Bode plot

Overview

Among his several important contributions to circuit theory and control theory, engineer Hendrik Wade Bode, while working at Bell Labs in the United States in the 1930s, devised a simple but accurate method for graphing gain and phase-shift plots. These bear his name, Bode gain plot and Bode phase plot. "Bode" is pronounced /ˈboʊdi/ BOH-dee (Dutch: [ˈboːdə]).

Bode was faced with the problem of designing stable amplifiers with feedback for use in telephone networks. He developed the graphical design technique of the Bode plots to show the gain margin and phase margin required to maintain stability under variations in circuit characteristics caused during manufacture or during operation. The principles developed were applied to design problems of servomechanisms and other feedback control systems. The Bode plot is an example of analysis in the frequency domain.

Definition

The Bode plot for a linear, time-invariant system with transfer function H ( s ) ( s being the complex frequency in the Laplace domain) consists of a magnitude plot and a phase plot.

The Bode magnitude plot is the graph of the function | H ( s = j ω ) | of frequency ω (with j being the imaginary unit). The ω -axis of the magnitude plot is logarithmic and the magnitude is given in decibels, i.e., a value for the magnitude | H | is plotted on the axis at 20 log 10 ⁡ | H | .

The Bode phase plot is the graph of the phase of the transfer function arg ⁡ ( H ( s = j ω ) ) as a function of ω , commonly expressed in degrees. The phase is plotted on the same logarithmic ω -axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.

Highpass Filter

The one-pole highpass RC-filter has the transfer function

where τ c = 1 / ω c is the inverse of the cut-off frequency ω c = 2 π f c = 1 R C .

Magnitude Plot

To obtain the Bode magnitude plot one calculates the magnitude of the transfer function along the imaginary axis

The last equality is obtained using the rules for complex numbers. On the Bode magnitude plot, decibels are used, and the plotted magnitude is:

Phase Plot

To obtain the Bode phase plot one calculates the argument of H h p ( j ω ) :

Care must be taken that the inverse tangent is set up to return degrees, not radians. The resulting Bode plot is shown in Figure 1(a) for f c = 100 H z .

Lowpass Filter

A passive (unity pass band gain) lowpass RC filter has the following transfer function:

where, again, τ c is the inverse of the cut-off frequency. The corresponding Bode Plot is shown in Figure 1(b).

The magnitude and the phase of the transfer function are related. For minimum-phase systems there is a unique relationship between the magnitude and the phase, the Bode gain-phase relationship.

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.

Frequency Response and Bode Plot

This section illustrates that a Bode Plot is a visualization of the frequency response of a system.

Consider a linear, time-invariant system with transfer function H ( s ) . Assume that the system is subject to a sinusoidal input with frequency ω ,

that is applied persistently, i.e. from a time − ∞ to a time t . The response will be of the form

i.e., also a sinusoidal signal with amplitude y 0 shifted in phase with respect to the input by a phase φ .

It can be shown that the magnitude of the response is

and that the phase shift is

A sketch for the proof of these equations is given in the appendix.

In summary, subjected to an input with frequency ω the system responds at the same frequency with an output that is amplified by a factor | H ( j ω ) | and phase-shifted by arg ⁡ ( H ( j ω ) ) . These quantities, thus, characterize the frequency response and are shown in the Bode plot.

Rules for handmade Bode plot

For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are asymptotes of the precise response. The effect of each of the terms of a multiple element transfer function can be approximated by a set of straight lines on a Bode plot. This allows a graphical solution of the overall frequency response function. Before widespread availability of digital computers, graphical methods were extensively used to reduce the need for tedious calculation; a graphical solution could be used to identify feasible ranges of parameters for a new design.

The premise of a Bode plot is that one can consider the log of a function in the form:

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

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 d B = 20 log 10 ⁡ ( X ) to define decibels. Given a transfer function in the form

where x n and y n are constants, s = j ω , a n , b n > 0 , and H is the transfer function:

To handle irreducible 2nd order polynomials, a x 2 + b x + c can, in many cases, be approximated as ( a x + 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 ( j ω ) , and since it is a complex function, | H ( j ω ) | = H ⋅ H ∗ . Thus at any place where there is a zero or pole involving the term ( s + x n ) , the magnitude of that term is ( x n + j ω ) ⋅ ( x n − j ω ) = x n 2 + ω 2 .

Corrected amplitude plot

To correct a straight-line amplitude plot:

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:

the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by − arctan ⁡ ( I m [ H ( s ) ] R e [ H ( s ) ] ) .

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

Example

To create a straight-line plot for the lowpass filter, introduced above, one considers the transfer function in terms of the angular frequency:

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 v d B :

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

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.

Phase plot

The phase Bode plot is obtained by plotting the phase angle of the transfer function given by

versus ω , where ω and ω c are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio ω ω 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 ω = ω 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 ω ω 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 ω 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 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

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 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 f₁₈₀, is the frequency where the open-loop gain flips sign. The second, labeled here f_0 dB, 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:

where vertical bars denote the magnitude of a complex number (for example, | a + j b | = [ a 2 + b 2 ] 1 2 ), and frequency f_0 dB is determined by the condition:

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 A_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 log 10 ⁡ ( | β A O L | 180 ) = 20 log 10 ⁡ ( | A O L | ) − 20 log 10 ⁡ ( β − 1 ) .

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_0 dB. Using this frequency, the Bode phase plot finds the phase of βA_OL. If the phase of βA_OL( f_0 dB) > −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_0 dB 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_0 dB < 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. Optimal gain and phase margins may be computed using Nevanlinna–Pick interpolation theory.

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.

In this example, A_OL = 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies, A_FB ≈ 58 dB as well.

Because the open-loop gain A_OL is plotted and not the product β A_OL, the condition A_OL = 1 / β decides f_0 dB. 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_0 dB is to look where the feedback gain intersects the open-loop gain. (Frequency f_0 dB is needed later to find the phase margin.)

Near this crossover of the two gains at f_0 dB, 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_0 dB, 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_0 dB and the phase-flip frequency f₁₈₀ are very nearly equal in this amplifier, f₁₈₀ ≈ f_0 dB ≈ 3.332 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_OL | = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies A_FB ≈ 77 dB as well.

Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and A_OL occurs at f_0 dB = 1 kHz. Notice that the peak in the gain A_FB near f_0 dB is almost gone.

Figure 9 is the phase plot. Using the value of f_0 dB = 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at f_0 dB 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.

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.

Bode plotter

The Bode plotter is an electronic instrument resembling an oscilloscope, which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against frequency in a feedback control system or a filter. An example of this is shown in Figure 10. It is extremely useful for analyzing and testing filters and the stability of feedback control systems, through the measurement of corner (cutoff) frequencies and gain and phase margins.

This is identical to the function performed by a vector network analyzer, but the network analyzer is typically used at much higher frequencies.

For education/research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).

Related plots

Two related plots that display the same data in different coordinate systems are the Nyquist plot and the Nichols plot. These are parametric plots, with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in polar coordinates, with magnitude mapping to radius and phase to argument (angle). The Nichols plot displays these in rectangular coordinates, on the log scale.

Proof Sketch for the Relation to Frequency Response

This section sketches a proof for the relation between frequency response and the magnitude and phase of the transfer function in Eqs.(1)-(2).

Slightly changing the requirements for Eqs.(1)-(2) one assumes that the input has been applied starting at time t = 0 and one calculates the output in the limit t → ∞ . In this case, the output is given by the convolution

of the input signal with the inverse Laplace transform of the transfer function h ( t ) . The integral can be split in the following way

where the second term vanishes in the limit t → ∞ if h has certain properties. Thus, inserting the sinusodial input signal one obtains

Since h ( t ) is a real function this can be written as

The term in brackets is the definition of the Laplace transform of h at s = j ω . Inserting the definition in the form H ( j ω ) = | H ( j ω ) | exp ⁡ ( arg ⁡ H ( j ω ) ) one obtains the output signal

stated in Eqs.(1)-(2).