Bode Analysis of Feedback Circuits

H. W. Bode developed a quick, accurate, and easy method of analyzing feedback amplifiers, and he published a book about his techniques in 1945.[2] Operational amplifiers had not been developed when Bode published his book, but they fall under the general classification of feedback amplifiers, so they are easily analyzed with Bode techniques. The mathematical manipulations required to analyze a feedback circuit are complicated because they involve multiplication and division. Bode developed the Bode plot, which simplifies the analysis through the use of graphical techniques. 

The Bode equations are log equations that take the form 20LOG(F(t)) = 20LOG(|F(t)|) + phase angle. Terms that are normally multiplied and divided can now be added and subtracted because they are log equations. The addition and subtraction is done graphically, thus easing the calculations and giving the designer a pictorial representation of circuit performance. Equation 5–8 is written for the low pass filter shown in Figure 5–8.

 

Figure 5–8. Low-Pass Filter

 

This magnitude, |V OUT /V IN | ≅ 1 when ω = 0.1/τ, it equals 0.707 when ω = 1/τ, and it is approximately = 0.1 when ω = 10/τ. These points are plotted in Figure 5–9 using straight line approximations.

The negative slope is –20 dB/decade or –6 dB/octave. The magnitude curve is plotted as a horizontal line until it intersects the breakpoint where ω = 1/τ. The negative slope begins at the breakpoint because the magnitude starts decreasing at that point. The gain is equal to 1 or 0 dB at very low frequencies, equal to 0.707 or –3 dB at the break frequency, and it keeps falling with a –20 dB/decade slope for higher frequencies.

The phase shift for the low pass filter or any other transfer function is calculated with the aid of Equation 5–9.

The phase shift is much harder to approximate because the tangent function is nonlinear. Normally the phase information is only required around the 0 dB intercept point for an active circuit, so the calculations are minimized. The phase is shown in Figure 5–9, and it is approximated by remembering that the tangent of 90° is 1, the tangent of 60° is √3 , and the tangent of 30° is √3/3.

 

Figure 5–9. Bode Plot of Low-Pass Filter Transfer Function

A breakpoint occurring in the denominator is called a pole, and it slopes down. Conversely, a breakpoint occurring in the numerator is called a zero, and it slopes up. When the transfer function has multiple poles and zeros, each pole or zero is plotted independently, and the individual poles/zeros are added graphically. If multiple poles, zeros, or a pole/zero combination have the same breakpoint, they are plotted on top of each other. Multiple poles or zeros cause the slope to change by multiples of 20 dB/decade.

An example of a transfer function with multiple poles and zeros is a band reject filter (see Figure 5–10). The transfer function of the band reject filter is given in Equation 5–10.

Figure 5–10. Band Reject Filter

 

The pole zero plot for each individual pole and zero is shown in Figure 5–11, and the combined pole zero plot is shown in Figure 5–12.

Figure 5–11.Individual Pole Zero Plot of Band Reject Filter

Figure 5–12. Combined Pole Zero Plot of Band Reject Filter

The individual pole zero plots show the dc gain of 1/2 plotting as a straight line from the –6 dB intercept. The two zeros occur at the same break frequency, thus they add to a 40-dB/decade slope. The two poles are plotted at their breakpoints of ω = 0.44/τ and ω = 4.56/τ. The combined amplitude plot intercepts the amplitude axis at –6 dB because of the dc gain, and then breaks down at the first pole. When the amplitude function gets to the double zero, the first zero cancels out the first pole, and the second zero breaks up. The upward slope continues until the second pole cancels out the second zero, and the amplitude is flat from that point out in frequency.

When the separation between all the poles and zeros is great, a decade or more in frequency, it is easy to draw the Bode plot. As the poles and zeros get closer together, the plot gets harder to make. The phase is especially hard to plot because of the tangent function, but picking a few salient points and sketching them in first gets a pretty good approximation.[3] The Bode plot enables the designer to get a good idea of pole zero placement, and it is valuable for fast evaluation of possible compensation techniques. When the situation gets critical, accurate calculations must be made and plotted to get an accurate result.

Consider Equation 5–11.

Taking the log of Equation 5–11 yields Equation 5–12.

If A and β do not contain any poles or zeros there will be no break points. Then the Bode plot of Equation 5–12 looks like that shown in Figure 5–13, and because there are no poles to contribute negative phase shift, the circuit cannot oscillate.

Figure 5–13. When No Pole Exists in Equation (5–12)

 

All real amplifiers have many poles, but they are normally internally compensated so that they appear to have a single pole. Such an amplifier would have an equation similar to that given in Equation 5–13.

 

The plot for the single pole amplifier is shown in Figure 5–14.

Figure 5–14. When Equation 5–12 has a Single Pole

The amplifier gain, A, intercepts the amplitude axis at 20Log(A), and it breaks down at a slope of –20 dB/decade at ω = ω a . The negative slope continues for all frequencies greater than the breakpoint, ω = ω a . The closed loop circuit gain intercepts the amplitude axis at 20Log(V OUT /V IN ), and because β does not have any poles or zeros, it is constant until its projection intersects the amplifier gain at point X. After intersection with the amplifier gain curve, the closed loop gain follows the amplifier gain because the amplifier is the controlling factor.

Actually, the closed loop gain starts to roll off earlier, and it is down 3 dB at point X. At point X the difference between the closed loop gain and the amplifier gain is –3 dB, thus according to Equation 5–12 the term –20Log(1+Aβ) = –3 dB. The magnitude of 3 dB is √2 , hence

 

, and elimination of the radicals shows that Aβ = 1. There is a method [4] of relating phase shift and stability to the slope of the closed loop gain curves, but only the Bode method is covered here. An excellent discussion of poles, zeros, and their interaction is given by M. E Van Valkenberg,[5] and he also includes some excellent prose to liven the discussion.