Bode Plots
Question 4c
By introducing an additional pole 1/(s+alpha) to the system,
The original transfer function G(s) has a DC gain of 1 and the additional pole has a DC gain of 1/alpha. Therefore, the new transfer function G1(s) has a DC gain of 1/alpha The DC gain of the new transfer function = 1/alpha . DC gain Is obtained by applying the final value theorem where limit of s goes to 0. Final value theorem can be applied as it is a stable system. Therefore the system has a larger DC gain when alpha is smaller.
In the homogeneous or transient response of a given linear system, each pole or complex pair of poles contribute terms of the form exp(a*t) where “a “ is the value of the real part of the pole in question. As our system is stable where all our poles are in the left-half plane, the contribution the response for a given value of “a” dies out faster if the pole is farther to the left ( since exp(-a*t) approaches zero faster for larger “a”. Therefore, the pole closest to the imaginary axis are the ones that tend to dominate the response since their contribution takes a longer time to die out.
Our additional pole 1/(1+alpha ) is a first order system while our original transfer function G(s) is a second order system In this case the test for the dominant pole compare "α" against "ζω0". This is because ζω0 is the real part of the complex conjugate root (we only compare the real parts of the roots when determining dominance because it is the real part that determines how fast the response decreases). in our case our ζωn (0.358 x 2.24)= 0.801 if alpha is much greater than ζω0, ie alpha = 1000, then the response will exhibits dominant second order behaviour due to 0.801 is much closer to the imaginary axis compared to 1000.
If alpha is 0.001 , it will be much closer to the imaginary axis than our real part of complex conjugate root, therefore it exhibits dominant first order behaviour due to the first order of our additional pole.
Also, the rise
By introducing an additional pole 1/(s+alpha) to the system,
The original transfer function G(s) has a DC gain of 1 and the additional pole has a DC gain of 1/alpha. Therefore, the new transfer function G1(s) has a DC gain of 1/alpha The DC gain of the new transfer function = 1/alpha . DC gain Is obtained by applying the final value theorem where limit of s goes to 0. Final value theorem can be applied as it is a stable system. Therefore the system has a larger DC gain when alpha is smaller.
In the homogeneous or transient response of a given linear system, each pole or complex pair of poles contribute terms of the form exp(a*t) where “a “ is the value of the real part of the pole in question. As our system is stable where all our poles are in the left-half plane, the contribution the response for a given value of “a” dies out faster if the pole is farther to the left ( since exp(-a*t) approaches zero faster for larger “a”. Therefore, the pole closest to the imaginary axis are the ones that tend to dominate the response since their contribution takes a longer time to die out.
Our additional pole 1/(1+alpha ) is a first order system while our original transfer function G(s) is a second order system In this case the test for the dominant pole compare "α" against "ζω0". This is because ζω0 is the real part of the complex conjugate root (we only compare the real parts of the roots when determining dominance because it is the real part that determines how fast the response decreases). in our case our ζωn (0.358 x 2.24)= 0.801 if alpha is much greater than ζω0, ie alpha = 1000, then the response will exhibits dominant second order behaviour due to 0.801 is much closer to the imaginary axis compared to 1000.
If alpha is 0.001 , it will be much closer to the imaginary axis than our real part of complex conjugate root, therefore it exhibits dominant first order behaviour due to the first order of our additional pole.
Also, the rise