Coursera Specialization on Embedding
Sensors and Motors
ESM_C4M1V2
Rapid Review of Second Order Transfer Functions
= Md2x/dt2 + b dx/dt + k x
[5]
[1]
[3]
C
[2]
2nd Order Differential Equation
for spring-mass damper system
[4]
R
L
2nd Order Differential Equation for
resistor-inductor-capacitor system
Natural Response for 2nd Order Equations
= Md2x/dt2 + b dx/dt + k x
[5]
[1]
Let x(t) = Xest
Quadratic formula
Let i(t) = Iest
[7]
[6]
R
Solutions for natural frequencies (i.e. poles)
L
Spring-mass-damper
Criteria for Oscillation Cases
Series RLC circuit
Damping Coefficient
Damping Ratio
[10]
b
[11]
[8]
Natural Frequency
Natural Frequency
n
= -α
[9]
[12]
Damped Natural Frequency
Damped Natural Frequency
n
[13]
Case
Condition
Undamped
ζ=b=0
Underdamped
b2 – 4mk <0
•
•
Critically
damped
b2
•
Overdamped
b2 – 4mk >0
– 4mk = 0
Pole location for underdamped
R case
Poles are complex conjugates in the
left half of the s-plane
0 < ζ <1 or for
j
Case
Condition
Undamped
α=R=0
Underdamped
<0
Critically
damped
=0
Overdamped
>0
Solution for Underdamped Case
Location of poles in
spring-mass-damper
system
[10]
• Let σ = b/2m = ζωn
• then s1 = - σ + jωd
• and s2 = - σ - jωd
Location of poles in
series RLC circuit
[15]
n
= -α
n
Solution for position
Solution for current
+
[14]
α = x0
β = (v0 + σx0) / ωd)
•
•
x0 = initial position
v0 = initial velocity
•
Pole location for underdamped
R case
Poles are complex conjugates in the
left half of the s-plane
0 < ζ <1 or for
B1 = i0
B2 = (di0/dt + αi0) / ωd)
i0 = initial current
di0/dt = initial slope of current
Response Time for Step Function
[16]
Critical damping gives the fastest system response
time that also does not overshoot the final setpoint
Decay Rate Depends on Natural Frequency and Damping Ratio
[17]
• Higher natural frequency leads to
faster speed of response
• Decay rate, τ = 1 / ζωn
• Higher damping minimizes
overshoot
• Decay rate, τ = 1 / ζωn
[18]
Citations
[1][5][7][11][15] www.slideplayer.net
[2] www.azimadli.com
[3] www.electronics-tutorials.com
[4][8] www.wikipedia.com
[6][9][10][14][17] www.ocw.mit.edu
[12] www.hyperphysics.phy-astr.gsu.edu
[13] www.kaztechnologies.com
[16] www.dummies.com
[18] www.chegg.com
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