Pole, zero and Bode plot EC04 305 Lecture notes YESAREKEY December 12, 2007 Authored by: Ramesh.K
Pole, zero and Bode plot EC04 305 Lecture notes A rational transfer function H (S) can be expressed as a ratio of a numerator polynomial n(s) divided by a denominator polynomial d(s), Mathematically, Where s=pi is a finite of H( S ), s=z i is called a finite zero of H(S), and K is the gain constant of transfer function H( S ). Generally z 1..z n are called the transfer function zeros and p 1..p n are called transfer function poles. They can be either real or complex. The complex poles or zeros must occur in conjugate pairs. A finite pole satisfies as, and a finite zero satisfies. BODE PLOTS H. bode developed a technique for computing approximate or asymptotic frequency response curves-called Bode plots. The technique is particularly useful in the case of real poles and zeros. A Bode magnitude plot is a graph, with a linear scale for the db values on the vertical axis and a logarithmic scale for ω on the horizontal 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. 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. The Amplifier Transfer function Amplifiers have frequency gain function either of the following two forms. The only diff b/w the two is that gain of ac amplifiers falls at low frequencies. The Frequency Bands As shown in the fig.b the amplifier gain is almost constant over a wide frequency range called the mid band. In this frequency almost all capacitors (coupling, bypass, internal capacitance have negligible effect and can be ignored in the gain calculations. At high frequencies the gain falls due to the internal capacitance of the device. Also at low frequencies the bypass capacitors will not act (a) (b) 1
as a short circuit for ac, hence the gain falls. The extend of frequencies are defined by the two frequencies, these are frequencies at which the gain drops by 3dB below the value at midband. The amplifier band width can be defined as Since Hence the gain band width product : where is the amp gain. The low freq amp gain can be expressed by, The high freq gain For and EFFECT OF FEEDBACK ON THE AMPLIFIER POLES : The amplifier frequency response and stability are directly determined by its poles. Stability and Pole location: The poles of the feedback amp can be obtained by solving the characteristic equation, 0 For an amplifier to be stable, its pole should lie in the left half of the s- plane. A pair of conjugate poles on the axis gives rise to sustained sinusoidal oscillations. Poles in the right half of the s plane gives rise to growing oscillations, as shown below. STUDY OF STABLILITY OF FEED BACK AMPLIFIERS USING BODE PLOTS If the feedback is termed as negative feedback and if the feedback is termed as positive or regenerative. Under these circumstances, the resultant transfer gain will be greater than, 2
the nominal gain without feed back. Since amplifier with positive feedback, this method is seldom used. (used in oscillators). Because of the reduced stability of an To illustrate the instability of amplifiers with positive feedback, consider the following situation: no signal is applied, but because of some transient disturbances, a signal X o appears at the o/p. a portion of this signal, back to the i/p circuit, and will appear in the o/p as an increased signal spurious o/p has regenerated itself. In other words, if, will be fed. If this term just equals X o, then the, the amplifier will oscillate. Hence, if an attempt is made to obtain large gain by making almost equals to unity, there is the possibility that the amplifier may break put into spontaneous oscillation. There is little point in attempting to achieve amplification at the expense of stability. The condition for Stability: In the design of a feedback amplifier, it must be ascertained that the circuit is stable at all frequencies. i.e., the system is stable if a transient disturbance results in a response which dies out. The stability of an amplifier depends on pole of its transfer function. As described above, if a pole exists with a positive real part, this will result in a disturbance increasing exponentially with time. Hence the condition which must be satisfied, if a system is to be stable, is that the pole of the transfer function must all lie on the left hand half of the s plane. If a system without feedback is stable, the poles of A do lie in the left hand half plane. It follows from the Eqn that the stability condition requires that the zeros of all lie in the left hand half of the s- plane. Gain and phase margins: It should be noted that instead of plotting the product in the complex plane, it is more convenient to plot the magnitude, usually in decibels, and also the phase of as a function of frequency. If we can show that is less than unity (<0 db) when the phase angle of is 180 o, the closed loop amplifier will be stable. The gain margin is defined as the value of in db at the frequency at which the phase angle of is 180 o.if the gain margin is negative this gives the db rise in open loop gain, which is theoretically permissible without oscillation. If the gain margin is positive, the amplifier is potentially unstable. Feedback amplifiers are usually designed to have sufficient gain margin to allow for the inevitable changes in the loop gain with temp, time and so on. The phase margin is 180 o minus the magnitude of the angle of at the frequency at which is unity (0 db).the magnitude of these quantities gives an indication of how stable Bode plots relating to the defn: of gain and phase margins an amplifier is. For E.g., a linear amp of good stability requires gain and phase margins of at least 10 db and 50 o respectively. The amount of phase margin has a profound effect on the shape of the closed loop magnitude response. 3
References Micro Electronic Circuits, Sedra/Smith, 4 th edition. Integrated Electronics: Analog and Digital Circuits And Systems ; Millman & Halkias Wikipedia, the free encyclopedia Lecture Notes 4
S.R.K Ramesh.K /ECE/MEA EC 12/20/2007 Steps to draw Bode plots Lecture Notes-EC04 # 303#ECNT www.edutalks.org
Ramesh.K /ECE/MEA EC rameshpkd@gmail.com Steps to draw Bode plots Lecture Notes-EC04 # 303#ECNT Steps to Plot Bode Magnitude Plot Let H(s) be the transfer function.covert H(s) into the form, then replace s by jω. List the corner frequencies (i.e, the coefficients of jω s of H(s)) in the increasing order and prepare a table as given below. Terms Corner Frequencies Change in slope In the table enter K or or K as the first term and the other term as the increasing order of the corner frequencies. Choose an arbitrary frequency ω L, less than the lowest corner frequency. Calculate the gain in db magnitude of K or or K at ω L and at the lower corner frequency. Calculate the gain at every corner frequency one by one using the formula given below. Chose an arbitrary frequency ω H, greater than the highest corner frequency. Calculate the gain at ω H using the above formula. Plot the Bode magnitude plot [A (db).vs. log (ω)] in a semi log graph. Steps to Plot Bode Phase Plot: Calculate the phase angle of H(jω) for various values of ω and are tabulated. The choices of frequencies are preferably the frequency selected for magnitude plot. Magnitude and phase of a complex umber: Phase: 1 www.edutalks.org