associated with this part, information technology is always drawn in exactly the same mode.

Magnitude of Pole at the Origin: Draw a line with a slope of -twenty dB/decade that goes through 0 dB at 1 rad/sec.

Phase

The stage of a simple zip is given past (H(jω) is a negative imaginary number for all values of ω so the stage is always -90°):

$$\bending H\left( {j\omega } \right) = \angle \left( { - {j \over \omega }} \right) = - 90^\circ $$

Stage of pole at the origin: The phase for a pole at the origin is -90°.

Instance: Pole at Origin

This case shows a simple pole at the origin.  The exact (dotted black line) is the same as the approximation (magenta).

Fundamental Concept: Bode Plot for Pole at Origin

No interactive demo is provided because the plots are always drawn in the aforementioned way.

• For a simple pole at the origin depict a straight line with a slope of -20 dB per decade and going through 0 dB at i rad/ sec.
• The phase plot is at -ninety°.
• The magnitude of an n^th order pole has a gradient of -20·n dB/decade and a abiding stage of -90°·north.

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A Zero at the Origin

A zero at the origin is just similar a pole at the origin but the magnitude increases with increasing ω, and the phase is +90° (i.e. only mirror the graphs for the pole around the origin around 0dB or 0°).

Instance: Goose egg at Origin

This instance shows a unproblematic goose egg at the origin.  The exact (dotted black line) is the same as the approximation (magenta).

Key Concept: Bode Plot for Zero at Origin

• The plots for a zero at the origin are similar those for the pole but mirrored about 0dB or 0°.
• For a elementary null at the origin draw a direct line with a slope of +twenty dB per decade and going through 0 dB at 1 rad/ sec.
• The stage plot is at +xc°.
• The magnitude of an n^th order aught has a slope of +xx·n dB/decade and a constant stage of +90°·due north.

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A Complex Conjugate Pair of Poles

The magnitude and stage plots of a complex conjugate (underdamped) pair of poles is more than complicated than those for a simple pole.  Consider the transfer role (with 0<ζ<1):

$$H(s) = {{\omega _0^2} \over {{southward^2} + ii\zeta {\omega _0}due south + \omega _0^ii}} = {1 \over {{{\left( {{s \over {{\omega _0}}}} \correct)}^two} + ii\zeta \left( {{s \over {{\omega _0}}}} \right) + 1}}$$

The analysis given below assumes the ζ is positive. For negative ζ come across here.

Magnitude

The magnitude is given past

$$\eqalign{
\left| {H(j\omega )} \right| &= \left| {{ane \over {{{\left( {{{j\omega } \over {{\omega _0}}}} \correct)}^2} + 2\zeta \left( {{{j\omega } \over {{\omega _0}}}} \right) + i}}} \correct| = \left| {{one \over { - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2} + j2\zeta \left( {{\omega \over {{\omega _0}}}} \right) + 1}}} \right| = \left| {{1 \over {\left( {1 - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} \right) + j\left( {2\zeta \left( {{\omega \over {{\omega _0}}}} \right)} \right)}}} \right| \cr
&= {1 \over {\sqrt {{{\left( {i - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} \right)}^2} + {{\left( {2\zeta {\omega \over {{\omega _0}}}} \right)}^ii}} }} \cr
{\left| {H(j\omega )} \right|_{dB}} &= - 20 \cdot {\log _{10}}\left( {\sqrt {{{\left( {one - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} \right)}^two} + {{\left( {ii\zeta {\omega \over {{\omega _0}}}} \right)}^2}} } \right) \cr} $$

As before, allow's consider three cases for the value of the frequency:

Case 1) ω<<ω₀.  This is the low frequency case.  We can write an approximation for the magnitude of the transfer function

$${\left| {H(j\omega )} \right|_{dB}} = - xx \cdot {\log _{ten}}\left( i \right) = 0$$

The low frequency approximation is shown in red on the diagram below.

Case 2) ω>>ω₀.  This is the high frequency case.  We tin write an approximation for the magnitude of the transfer function

$${\left| {H(j\omega )} \right|_{dB}} = - 20 \cdot {\log _{10}}\left( {{{\left( {{\omega \over {{\omega _0}}}} \right)}^2}} \right) = - 40 \cdot {\log _{10}}\left( {{\omega \over {{\omega _0}}}} \right)$$

The high frequency approximation is at shown in green on the diagram below.  It is a straight line with a slope of -40 dB/decade going through the break frequency at 0 dB.  That is, for every cistron of x increase in frequency, the magnitude drops past xl dB.

Case 3) ω≈ω₀.  It can be shown that a peak occurs in the magnitude plot well-nigh the suspension frequency.  The derivation of the approximate amplitude and location of the peak are given here.   Nosotros make the approximation that a tiptop exists only when

0<ζ<0.v

and that the peak occurs at ω₀ with elevation 1/(2·ζ).

To draw a piecewise linear approximation, utilize the low frequency asymptote up to the break frequency, and the loftier frequency asymptote thereafter.  If ζ<0.5, and so depict a meridian of aamplitude 1/(2·ζ)  Describe a smooth bend between the low and high frequency asymptote that goes through the top value.

As an example for the curve shown below ω₀=10, ζ=0.i,

$$H(s) = {1 \over {{{{southward^2}} \over {100}} + 0.02\zeta s + 1}} = {ane \over {{{\left( {{southward \over {10}}} \right)}^2} + 0.2\left( {{s \over {ten}}} \right) + 1}} = {1 \over {{{\left( {{s \over {{\omega _0}}}} \right)}^2} + 2\zeta \left( {{s \over {{\omega _0}}}} \right) + 1}}$$

The peak will have an aamplitude of ane/(2·ζ)=5.00 or xiv dB.

The resulting asymptotic approximation is shown as a black dotted line, the exact response is a black solid line.

Magnitude of Underdamped (Complex) poles: Draw a 0 dB at depression frequencies until the break frequency, ω₀, then drops with a gradient of -twoscore dB/decade. If ζ<0.five we draw a peak of height at ω₀, otherwise no peak is fatigued.

$$\left| {H(j{\omega _0})} \right| \approx {one \over {2\zeta }},\quad {\left| {H(j{\omega _0})} \right|_{dB}} \approx - 20 \cdot {\log _{10}}\left( {2\zeta } \right)$$

Notation: The actual peak of the superlative and its frequency are both slightly less than the approximations given above. An in depth give-and-take of the magnitude and phase approximations (along with some alternating approximations) are given here.

Stage

The phase of a complex conjugate pole is given by is given by

$$\eqalign{
\angle H(j\omega ) &= \angle \left( {{1 \over {{{\left( {{{j\omega } \over {{\omega _0}}}} \right)}^two} + ii\zeta \left( {{{j\omega } \over {{\omega _0}}}} \correct) + 1}}} \right) = - \bending \left( {{{\left( {{{j\omega } \over {{\omega _0}}}} \right)}^2} + 2\zeta \left( {{{j\omega } \over {{\omega _0}}}} \right) + i} \correct) = - \bending \left( {i - {{\left( {{\omega \over {{\omega _0}}}} \correct)}^two} + ii\zeta \left( {{{j\omega } \over {{\omega _0}}}} \right)} \right) \cr
&= - \arctan \left( {{{2\zeta {\omega \over {{\omega _0}}}} \over {1 - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2}}}} \correct) \cr} $$

Let us again consider three cases for the value of the frequency:

Case i) ω<<ω₀.  This is the low frequency case.  At these frequencies We tin can write an approximation for the phase of the transfer function

$$\angle H\left( {j\omega } \right) \approx -\arctan \left({{2 \zeta \omega \over {\omega _0}}} \right) \approx -\arctan(0) = 0^\circ = 0\;rad$$

The depression frequency approximation is shown in red on the diagram below.

Case 2) ω>>ω₀.  This is the high frequency case.  We tin can write an approximation for the stage of the transfer function

$$\bending H\left( {j\omega } \right) \approx - 180^\circ = -\pi\;rad$$

Note: this event makes use of the fact that the arctan part returns a result in quadrant 2 since the imaginary part of H(j&omega;) is negative and the real role is positive.

The loftier frequency approximation is at shown in green on the diagram below.  Information technology is a directly line at -180°.

Case three) ω=ω₀.  The pause frequency.  At this frequency

$$\bending H(j\omega_0 ) = - xc^\circ $$

The asymptotic approximation is shown below for ω₀=10, ζ=0.1, followed by an caption

$$H(south) = {1 \over {{{{s^2}} \over {100}} + 0.02\zeta due south + 1}} = {1 \over {{{\left( {{due south \over {x}}} \right)}^2} + 0.ii\left( {{southward \over {10}}} \right) + one}} = {ane \over {{{\left( {{s \over {{\omega _0}}}} \correct)}^2} + two\zeta \left( {{south \over {{\omega _0}}}} \right) + 1}}$$

A piecewise linear approximation is a bit more complicated in this case, and there are no difficult and fast rules for cartoon information technology.  The most mutual style is to look up a graph in a textbook with a chart that shows phase plots for many values of ζ.  3 asymptotic approximations are given hither.  We will use the approximation that connects the the low frequency asymptote to the high frequency asymptote starting at

$$\omega = {{{\omega _0}} \over {{{ten}^\zeta }}} = {\omega _0} \cdot {x^{ - \zeta }}$$

and catastrophe at

$$\omega = {\omega _0} \cdot {10^\zeta }$$

Since ζ=0.ii in this case this means that the stage starts at 0° and so breaks downward at ω=ω₀/x^ζ=7.9 rad/sec. The stage reaches -180° at ω=ω₀·x^ζ=12.6 rad/sec.

As a applied matter If ζ<0.02, the approximation can be simply a vertical line at the intermission frequency. One advantage of this approximation is that it is very easy to plot on semilog paper. Since the number 10·ω₀ moves upwards by a total decade from ω₀, the number ten^ζ·ω₀ will be a fraction ζ of a decade above ω₀. For the example to a higher place the corner frequencies for ζ=0.one fall about ω₀ i tenth of the way between ω₀ and ω₀/10 (at the lower break frequency) to i tenth of the way between ω₀ and ω₀·10 (at the higher frequency).

Phase of Underdamped (Complex) Poles: Follow the low frequency asymptote at 0° until

$$\omega = {{{\omega _0}} \over {{{ten}^\zeta }}}$$

and so decrease linearly to meet the loftier frequency asymptote at -180° at

$$\omega = {\omega _0} \cdot {10^\zeta }$$

Other magnitude and phase approximations (along with exact expressions) are given hither.

Fundamental Concept: Bode Plot for Complex Conjugate Poles

• For the magnitude plot of circuitous conjugate poles draw a 0 dB at depression frequencies, go through a peak of acme,

  $$\left| {H(j{\omega _0})} \right| \approx {1 \over {two\zeta }},\quad {\left| {H(j{\omega _0})} \right|_{dB}} \approx - 20 \cdot {\log _{10}}\left( {2\zeta } \right)$$

  at the pause frequency and then drop at 40 dB per decade (i.eastward., the slope is -40 dB/decade).  The loftier frequency asymptote goes through the break frequency.  Notation that in this approximation the superlative only exists for

  0 < ζ < 0.v

• To draw the phase plot just follow low frequency asymptote at 0° until

  $$\omega = {{{\omega _0}} \over {{{10}^\zeta }}} = {\omega _0} \cdot {10^{ - \zeta }}$$

  then decrease linearly to run into the high frequency asymptote at -180° at

  $$\omega = {\omega _0} \cdot {10^\zeta }$$

  If ζ<0.02, the approximation can be simply a vertical line at the interruption frequency.
• Note that the shape of the graphs (magnitude peak height, steepness of phase transition) are determined solely by ζ, and the frequency at which the magnitude elevation and stage transition occur are determined solely past ω₀.
  

Note: Other magnitude and phase approximations (along with exact expressions) are given here.
The analysis given in a higher place assumes the ζ is positive. For negative ζ see here

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A Complex Conjugate Pair of Zeros

Not surprisingly a complex pair of zeros yields results similar to that for a circuitous pair of poles.  The magnitude and phase plots for the complex zero are the mirror epitome (effectually 0dB for magnitude and around 0° for phase) of those for the circuitous pole. Therefore, the magnitude has a dip instead of a summit, the magnitude increases to a higher place the break frequency and the phase increases rather than decreasing. The results will not be derived here, but closely follow those for circuitous poles.

Note: The analysis given below assumes the ζ is positive. For negative ζ run across here

Instance: Complex Conjugate Zero

The graph below corresponds to a complex cohabit zilch with ω₀=3, ζ=0.25

$$H\left( s \right) = {\left( {{southward \over {{\omega _0}}}} \right)^2} + ii\zeta \left( {{southward \over {{\omega _0}}}} \right) + i$$

The dip in the magnitude plot will have a magnitude of 0.v or -6 dB. The break frequencies for the phase are at ω=ω₀/10^ζ=1.7 rad/sec and ω=ω₀·x^ζ=v.3 rad/sec.

Key Concept: Bode Plot of Complex Cohabit Zeros

• The plots for a circuitous conjugate pair of zeros are very much like those for the poles but mirrored about 0dB or 0°.
• For the magnitude plot of complex conjugate zeros draw a 0 dB at depression frequencies, go through a dip of magnitude:

  $$\left| {H(j{\omega _0})} \right| \approx {2\zeta},\quad {\left| {H(j{\omega _0})} \correct|_{dB}} \approx xx \cdot {\log _{10}}\left( {2\zeta } \right)$$

  at the break frequency and then rise at +xl dB per decade (i.e., the slope is +forty dB/decade).  The high frequency asymptote goes through the break frequency.  Note that the tiptop only exists for

  0 < ζ < 0.five

• To draw the phase plot simply follow depression frequency asymptote at 0° until

  $$\omega = {{{\omega _0}} \over {{{10}^\zeta }}} = {\omega _0} \cdot {10^{ - \zeta }}$$

  then increase linearly to meet the high frequency asymptote at 180° at

  $$\omega = {\omega _0} \cdot {10^\zeta }$$

• Note that the shape of the graphs (magnitude peak height, steepness of stage transition) are determined solely by ζ, and the frequency at which the magnitude peak and phase transition occur are determined solely past ω₀.
  

Note: Other magnitude and phase approximations (along with exact expressions) are given here.
The assay given below assumes the ζ is positive. For negative ζ encounter here.

---

Non-Minimum Phase Systems

All of the examples above are for minimum phase systems. These systems take poles and zeros that practice not have positive real parts. For example the term (due south+2) is zero when southward=-2, so information technology has a negative real root. Outset order poles and zeros have negative real roots if ω₀ is positive. Second order poles and zeros have negative existent roots if ζ is positive. The magnitude plots for these systems remain unchanged, but the phase plots are inverted. See here for discussion.

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Interactive Demos:

Below you lot volition find interactive demos that prove how to describe the asymptotic approximation for a abiding, a first lodge pole and naught, and a 2d order (underdamped) pole and aught. Note there is no demo for a pole or zero at the origin because these are e'er drawn in exactly the same manner; there are no variable parameters (i.e., ω₀ or ζ).

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Interactive Demo: Bode Plot of Constant Term

This demonstration shows how the gain term affects a Bode plot. To run the sit-in either enter the value of One thousand, or |K| expressed in dB, in one of the text boxes below. If you lot enter |K| in dB, then the sign of K is unchanged from its electric current value. You can also prepare |Thousand| and ∠Grand by either clicking and dragging the horizontal lines on the graphs themselve. The magnitude of Thousand must be between 0.01 and 100 (-40dB and +40dB). The stage of Grand (∠K) can but be 0° (for a positive value of K) or ±180° (for negative K).

Enter a value for proceeds, 1000: ,

or enter |K| expressed in dB: dB.

Notation that for the instance of a constant term, the approximate (magenta line) and verbal (dotted black line) representations of magnitude and phase are equal.

---

Interactive Demo: Bode Plot of a Real Pole

This sit-in shows how a showtime guild pole expressed every bit:

\[H(south) = \frac{i}{1+\frac{south}{\omega_0}}=\frac{ane}{1+j\frac{\omega}{\omega_0}},\]

is displayed on a Bode plot. To alter the value of ω₀, you can either change the value in the text box, below, or drag the vertical line showing ω₀ on the graphs to the right. The exact values of magnitude and phase are shown as black dotted lines and the asymptotic approximations are shown with a thick magenta line. The value of ω₀ is constrained such that 0.one≤ω₀≤10 rad/second.

Enter a value for ω_o:

Asymptotic Magnitude: The asymptotic magnitude plot starts (at low frequencies) at 0 dB and stays at that level until it gets to ω₀. At that point the gain starts dropping with a slope of -20 dB/decade.

Asymptotic Stage: The asymptotic stage plot starts (at low frequencies) at 0° and stays at that level until information technology gets to 0.1·ω₀ (0.ane rad/sec). At that point the stage starts dropping at -45°/decade until it gets to -90° at x·ω₀ (10  rad/sec), at which point it becomes constant at -90° for high frequencies. Phase goes through -45° at ω=ω₀.

---

Interactive Demo: Bode Plot of a Real zero

This demonstration shows how a showtime order goose egg expressed as:

\[H(s) = 1+\frac{south}{\omega_0}= 1+j\frac{\omega}{\omega_0},\]

is displayed on a Bode plot. To modify the value of ω₀, you can either change the value in the text box, below, or drag the vertical line showing ω₀ on the graphs to the right. The verbal values of magnitude and phase are shown every bit black dotted lines and the asymptotic approximations are shown with a thick magenta line. The value of ω₀ is constrained such that 0.one≤ω₀≤10 rad/second.

Enter a value for ω_o:

Asymptotic Magnitude: The asymptotic magnitude plot starts (at low frequencies) at 0 dB and stays at that level until it gets to ω₀. At that point the proceeds starts rising with a slope of +20 dB/decade.

Asymptotic Magnitude: The asymptotic stage plot starts (at low frequencies) at 0° and stays at that level until it gets to 0.one·ω₀ . At that bespeak the phase starts ascension at +45°/decade until it gets to +xc° at 10·ω₀, at which point information technology becomes constant at +xc° for high frequencies. Phase goes through +45° at ω=ω₀.

---

Interactive Demo: Bode Plot of a Pair of Complex Conjugate Poles

This demonstration shows how a second order pole (complex cohabit roots) expressed every bit:

\[H(s)={one \over {{{\left( {{southward \over {{\omega _0}}}} \correct)}^2} + 2\zeta \left( {{due south \over {{\omega _0}}}} \right) + i}} = {1 \over {1 - {{\left( {{\omega \over {{\omega _0}}}} \correct)}^ii} + j2\zeta {\omega \over {{\omega _0}}}}},\]

is displayed on a Bode plot. You tin can modify ω₀, and ζ. The value of ω₀ is constrained such that 0.one≤ω₀≤10 rad/second, and 0.05≤ζ≤0.99.

Enter value for ω_o: or click and drag on graph to set up ω₀.,

and utilise text-box or slider, below, for ζ.

Asymptotic Magnitude: The asymptotic magnitude plot starts (at low frequencies) at 0 dB and stays at that level until information technology gets to ω₀. At that signal the gain starts dropping with a slope of -40 dB/decade. Note: information technology is -xl dB per decade because there are 2 poles in the denominator.
If ζ<0.five we gauge the superlative height equally $|H(j\omega_{height})|\approx\frac{1}{2\zeta}$ (exact tiptop is $|H(j\omega_{peak})|=\frac{i}{ii\zeta \sqrt{i-\zeta^2}}$). We approximate the peak location at $\omega_{elevation}\approx\omega_0$ (exact peak location is at $\omega_{superlative}=\omega_0\sqrt{1-2\zeta^2}$ ).
All the same, if ζ≥0.five, the peak is sufficiently small-scale that we don't include it in our plot.

Since ζ≥0.five, we do not depict a peak.

Since ζ<0.5, nosotros draw a peak. Annotation how close together the estimate and verbal values are for ω_superlative and |H(jω_summit)|.

	ωpeak	|H(jωpeak)|	|H(jωpeak)|dB
Approximate
Exact

Asymptotic Phase: The asymptotic phase plot starts (at depression frequencies) at 0° and stays at that level until information technology gets to ω₀/10^ζ. At that betoken the phase starts dropping at -ninety°/decade until it gets to -180°, at which point it becomes abiding at -180° for high frequencies. Phase goes through -90° at ω=ω₀. If ζ<0.02 the phase transition betwixt 0 and -180° can be approximated by a vertical line.

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Interactive Demo: Bode Plot of a Pair of Complex Conjugate Zeros

This demonstration shows how a 2nd society zero (complex conjugate roots) expressed as:

\[H(s)= {{\left( {{due south \over {{\omega _0}}}} \right)}^2} + 2\zeta \left( {{south \over {{\omega _0}}}} \correct) + i = 1 - {{\left( {{\omega \over {{\omega _0}}}} \right)}^2} + j2\zeta {\omega \over {{\omega _0}}},\]

is displayed on a Bode plot. You can alter ω₀, and ζ. The value of ω₀ is constrained such that 0.i≤ω₀≤10 rad/second, and 0.05≤ζ≤0.99.

Enter value for ω_o: or click and drag on graph to set ω₀.,

and use text-box or slider, below, for ζ.

Asymptotic Magnitude: The asymptotic magnitude plot starts (at low frequencies) at 0 dB and stays at that level until information technology gets to ω₀. At that point the proceeds starts ascension with a gradient of +twoscore dB/decade.
If ζ<0.five we estimate the peak elevation every bit $|H(j\omega_{superlative})|\approx\frac{1}{2\zeta}$ (exact peak is $|H(j\omega_{peak})|=\frac{ane}{2\zeta \sqrt{1-\zeta^2}}$). We approximate the peak location at $\omega_{peak}\approx\omega_0$ (exact top location is at $\omega_{peak}=\omega_0\sqrt{1-2\zeta^2}$ ).
Notwithstanding, if ζ≥0.five, the height is sufficiently small that we don't include it in our plot.

Since ζ≥0.five, so we do non describe a tiptop.

Since ζ<0.5, we draw a elevation. Annotation how close together the gauge and verbal values are for ω_pinnacle and |H(jω_summit)|.

	ωsuperlative	|H(jωsummit)|	|H(jωpeak)|dB
Approximate
Exact

Asymptotic Phase: The asymptotic stage plot starts (at depression frequencies) at 0° and stays at that level until it gets to ω₀/10^ζ. At that point the phase starts ascension at +90°/decade until it gets to +180°, at which point it becomes constant at +180° for loftier frequencies. Phase goes through +90° at ω=ω₀. If ζ<0.02 the phase transition between 0 and +180° can be approximated past a vertical line.

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Brief review of page:   This document derived piecewise linear approximations that can exist used to depict different elements of a Bode diagram.  A synopsis of these rules tin be institute in a separate document.

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References

Replace

Source: https://lpsa.swarthmore.edu/Bode/BodeHow.html

Posted by: middletonupostink.blogspot.com

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