The Asymptotic Bode Diagram: Derivation of Approximations

A Real Zero

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

Magnitude

The development of the magnitude plot for a zero follows that for a pole.  Refer to the previous section 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, as with the case of the real pole, there are three cases:

1. At  low 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 gain is about 3 dB.

Magnitude of a Real Zero: For a simple real zero the piecewise linear asymptotic Bode plot for magnitude is at 0 dB until the break frequency and then increases at 20 dB per decade (i.e., the slope is +20 dB/decade).

Phase

The phase 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 can be derived similarly to those for the real pole, given above):

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

Phase of a Real Zero: Follow the low frequency asymptote at 0° until one tenth the break frequency (0.1 ω₀) then increase linearly to meet the high frequency asymptote at ten times the break frequency (10 ω₀).

A Pole at the Origin

A pole at the origin is easily drawn exactly.  Consider

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

Magnitude

The magnitude is given by

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

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 1 rad/ sec.  It also goes through 20 dB at 0.1 rad/sec, -20 dB at 10 rad/sec... Since there are no parameters (i.e., ω₀) associated with this function, it is always drawn in exactly the same manner.

Magnitude of Pole at the Origin: Draw a line with a slope of -20 dB/decade that goes through 0 dB at 1 rad/sec.

Phase

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

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

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

A Zero at the Origin

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

A Complex Conjugate Pair of Poles

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

$$H(s) = {{\omega _0^2} \over {{s^2} + 2\zeta {\omega _0}s + \omega _0^2}} = {1 \over {{{\left( {{s \over {{\omega _0}}}} \right)}^2} + 2\zeta \left( {{s \over {{\omega _0}}}} \right) + 1}}$$

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

Magnitude

The magnitude is given by

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

As before, let'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

$${\left| {H(j\omega )} \right|_{dB}} = - 20 \cdot {\log _{10}}\left( 1 \right) = 0$$

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

Case 2) ω>>ω₀.  This is the high frequency case.  We can 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 factor of 10 increase in frequency, the magnitude drops by 40 dB.

Case 3) ω≈ω₀.  It can be shown that a peak occurs in the magnitude plot near the break frequency.  The derivation of the approximate amplitude and location of the peak are given here.   We make the approximation that a peak exists only when

0<ζ<0.5

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

To draw a piecewise linear approximation, use the low frequency asymptote up to the break frequency, and the high frequency asymptote thereafter.  If ζ<0.5, then draw a peak of amplitude 1/(2·ζ)  Draw a smooth curve between the low and high frequency asymptote that goes through the peak value.

As an example for the curve shown below ω₀=10, ζ=0.1,

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

The peak will have an amplitude of 1/(2·ζ)=5.00 or 14 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 low frequencies until the break frequency, ω₀, and then drops with a slope of -40 dB/decade. If ζ<0.5 we draw a peak of height at ω₀, otherwise no peak is drawn.

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

Note: The actual height of the peak and its frequency are both slightly less than the approximations given above. An in depth discussion of the magnitude and phase approximations (along with some alternate approximations) are given here.

Phase

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

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

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

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

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

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

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

The high frequency approximation is at shown in green on the diagram below.  It is a straight line at -180°.

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

$$\angle H(j\omega_0 ) = - 90^\circ $$

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

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

A piecewise linear approximation is a bit more complicated in this case, and there are no hard and fast rules for drawing it.  The most common way is to look up a graph in a textbook with a chart that shows phase plots for many values of ζ.  Three asymptotic approximations are given here.  We will use the approximation that connects the the low frequency asymptote to the high frequency asymptote starting at

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

and ending at

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

Since ζ=0.2 in this case this means that the phase starts at 0° and then breaks downward at ω=ω₀/10^ζ=7.9 rad/sec. The phase reaches -180° at ω=ω₀·10^ζ=12.6 rad/sec.

As a practical matter If ζ<0.02, the approximation can be simply a vertical line at the break frequency. One advantage of this approximation is that it is very easy to plot on semilog paper. Since the number 10·ω₀ moves up by a full decade from ω₀, the number 10^ζ·ω₀ will be a fraction ζ of a decade above ω₀. For the example above the corner frequencies for ζ=0.1 fall near ω₀ one tenth of the way between ω₀ and ω₀/10 (at the lower break frequency) to one 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

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

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

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

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

A Complex Conjugate Pair of Zeros

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

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

Non-Minimum Phase Systems

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

Interactive Demos:

Below you will find interactive demos that show how to draw the asymptotic approximation for a constant, a first order pole and zero, and a second order (underdamped) pole and zero. Note there is no demo for a pole or zero at the origin because these are always drawn in exactly the same way; there are no variable parameters (i.e., ω₀ or ζ).

Interactive Demo: Bode Plot of Constant Term

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

Enter a value for gain, K: ,

or enter |K| expressed in dB: dB.

Note that for the case of a constant term, the approximate (magenta line) and exact (dotted black line) representations of magnitude and phase are equal.

Interactive Demo: Bode Plot of a Real Pole

This demonstration shows how a first order pole expressed as:

is displayed on a Bode plot. To change 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 0.1≤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 Phase:  The asymptotic phase plot starts (at low frequencies) at 0° and stays at that level until it gets to 0.1·ω₀ (0.1 rad/sec). At that point the phase starts dropping at -45°/decade until it gets to -90° at 10·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 first order zero expressed as:

is displayed on a Bode plot. To change 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 0.1≤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 rising with a slope of +20 dB/decade.

Asymptotic Magnitude:  The asymptotic phase plot starts (at low frequencies) at 0° and stays at that level until it gets to 0.1·ω₀ . At that point the phase starts rising at +45°/decade until it gets to +90° at 10·ω₀, at which point it becomes constant at +90° 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 conjugate roots) expressed as:

is displayed on a Bode plot. You can change ω₀, and ζ. The value of ω₀ is constrained such that 0.1≤ω₀≤10 rad/second, and 0.05≤ζ≤0.99.

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 -40 dB/decade. Note: it is -40 dB per decade because there are two poles in the denominator.
If ζ<0.5 we estimate the peak height as $|H(j\omega_{peak})|\approx\frac{1}{2\zeta}$ (exact height is $|H(j\omega_{peak})|=\frac{1}{2\zeta \sqrt{1-\zeta^2}}$). We approximate the peak location at $\omega_{peak}\approx\omega_0$ (exact peak location is at $\omega_{peak}=\omega_0\sqrt{1-2\zeta^2}$ ).
However, if ζ≥0.5, the peak is sufficiently small that we don't include it in our plot.

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

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

This demonstration shows how a second order zero (complex conjugate roots) expressed as:

is displayed on a Bode plot. You can change ω₀, and ζ. The value of ω₀ is constrained such that 0.1≤ω₀≤10 rad/second, and 0.05≤ζ≤0.99.

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 rising with a slope of +40 dB/decade.
If ζ<0.5 we estimate the peak height as $|H(j\omega_{peak})|\approx\frac{1}{2\zeta}$ (exact height is $|H(j\omega_{peak})|=\frac{1}{2\zeta \sqrt{1-\zeta^2}}$). We approximate the peak location at $\omega_{peak}\approx\omega_0$ (exact peak location is at $\omega_{peak}=\omega_0\sqrt{1-2\zeta^2}$ ).
However, if ζ≥0.5, the peak is sufficiently small that we don't include it in our plot.

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

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