Bode plot Totally Explained

Everything about Bode Plot totally explained

A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot:
A Bode magnitude plot is a graph of log magnitude versus frequency, plotted with a log-frequency axis, to show the transfer function or frequency response of a linear, time-invariant system.
The magnitude axis of the Bode plot is usually expressed as decibels, that is, 20 times the common logarithm of the amplitude gain. With the magnitude gain being logarithmic, Bode plots make multiplication of magnitudes a simple matter of adding distances on the graph (in decibels), since ť

log(a cdot b) = log(a) + log(b).

A Bode phase plot is a graph of phase versus frequency, also plotted on a log-frequency axis, usually used in conjunction with the magnitude plot, to evaluate how much a frequency will be phase-shifted. For example a signal described by: Asin(ωt) may be attenuated but also phase-shifted. If the system attenuates it by a factor x and phase shifts it by −Φ the signal out of the system will be (A/x) sin(ωt − Φ). The phase shift Φ is generally a function of frequency.
Phase can also be added directly from the graphical values, a fact that's mathematically clear when phase is seen as the imaginary part of the complex logarithm of a complex gain.
In Figure 1(a), the Bode plots are shown for the one-pole highpass filter function:
ť :

mathrm 
ight) | = 1.

One measure of proximity to instability is the gain margin. The Bode phase plot locates the frequency where the phase of βA_OL reaches −180°, denoted here as frequency f₁₈₀. Using this frequency, the Bode magnitude plot finds the magnitude of βA_OL. If |βA_OL|₁₈₀ = 1, the amplifier is unstable, as mentioned. If |βA_OL|₁₈₀ < 1, instability doesn't occur, and the separation in dB of the magnitude of |βA_OL|₁₈₀ from |βA_OL| = 1 is called the gain margin. Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log₁₀(|βA_OL|₁₈₀) = 20 log₁₀(|A_OL|₁₈₀) − 20 log₁₀(1 / β ).
Another equivalent measure of proximity to instability is the phase margin. The Bode magnitude plot locates the frequency where the magnitude of |βA_OL| reaches unity, denoted here as frequency f_0dB. Using this frequency, the Bode phase plot finds the phase of βA_OL. If the phase of βA_OL(f_0dB) > −180°, the instability condition can't be met at any frequency (because its magnitude is going to be < 1 when f = f₁₈₀), and the distance of the phase at f_0dB in degrees above −180° is called the phase margin.
If a simple yes or no on the stability issue is all that's needed, the amplifier is stable if f_0dB < f₁₈₀. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (minimum phase systems). Although these restrictions usually are met, if they're not another method must be used, such as the Nyquist plot.

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_0dB. The feedback gain at low frequencies and for large A_OL is A_FB ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain A_OL), so an equivalent way to find f_0dB is to look where the feedback gain intersects the open-loop gain. (Frequency f_0dB is needed later to find the phase margin.)
   Near this crossover of the two gains at f_0dB, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β A_OL = −1). Beyond the unity gain frequency f_0dB, the open-loop gain is sufficiently small that A_FB ≈ A_OL (examine the formula at the beginning of this section for the case of small A_OL).
   Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency f₁₈₀ where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, A_FB ≈ A_OL for small A_OL.)
   Comparing the labeled points in Figure 6 and Figure 7, it's seen that the unity gain frequency f_0dB and the phase-flip frequency f₁₈₀ are very nearly equal in this amplifier, f₁₈₀ ≈ f_0dB ≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.
   Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the the condition | β A_OL | = 1 to lower frequency. 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_0dB = 1 kHz. Notice that the peak in the gain A_FB near f_0dB is almost gone. Figure 9 is the phase plot. Using the value of f_0dB = 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at f_0dB is −135°, which is a phase margin of 45° above −180°.
   Using Figure 9, for a phase of −180° the value of f₁₈₀ = 3.332 kHz (the same result as found earlier, of course). The open-loop gain from Figure 8 at f₁₈₀ is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.
   As an aside, it should be noted that stability isn't 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. 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 usage of applications for plotting Bode diagrams for given transfer functions helps better understanding and faster getting of results (see external links).

References and notes

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