Pole Location

8/3/2019 Pole Location

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Frequency response of feedback amplifiers

• In previous context of discussion for feedback amplifiers, itis assumed that open-loop gain and feedback ratio are

independent of frequency.• But open-loop gain of real amplifiers is a function of

frequency. Magnitude response drops off and phase shiftincreases at high frequencies.

•

When feedback is applied to the open-loop amplifier,undesirable frequency response (also transient response)can result.

• Considering frequency dependence, the closed-loop gainof a feedback amplifier should be re-formatted as function

of Laplace variable S as follows:

• The zeros and poles of the above transfer function are thekey to understand the behavior of the feedback amplifier

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Transient response in terms of pole location

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• Remember from the Circuit Analysis Course, the mathematical form ofthe transient response is related to the location of the poles in the

complex domain.

• Obviously, we do not want to have poles on the positive real axis,because the transient response eventually drives the amplifier intovoltage limits, resulting in nonlinear distortion.

• Approximately within 5 time constant, the amplitude of exp termsdecays to negligible value compared to initial amplitude.

•

The greater the distance of the pole from the origin, the faster thetransient response decays

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Effects of feedback on pole: one pole I

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• Negative feedback has dramatic effects on pole locationsof amplifiers (OpAmp), which in turn affects transientresponse and frequency response of the amplifiers

• First, considering a one-pole (or dominant pole) amplifier,the open loop gain is of the form

• With feedback in this amplifier, then the closed-loop gain

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Effects of feedback on pole: one pole II

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• Transient and frequency response of feedback amplifiers are relatedto the pole location , so we need to consider how the pole wouldchange as feedback ratio changes

• For single-pole (dominant-pole) amplifier, the pole for the closed-loopgain becomes

• So, the above pole is still on the negative real axis, but moves furtherfrom the origin as increases

• Real amplifiers usually have more than one pole. However, sometimesone pole is much closer to the origin than others (called dominantpole). In that case, we can ignore the rest of the poles.

• Summary: for one-pole amplifier, the feedback back leads to a smoothroll-off of the frequency response and fast decaying of the transient

response without ringing



Example of a one-pole amplifier:



Effects of feedback on pole: two pole I

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• Now, considering a two-pole amplifier, the open loop gainis of the form

• Again assume feedback ratio is constant (not a function offrequency) and evaluate the poles of the closed loop

transfer function

• For the poles, as increases, the poles move together untilthey meet at the point in the middle. Then, further increasecauses the poles to become complex, moving away from thereal axis along the vertical line across the meeting point. (the

path followed by the poles is called a root locus)



Effects of feedback on pole: two pole II

0 A

• Usually, feedback amplifiers aredesigned so that is muchlarger than unity, which is usuallynecessary to achieve gainstabilization, impedance control,nonlinear distortion reduction etc.

• From the root locus, it can beseen that a too large value of

might move the poles outside thedesirable region of the s-plane(the 45 degree negative axis). Inthat case, undesirable frequencyresponse peaks and transientringing occurs.

0 A



Example of a two-pole amplifier:

Transient response Frequency response



Effects of feedback on pole: three poles• An amplifier with three or more poles can be analyzed using the

same method as in the one pole and two poles case, but math

analysis get much more complicated.• Qualitatively, in three poles case, feedback can cause the poles to

move even to the right half of the complex plane, thus making theamplifier instable.

• Example root lotus for 3 poles and 4 poles are shown below:



Example of a two-pole amplifier:

Transient response

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Pole Location