Time Response Flashcards
Values of Laplace variable s that causes the transfer function infinite (based on denominator)
Poles
Values of Laplace variable s that causes the transfer function infinite (based on numerator)
Zeros
Functions without zeros
first order system
the coefficient in response equation (eg. 1)
forced response
the other parts of response equation (eg. e^-t, etc.)
natural response
General First Order Equation
a/s+a
time for e^-at to decay at 37% of initial value
Time constant
time it takes for step response to rise to 63% of the final value
Time constant
parameter a means
exponential frequency
Time constant (Tc) of first order = ?
1/a
The initial slope based on first order function = ?
1/Tc = a
time for wave to go from 0.1 to 0.9 of final value
Rise Time
Rise Time (Tr) for first order = ?
2.2/a
time for response to reach and stay within 2% of the final value
Settling Time
Settling Time (Ts) = ?
4/a
General equation for 2nd Order systems
b/ s^2+as+b
When Overdamped response (Distinct & Real Poles = x1, x2)
= A + Be^-x1t +Ce^-x2t
When Underdamped response (Complex Poles= x1+- jω)
= A + Be^=x1t cos(ωt-ϕ)
When Undamped response (Imaginary Poles= +- jω)
= A + Bcos(ωt-ϕ)
When Underdamped response (Repeating/ Real Poles= x1, x1 )
= A + Be^-x1t +Cte^-x1t
Frequency of
oscillation of the system without damping
Natural Frequency (ωn)
measure of how
oscillations die down after a disturbance
Damping Ratio (ζ)
Natural Frequency (ωn)
= sqrt(b)
Damping Ratio (ζ)
= a/2(ωn)
Equation of 2nd Order with Natural Frequency and Damping Ratio
=(ωn)^2 / s^2 + 2ζ(ωn)s + (ωn)^2
ζ > 1
Overdamped
ζ = 1
Critically Damped
0 < ζ < 1
Underdamped
ζ = 0
Undamped
Rise Time (Tr) for 2nd Order
Tr = (1.76(ζ)^3 - 0.417(ζ)^2 + 1.039(ζ) + 1) / ωn
Peak Time (Tp) for second order, time required to reach the first or max peak
Tp = pi / ωn(sqrt(1-(ζ)^2))
Settling Time (Ts)
Ts = 4 / (ζ)ωn
% Overshoot, amount in which overshoots the steady state value at peak time as % of steady-state value
e^ -(ζpi /(sqrt(1-(ζ)^2)) x 100
