so far, we have looked at the asymptotic behavior of first order constructs, like pure integrators or single poles and zeros. once you start working with typical dynamic systems, it is very likely that you will have to deal with higher order polynomial expressions. the trick to dealing with those is remembering that any polynomial, no matter the order, can always be factored into a bunch of first order constructs which will correspond to the real roots, and a number of second order constructs which will correspond to complex conjugate pairs of roots.
typical examples of second order systems are mass spring dampers and rlc circuits. both of this, depending on the ratio of either the damping to the mass or the resistance to the inductance, will have a pair of complex conjugate roots. in general, any complex conjugate pole pair can be written in a standard second order transfer function form like this, where w_n is called the natural frequency and zeta is called the damping ratio.
notice that for our couple of examples, the natural frequency is equal to the square root of k/m for our basic mechanical system and square root of 1/lc for our basic electrical system. anyways, if we calculate the roots of that quadratic polynomial using the standard formula-- minus b plus minus square root of whatever-- we find that the complex conjugate pair of roots will have this form. note that these roots will only be a complex conjugate pair as long as the damping ratio zeta is less than 1. anything greater than 1, and both roots will become real numbers, which means that the system will behave as the product of two first order poles.
as we did before to calculate the frequency response, we replace s by jw in the transfer function. and to do our asymptotic approximations, we will look at the tendencies before and after w_n. when the frequency w is much less than the natural frequency, the 1 will dominate the denominator. so both the magnitude and the phase will be approximately 0. when the frequency w matches the natural frequency, that second order term becomes -1 and cancels with the 1, and the middle term becomes a pure imaginary constant with a magnitude of 1/(2*zeta) and a phase of -90 degrees because the j is in the denominator, so g falls on the negative imaginary axis.
finally, when the frequency w is much larger than the natural frequency the quadratic term will dominate. when taking the log, the square will come out and multiply the 20, so the magnitude will asymptotically approach a straight line with a slope of -40 dbs per decade. the phase will go to -180 degrees because g will now fall on the negative real axis.
notice that the actual resonant peak falls a little to the left of w_n because, if you look at the imaginary part of the poles, the frequency component is w_n multiplied by square root of (1-zeta)^2. this adjusted value is what is called a damped natural frequency. only when the damping ratio zeta approaches 0, the damped natural frequency will approach w_n. and note that, in that case, the magnitude of the resonant peak will go towards infinity.
a small value of the damping ratio means a higher and sharper resonant peak as well as a sharper shift in the phase. you can see that as we increase zeta, the magnitude of the resonant peak comes down and the phase transition becomes smoother. here i want to highlight the damping ratio of 0.707, ((2)^1/2)/2, which is normally known as the critical damping. this damping makes the magnitude -3 dbs at the natural frequency. also note that for zeta equal to 1 the imaginary part of the poles goes to 0, and our second order system becomes a product of two single real poles located at w_n.
at this point let me go back to our interactive design tool because there are a couple of additional things i want to highlight. first, let me bring in a pair of complex conjugate poles, and i am going to place them close to 10 radians per second. let me just make sure that i set the natural frequency to exactly 10.
i notice that, because i am starting with a damping ratio of 1, my polynomial is the product of two real roots. as soon as i change the damping to any value lower than 1, let's say 0.5, my roots become a pair of complex conjugate values. notice that for this particular case, the magnitude value of the natural frequency equal to 1/(2*zeta) becomes 1/1, which is 0 in dbs, so the crossover will occur at the natural frequency.
if i choose a smaller damping ratio, what you're going to see is a sharper and higher magnitude peak. in this particular case, i picked 0.05, which means 1/(2*zeta) will be 10. log of 10 is 1, times 20 is 20 dbs. if i change the natural frequency, all i am doing is shifting the location of that peak.
now, if i want to increase or decrease the gain because of the properties of logarithms, we're just superimposing the effects of these two constructs on one graph. in this case, low of a gain of 10 is 1, times 20 is 20 dbs. so all that has happened is the entire magnitude trace has been shifted up by 20. notice that the phase does not get affected at all.
let me add a zero now, at around 10 radians per second. let me just make sure that my 0 is exactly at 10 radians. and what has happened now is that -40 db slope that i had originally has been shifted up by the 20 dbs per decade of the 0. and the phase that was -180 climbs up to -90. similarly, if i add a pole, let's say at 100 radians per second-- again, let me make sure it is actually at 100 -- now the -40 dbs per decade became -20 at that 0, and then goes back to -40 again after the new pole. so, using this concept of superposition i can easily construct any transfer function that i am interested in studying. all i need to do is break down or factor the transfer function into smaller constructs, and then graphically add all of those traces together.
understanding this simple concept can be quite powerful because it allows us to get a good idea of the primary dynamics of our system just by observing the magnitude and phase traces in a bode diagram.
