Investing amplifier bode plot example

The Bode plot of Figure 1, for example, shows the interac- tion of the magnitude response of the open-loop gain (|A|) and the reciprocal of the feedback. Feedback Amplifier Design ; s = tf('s'); a = a0/(1+s/w1)/(1+s/w2) ; h = bodeplot(a,'r'); setoptions(h,'FreqUnits','rad/s','MagUnits','dB','PhaseUnits','deg', '. I understood the question as being a standard text book inverting opamp, with R1 and R2 replaced by capacitors C1 and C2.
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To express a ratio between two numbers A and B as a decibel we apply the following formula for numbers that represent amplitudes numbers that represent a power measurement use a factor of 10 rather than 20 :. If we have a system transfer function T s , we can separate it into a numerator polynomial N s and a denominator polynomial D s.

We can write this as follows:. To get the magnitude gain plot, we must first transit the transfer function into the frequency response by using the change of variables:. From here, we can say that our frequency response is a composite of two parts, a real part R and an imaginary part X:. The Bode magnitude and phase plots can be quickly and easily approximated by using a series of straight lines.

These approximate graphs can be generated by following a few short, simple rules listed below. Once the straight-line graph is determined, the actual Bode plot is a smooth curve that follows the straight lines, and travels through the breakpoints.

We say that the values for all z n and p m are called break points of the Bode plot. These are the values where the Bode plots experience the largest change in direction. Bode Gain Plots , or Bode Magnitude Plots display the ratio of the system gain at each input frequency. Luckily, our decibel calculation comes in handy. Let's say we have a frequency response defined as a fraction with numerator and denominator polynomials defined as:. If we convert both sides to decibels, the logarithms from the decibel calculations convert multiplication of the arguments into additions, and the divisions into subtractions:.

And calculating out the gain of each term and adding them together will give the gain of the system at that frequency. The line is straight until it reaches the next break point. Double, triple, or higher amounts of repeat poles and zeros affect the gain by multiplicative amounts. Here are some examples:. Bode phase plots are plots of the phase shift to an input waveform dependent on the frequency characteristics of the system input.

Again, the Laplace transform does not account for the phase shift characteristics of the system, but the Fourier Transform can. Because the gain value is constant, and is not dependent on the frequency, we know that the value of the magnitude graph is constant at all places on the graph. There are no break points, so the slope of the graph never changes. We can draw the graph as a straight, horizontal line at 6dB:. We now have the slope of the line, and a point that it intersects, and we can draw the graph:.

Control Systems. From Wikibooks, open books for an open world. The Wikibook of: Control Systems. An example of a Bode magnitude and phase plot set. The Bode plot for a gain function that is the product of a pole and zero can be constructed by superposition, because the Bode plot is logarithmic, and the logarithm of a product of factors is sum of the individual, separate logarithms. The following two figures 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 is assumed to reside at higher frequency 1kHz than the pole Hz to make a more interesting example. Notice in the bottom phase plot 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.

In the limited range of frequencies between Hz and 1 kHz where both pole and zero are active contributors to the phase, the straight-line approximation is crude. 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. The open-loop gain A OL is a complex function of frequency, with both magnitude and phase.

Key to this determination are two frequencies. The first, labeled here as f , is the frequency where the open-loop gain flips sign. That is, frequency f is determined by the condition:. One measure of proximity to instability is the gain margin. Another equivalent measure of proximity to instability is the phase margin.

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. Two examples illustrate gain behavior and terminology. For a three-pole amplifier, gain and phase plots for a borderline stable and a stable amplifier are compared. These considerations also affect the amplifier step response.

Because the feedback amplifier does not have only real poles and zeroes, the straight-line Bode plot approximation does not work for the case with feedback. The first of two figures at the right compares the Bode plots 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. So, an equivalent way to find f 0dB is to look where the low-frequency asymptote to the feedback or closed-loop gain intersects the open-loop gain. Frequency f 0dB is needed later to find the phase margin.

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. The amplifier is borderline stable. The upper of the two figures shows the gain plot. Notice that the peak in the gain A FB near f 0dB seen in the borderline stable amplifier is almost gone.

The lower of the two figures is the phase plot. As an aside, it should be noted that stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good step response. Main Article Discussion Related Articles [? Unusual gain behavior can render the concepts of gain and phase margin inapplicable.

Then other methods such as the Nyquist plot have to be used to assess stability. Therefore, we could use the previous values found for the borderline stable amplifier. However, for clarity the procedure is described using only the curves for the stable amplifier. Technical Publications Pune, pp. ISBN

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