Bode Diagrams -- a Detailed Tutorial
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created: 7/08/17
revised: 8/3/17
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Bode diagrams   --- http://en.wikipedia.org/wiki/Bode_plot
History of the op amp (from Analog Devices)
http://www.kelm.ftn.uns.ac.rs/literatura/mpi/pdf/Op%20Amp%20Applications%20Handbook.pdf

Getting the Most from Bode Diagrams
Bode diagrams are very powerful for the design and analysis of feedback control system, much more useful than you would guess from the bare bones explanation of bode diagrams in most textbooks. In 2001 while still working I wrote up a detailed bode diagram tutorial reflecting what I had learned and used in 40 years of designing motor control systems. My message here is once Bode diagrams are well understood with a little practice you will be able to estimate in your head how the the placement of poles and zeros affects phase margin. This makes compensating a new loop easy, you can just sketch out options for pole/zero placement.

Much of this information I had originally learned from a little softcover book (long out of print) on bode diagrams from one of the early op amp companies (Burr Brown?) published in the 60s or early 70s. I recently rediscovered my bode diagram viewgraphs in my files. I think it is a nice piece of work, so I am putting it online for engineers and engineering students. Asymptotic (straight line) Bode Diagrams
A circuit analysis program can plot the exact magnitude and phase response of a circuit or control loop. While this can be useful to check a final design, to verify that it has the required BW (bandwidth) and phase margin, it is not the tool you want to design the control loop. The reason is that for a circuit of any complexity the real magnitude and phase responses will be too soft, too hard to adjust. For design purposes you want a straight line approximation to the transfer function so you can 'see' where the poles and zeros of the forward path (G) and the feedback path (H) are located. These frequency breaks and the gains will need to be adjusted to get the desired control loop response. Credit for the 'straight line asymptotic' trick to make loop easy goes to Hendrik Bode in 1938. The goal is to derive a straight line (asymptotic) approximation of the transfer function of your circuit. Each straight segment will be a plot of a dominant term of the transfer function over the frequency range where it is dominant. This will be a log-log plot of magnitude vs frequency so terms incorporating frequency (s^2, s, s^0, s^-1, s^-2, etc) will plot as straight lines with different slopes, the slope set by the exponent of 's'. In such a plot the frequency of the poles and zeros are easy to see as they sit at the intersections of the straight line segments.

What we want is an equation of the transfer function with the poles and zero visable. Often it can be written by inspection, but it may take some playing around to get a single transfer function if a block has several circuits in series expecially if the some of the poles or zeros are not too separated. Multiply out the individual transfer functions to get a single transfer function with polynomials in terms of power of 's' in the numerator and the denominator.

Key idea
The key idea is to separately take the numerator and denominator polynominals and assume that over some frequency range a single term in the equation is dominant. Just retain that term over that frequency range and drop the others. Plot it up and label each straight line segment with its simplified transfer function (as shown below). The intersections of the straight line segment will give the poles and zero in terms of the constants in the equation. Don't worry about the gain at this point, it will be adjusted during the design process. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Loop transfer function example
The figure immediately below shows a loop transfer function that arises again and again in my field of motion control. The power stage applies a torque (motor current) to a load inertia to control its velocity. Thus the load transfer function [torque => (rotational) velocity] includes an integration (1/s). For the velocity loop to be tight an electronic integrator (1/s) is almost always included. Hence the raw transfer function (k/s^2) has a double integration, each integrator contributing 90 degree lag or 180 degrees total, so without loop compensation this type of feedback loop will be unstable. ..
Breaking out the electronic integrator
The integration (1/s) of the load arises from the load inertia so it cannot be removed.  However the electronic integration can be 'broken out' (removed at higher frequencies) by (in principle) inserting a resistor in series with the integrating capacitor. In the transfer function above this creates the first order zero at w1. A loop crossover  (GH=1) at w1 would be stable, but the phase margin poor since only half the 90 degrees of lag from the electronic integrator is removed at w1 and the pole at w2 introduces additional lag.

Crossover frequency
Hence typically the gain is adjusted so the loop crossover is between w1 and w2. That is the electronic integrator break zero out is set below the loop crossover and the pole at w2 is set above the loop crossover. Ideally crossover would be set at the geometric mean of w1 and w2 [sqrt{w1^2 + w2^2}] because this frequency has the best phase margin, i.e. the lowest phase lag around the loop.

Noise filter pole
While a system with just an electronic integrator may be stable and tight, its real world performance is likely to be poor due to high frequency noise. Almost all real control systems need some noise filtering. For a motor to rotate smoothly and to minimize heating from high frequency noise currents a low pass 'noise filter' is usually required to attenuate noise spikes and other high frequency electrical noise that always exists. Adding a first order low pass 'noise filter' at w2 creates the first order pole in the bode diagram above at w2. A noise filter would normally be set to 'break in' after the loop has crossed over.

There you have it. A high performance control loop for a 1/s inertia load is tightened by an electronic integrator that is broken out below crossover, and noise is attenuated by a noise filter that breaks in above crossover. Even though the lag of the noise filter comes in strongly above its pole at the crossover frequency some of  its phase lag is still 'visible'. While the breakout of the electronic integrator zero is positioned below the crossover frequency, it is not fully broken out at crossover, so it can be thought of as creating (in effect) some phase lag that is 'visible' at crossover.  Hence both the (noise filter) pole and the (integrator break out) zero reduce phase margin at crossover. The closer they are to crossover the lower the phase margin, but the higher the loop performance.

For example if a slightly underdamped transient response is acceptable, then a 45 phase margin will do the job. The loop gain would be set so that the loop crosses over in -1 slope region approximately midway between the zero and pole (ideally at the geometric mean), where a -1 (or 1/w) slope on a bode diagram implies (ideally) 90 phase lag (see below). For 45 degree phase margin the sum of lags from the zero and pole can contribute an additional 45 degrees lag. If positioned symmetrically around crossover, this is 22.5 degrees each. Knowing their phase contributions at crossover allows the frequency of the pole and zero to be calculated. Since [tan(22.5 degrees) = .414 and 1/tan(22.5 degrees) = 2.41], the integrator breakout zero located at x.414 of loop crossover and the noise filter pole located at x2.41 loop crossover will give the desired loop stability with a 45 degree phase margin.
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Combining blocks
The straight line asymptote of usual (negative) feedback configuration [G forward path and H feedback path] is found by assuming that the closed loop (in/out) response follows the inverse of the feedback path straight line asymptote (1/H) when when [G > (1/H)] and follows the forward path straight line assumptote when [G < (1/H)]. This translates into plotting the 1/H and G straight line asymptotes and the smaller dominates. If the paths are additive, then the larger dominates. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Label each bode diagram straight line segment with its equation
It is very useful to label each bode straight line segment with its equation derived from the circuit blocks (via if necessary the overall transfer function). The straight line segments need to join up at each pole and zero, so equating the straight line equations at these pole/zero frequencies and solving yields expressions for the pole and zero frequencies in terms of circuit parameters. An equation for the close loop BW (bandwidth) is easily found by equating the straight line segment of G and 1/H where they cross.

Here's a worked out example. Note, '800/90 velocity loop' is just a reference to product families our company manufactured.
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How to Find Stability from Bode Diagrams
Deriving the exact in/out transfer function for the standard G and H control configuration, and showing the approximate in/out transfer function is 1/H when loop gain (GH >>1) and G when (GH <<1). .. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Criteria for output with no input -- an oscillator
What is the criteria for a closed loop control system having an output with no input, in other words it has become totally unstable, an oscillator? Simple, the denominator of the (exact) in/out transfer function [1 + GH] has to go to zero at some frequency. This frequency is usually around loop crossover frequency and is approximately the frequency where the loop will ring or oscillator if is unstable.

However, the denominator of a transfer function is a complex number with amplitude and phase. So for GH to equal -1, there are two criteria one based on amplitude [GH=1] and one based on phase [GH phase = 180 degrees]. [G = 1/H] is just the crossover of the straight line segments of forward and inverted feedback paths, and at this crossover frequency the phase shift around the loop, i.e. the phase shift of GH must be 180 degrees.

Phase margin
In modern control loop terminology a phase shift around the loop of 180 degrees at crossover means a 'phase margin' of zero degrees. Phase margin is [180 degrees minus the GH phase shift at crossover]. The degree of stability of a feedback loop is generally described in terms of its phase margin. A loop's (small signal) transient response (say to a small signal input step) can be predicted quite well from the loop's phase margin. My understanding from text books is that there is no simple exact relationship between phase margin and transient response, but relationship is close enough that in practice the the degree of of over or under shoot to a step can be predicted quite well.

This makes phase margin, which as shown below can be estimated from the loop's bode diagram, a key criteria in loop design not only for stability, but also for predicting the character of the loop's transient response. ..

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Bode slope reveals circuit phase
Slope of a bode diagram straight line segment reveals the phase shift of the underlying function, negative slope indicating lag and positive slope indicating lead. This follows directly from plotting the transfer function of simple functions like [1/s^2] (180 degree lag) or an integrator [1/s] (90 degree lag). ..
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Phase Margin & Transient Response
Phase margin
Phase margin is the difference in degrees between the phase shift around a control loop and 180 degrees at the frequency where the loop gain (GH = 1),  i.e. where forward path G crosses ideal feedback path 1/H. Transient response vs phase margin
Below shows the transient response from a circuit analysis program of the example control loop circuit adjusted for various phase margins. Notice that the bode diagrams with 45 degree and 67.5 degree phase margins have slightly different transient response. Above three systems all with 45 degree phase margin, yet with slighltly different step transient response
.. Above two systems both with 67.5 degree phase margin, yet with slighltly different step transient response
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Control circuit example with different phase margins
Below is a simple control circuit numerical example with the zero and pole symetriallty adjusted to give different phase margins. .. .. ..
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Estimating Phase Margin from Bode Diagrams
When compensating a control loop, it is very helpful to be able to quickly estimate how phase margin changes when adding a first order pole or zero into the loop. The phase a first order pole/zero contributes at crossover depends (in a ratio sense) on how far the pole/zero frequency is from crossover. The math for first order phase is just the inverse tangent function where the argument is the frequency ratio. For small values the angle of the inverse tangent function is approximated by the argument in radians. For example, a pole at 1/10th crossover frequency contributes a phase lag that is approximately 1/10th radian or 5.73 degree at crossover. The table below shows this approximating rule works pretty well down to a frequency ratio of one half.

The same approximating rule also works (with a 90 degree correction) above the pole/zero. For example, a zero at 1/3rd crossover frequency contributes a phase lead at crossover of [(90 - tan^-1 (0.33)) = (90 - 18.4) = 71.6 degree]. .. ..
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Bode diagrams associated with some simple closed loop configurations
I have plotted up the straight line response for G (forward path) and 1/H (feedback path) for some simple loop configurations that are stable. With both G and 1/H plotted all the poles and zeros of the system are visible. I have put in some numbers and show the phase margins indicating good stability.

The in/out bode diagram of the closed loop just following the lower of the two curves. This is easily understood as the feedback path controls the system when the loop gain > 1, so in this frequency region the closed loop response goes as 1/H, but when the gain in the forward path gets too low such that loop gain <1, the response of the closed loop follows G.

In the circuit below, which is just simple feedback path around an integrator, the phase margin is 90 degrees since the integrator's 90 degree lag at all frequencies is subtracted from 180 degrees. From the bode diagram the 90 degree phase margin can be seen from the relative slope between the two plots which is -1 both below and above crossover regardless of the gain constant 'k'. ..
In the circuit below a low pass (noise filter) is added to the forward path above crossover. This will degrade phase margin because some of its lag is visible at crossover, in this case with a pole at x3 crossover a loss of 18.4 degrees since [tan-1 (1/3) = 18.4 degree]. Above 3w0 the slope difference goes from -1 to -2. ..
The circuit below with the x3 low pass above crossover moved into the feedback path has the same loss of phase margin (18.4 degree) as the circuit above, but it may not be so obvious as to why. The key here is to focus on the difference between the G and 1/H plots. This is what is really important! Above 3w0 the slope difference goes from -1 to -2, the same as the circuit above, this is why loss of phase margin is the same. ..
The circuit below has low pass filters at x3 above crossover in both the forward and feedback paths, so it has a bode diagram that is sort of a combination of the two circuits above. In this case above 3w0 the slope difference changes from -1 to -3, telling us that each of the low pass filters is contributing a loss of 18.4 degree in phase margin resulting in a phase margin of 53.2 degrees. ..
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A very useful simple element for loop compensation is what is known as a 'lead-lag'. As shown by the magnitude curve below, this is nothing more than a first order zero followed by a first order pole. Its implementation is just a circuit with a rising gain that is terminated (or broken out) at a higher frequency. This produces a symmetrical phase lead as shown below peaking approximately (actually at the geometric mean) between the zero and the pole.

The interesting and useful feature of a lead-lag circuit for loop compensation is that its (lead) phase shift depends only on the ratio (alpha in the figure below) of its pole to zero frequency. I have tabulated its peak (lead) phase as a function of alpha in the figure below. ..
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The phase response of a lead-lag circut is perfect for compensating a double integral (1/s^2) load. One only need position the peak phase of the lead-lag at the loop crossover and set the alpha for the desired phase margin. The example below shows a lead-lag compensator with a rising gain of nine (alpha = nine) centered at crossover, which means a zero at x1/3rd crossover and a pole at x3 of crossover. This produces a well damped loop (phase margin = 53 degrees) for a double integrated load. .. ..
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Op Amp Stability Analysis
Of course a simple inverting op amp circuit is most easily analyzed by equating the input and output currents at the op amp's virtual ground. This works in the vast majority of cases where the op amp open loop gain is high relative to the gain of the circuit, but if the op amp's open loop response is coming into play, then an op amp circuit can be analyzed like any feedback system. It is only necessary to redraw the circuit in the standard G and H format. Here's an example for a simple op amp inverter circuit. .. .. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Vintage Bode Diagram reference handbook
I learned much of this bode diagram information as a young engineer from a wonderful little handbook that was published in the 60s or early 70s by one of the early op amp companies, Burr Brown (?) (or Philbrick or Analog Devices). This was the time (late 60s) before integrated circuit op amps were available commercially, where buying an op amp meant buying a module with PC board inside stuffed with transistors. The bode diagram handbook was published to show how to use their op amps in control circuits. This went far beyond what was in control textbooks where bode diagrams were barely mentioned. The handbook showed how with a little practice bode diagrams could be used to set the poles and zeros of a practical circuit that included noise filtering, and how phase margin could be estimated from the bode diagram to get the desired transient response and stability.

Even now I can see in my mind the cover of this little handbook. I kept it for years even as it fell apart. However, I cannot remember the company that put it out, nor can I find any reference to it online. A google search for "bode diagram handbook" gives a null result. If I had to guess, I would say the handbook was published by Burr Brown. I remember it did have a brown cover. I've made little progress in a google search. Mostly I find histories of the early op amp companies, or the history of op amp development, or the history of integrated op amps, all of which are interesting topics that I lived through.

I remember well the excitement of well attended seminars when designers of the early integrated circuit op amps, who were well known and to some extent 'stars' in the EE world, would come to Boston (from CA or Texas) to discuss their latest circuit design tricks used in their new op amps. While commercial circuit designers like me found these new circuit tricks interesting (they were just being invented by the integrated circuit guys), they were of little practical use to us, other than to allow us to evaluate and use effectively the new op amps, because they were nearly all very specific to integrated circuits.