ME 430/ECE 451 Equation Sheet
First Order systems:
T(s) = a
s+a
Rise Time:
Tr=2.2
a
Settling Time:
Ts=4.6
a(1%)
Ts=4
a(2%)
Second Order systems:
T(s) = ω2
n
s2+ 2ζωns+ω2
n
Settling Time:
Ts=4.6
ζωn
(1%)
Ts=4.0
ζωn
(2%)
Ts=3.0
ζωn
(5%)
Percent Overshoot:
%OS = 100 e−(πζ /√1−ζ2)
ζ=−ln(%OS/100)
pπ2+ln2(%OS/100)
Rise Time:
Tr=1.64
ωn
(ζ=.5)
Tr=1.86
ωn
(ζ=.6)
Tr=2.13
ωn
(ζ=.7)
Peak Time:
Tp=π
ωnp1−ζ2
Root Locus Refinement:
•Breakaway and break-in points: For G=
NG
DGfind ssuch that: N0
GDG−D0
GNG= 0
•jw-axis crossing: Form Routh Table (for
closed loop transfer function) and find gain
for obtaining a row of zeros. Form aux.
equation using that gain and solve for s.
•Arrival and Departure Angles:
Use : Pθzeros −Pθpoles =±180◦
Steady State Error:
For closed loop transfer function T(s):
E(s) = X(s)−Y(s)
=X(s)[1 −T(s)]
Useful Numbers:
5% overshoot leads to ζ'0.7
16% overshoot leads to ζ'0.5
α= tan−1(ζ
√1−ζ2)
ζ= 0.7 leads to α'45◦
ζ= 0.5 leads to α'30◦
182= 324 √18 '4.2
222= 484 √22 '4.7
262= 676 √26 '5.1
4×26 = 104
9×26 = 234
12 ×26 = 312
√12
2'1.73 √12
6'0.58
√220
6'2.47
