The response behavior of second-order systems is more complex than first- order systems. Their behavior is governed by the equation
1 ωn2 ¨ y + 2ζ ωn ˙y + y = KF (t), (4.46) where ωn = p
a0/a2 denotes the natural frequency and ζ = a1/2√a0a2
damping ratio of the system. Note that when 2ζ >> 1/ωn, the second
derivative term in Equation 4.46 becomes negligible with respect to the other terms, and the system behavior approaches that of a first-order system with a system time constant equal to 2ζ/ωn.
FIGURE 4.5
The time history of the thermocouple system.
Equation 4.46 could represent, among other things, a mechanical spring- mass-damper system or an electrical capacitor-inductor-resistor circuit, both with forcing. The solution to this type of equation is rather lengthy and is described in detail in many applied mathematics texts (see [24]). Now exam- ine where such an equation would come from by considering the following example.
A familiar situation occurs when a bump in the road is encountered by a car. If the car has a good suspension system it will absorb the effect of the bump. The bump hardly will be felt. On the other hand, if the suspension system is old, an up-and-down motion is present that may take several seconds to attenuate. This is the response of a linear, second-order system (the car with its suspension system) to an input forcing (the bump).
The car with its suspension system can be modeled as a mass (the body of the car and its passengers) supported by a spring (the suspension coil) and a damper (the shock absorber) in parallel (usually there are four sets of spring-dampers, one for each wheel). Newton’s second law can be applied, which states that the mass times the acceleration of a system is equal to the sum of the forces acting on the system. This becomes
md 2y dt2 = X i Fi= Fg+ Fs(t) + Fd(t) + F (t), (4.47)
Fs(t) is the spring force (= −k[L∗+ y]), where k is the spring constant and
L∗ the initial compressed length of the spring, F
d(t) is the damping force
(= −γdy/dt), where γ is the damping coefficient, and F (t) is the forcing function. Note that the spring and damping forces are negative because they are opposite to the direction of motion. The height of the bump as a function of time as dictated by the speed of the car would determine the exact shape of F (t). Now when there is no vertical displacement, which is the case just before the bump is encountered, the system is in equilibrium and y does not change in time. Equation 4.47 reduces to
0 = mg − kL∗. (4.48)
This equation can be used to replace L∗ in Equation 4.47 to arrive at
m k d2y dt2 + γ k dy dt + y = 1 kF (t). (4.49)
Comparing this equation to Equation 4.46 yields ωn=
p
k/m, ζ = γ/√4km, and K = 1/k.
Another example of a second-order system is an electrical circuit com- prised of a resistor, R, a capacitor, C, and an inductor, L, in series with a voltage source with voltage, Ei(t), that completes a closed circuit. The
voltage differences, ∆V , across each component in the circuit are ∆V = RI for the resistor, ∆V = LdI/dt for the inductor, and ∆V = Q/C for the capacitor, where the current, I, is related to the charge, Q, by I = dQ/dt. Application of Kirchhoff’s voltage law to the circuit’s closed loop gives
LCd 2I dt2 + RC dI dt + I = C dEi(t) dt . (4.50)
Comparing this equation to Equation 4.46 gives ωn =
p
1/LC, ζ = R/p4L/C, and K = C.
The approach to solving a nonhomogeneous, linear, second-order, ordi- nary differential equation with constant coefficients of the form of Equa- tion 4.46 involves finding the homogeneous, yh(t), and particular, yp(t),
solutions and then linearly superimposing them to form the complete solu- tion, y(t) = yh(t)+yp(t). The values of the arbitrary coefficients in the yh(t)
solution are determined by applying the specified initial conditions, which are of the form y(0) = yo and ˙y(0) = ˙yo. The values of the arbitrary coef-
ficients in the yp(t) solution are found through substitution of the general
form of the yp(t) solution into the differential equation and then equating
like terms.
The form of the homogeneous solution to Equation 4.46 depends upon roots of its corresponding characteristic equation
1 ωn2
r2+ 2ζ
ωn
which are
r1,2= −ζωn± ωn
p
ζ2− 1. (4.52)
Depending upon the value of the discriminant pζ2− 1, there are three
possible families of solutions (see the text web site for the step-by-step so- lutions):
• ζ2− 1 > 0: the roots are real, negative, and distinct. The general form
of the solution is
yh(t) = c1er1t+ c2er2t. (4.53)
• ζ2− 1 = 0: the roots are real, negative, and equal to −ω
n. The general
form of the solution is
yh(t) = c1ert+ c2tert. (4.54)
• ζ2− 1 < 0: the roots are complex and distinct. The general form of the
solution is
yh(t) = c1er1t+ c2er2t= eλt(c1cos µt + c2sin µt), (4.55)
using Euler’s formula eit= cos t + i sin t and noting that
r1,2= λ ± iµ, (4.56)
with λ = −ζωn and µ = ωn
p 1 − ζ2.
All three general forms of solutions have exponential terms that decay in time. Thus, as time increases, all homogeneous solutions tend toward a value of zero. Such solutions often are termed transient solutions. When 0 < ζ < 1 (whenpζ2− 1 < 0) the system is called under-damped; when
ζ = 1 (when pζ2− 1 = 0) it is called critically damped; when ζ > 1
(whenpζ2− 1 > 0) it is called over-damped. The reasons for these names
will be obvious later. Now examine how a second-order system responds to step and sinusoidal inputs.
4.6.1
Response to Step-Input Forcing
The responses of a second-order system to a step input having F (t) = A for t > 0 with the initial conditions y(0) = 0 and ˙y(0) = 0 are as follows:
• For the under-damped case (0 < ζ < 1) y(t) = KA ( 1 − e−ζωnt " 1 p 1 − ζ2sin(ωnt p 1 − ζ2+ φ) #) (4.57) where φ = sin−1(p1 − ζ2). (4.58)
As shown by Equation 4.57, the output initially overshoots the input, lags it in time, and is oscillatory. As time continues, the oscillations damp out and the output approaches, and eventually reaches, the input value. A special situation arises for the no-damping case when ζ = 0. For this situation the output lags the input and repeatedly overshoots and undershoots it forever.
• For the critically damped case (ζ = 1),
y(t) = KA1 − e−ωnt(1 + ω
nt)
. (4.59)
No oscillation is present in the output. Rather, the output slowly and monotonically approaches the input, eventually reaching it.
• For the over-damped case (ζ > 1), y(t) = KA · {1 − e−ζωnt[cosh(ω nt p ζ2− 1) +p ζ ζ2− 1sinh(ωnt p ζ2− 1)]}. (4.60)
The behavior is similar to the ζ = 1 case. Here the larger the value of ζ, the longer it takes for the output to reach the value of the input signal. Note that in the equations of all three cases the quantity ζωn in the
exponential terms multiplies the time. Hence, the quantity 1/ζωnrepresents
the time constant of the system. The larger the value of the time constant, the longer it takes the response to approach steady state. Further, because the magnitude of the step-input forcing equals KA, the magnitude ratio, M (t), for all three cases is obtained simply by dividing the right sides of Equations 4.57, 4.59, and 4.60 by KA.
Equations 4.57 through 4.60 appear rather intimidating. It is helpful to plot these equations rewritten in terms of their magnitude ratios and examine their form. The system response to step-input forcing is shown in Figure 4.6 for various values of ζ. The quickest response to steady state is when ζ = 0 (that is when the time constant 1/ζωn is minimum). However,
such a value of ζ clearly is not optimum for a measurement system because the amplitude ratio overshoots, then undershoots, and continues to oscillate
FIGURE 4.6
The magnitude ratio of a second-order system responding to step-input forc- ing.
about a value of M (ω) = 1 forever. The oscillatory behavior is known as ringing and occurs for all values of ζ < 1.
Shown in Figure 4.7 is the response of a second-order system having a value of ζ = 0.2 to step-input forcing. Note the oscillation in the response about an amplitude ratio of unity. In general, this oscillation is characterized by a period Td, where Td = 2π/ωd, with the ringing frequency ωd =
ωn
p
1 − ζ2. The rise time for a second-order system is the time required
for the system to initially reach 90 % of its steady-state value. The settling time is the time beyond which the response remains within ± 10 % of its steady-state value.
A value of ζ = 0.707 quickly achieves a steady-state response. Most second-order instruments are designed for this value of ζ. When ζ = 0.707, the response overshoot is within 5 % of M (t) = 1 within about one-half of the time required for a ζ = 1 system to achieve steady state. For values of ζ > 1, the system eventually reaches a steady-state value, taking longer times for larger values of ζ.
4.6.2
Response to Sinusoidal-Input Forcing
The response of a second-order system to a sinusoidal input having F (t) = KA sin(ωt) with the initial conditions y(0) = 0 and ˙y(0) = 0 is
FIGURE 4.7
The temporal response of a second-order system with ζ = 0.2 to step-input forcing.
yp(t) =
KA sin[ωt + φ(ω)] {[1 − (ω/ωn)2]2+ [2ζω/ωn]2}1/2
, (4.61)
where the phase lag in units of radians is φ(ω) = − tan−1 2ζω/ωn 1 − (ω/ωn)2 for ω ωn ≤ 1, (4.62) or φ(ω) = −π − tan−1 2ζω/ωn 1 − (ω/ωn)2 for ω ωn > 1. (4.63)
Note that Equation 4.61 is the particular solution, which also is the steady- state solution. This is because the homogeneous solutions for all ζ are tran- sient and tend toward a value of zero as time increases. Hence, the steady- state magnitude ratio based upon the input KA sin(ωt), Equation 4.61 be- comes
M (ω) = 1
{[1 − (ω/ωn)2]2+ [2ζω/ωn]2}1/2
These equations show that the system response will contain both magnitude and phase errors. The magnitude and phase responses for different values of ζ are shown in Figures 4.8 and 4.9, respectively. Note that the magnitude ratio is a function of frequency, ω, for the sinusoidal-input forcing case, whereas it is a function of time, t, for the step-input forcing case.
First examine the magnitude response shown in Figure 4.8. For low values of ζ, approximately 0.6 or less, and ω/ωn ≤ 1, the magnitude ra-
tio exceeds unity. The maximum magnitude ratio occurs at the value of ω/ωn =
p
1 − 2ζ2. For ω/ω
n≥∼ 1.5, the magnitude ratio is less than unity
and decreases with increasing values of ω/ωn.
Typically, magnitude attenuation is given in units of dB/decade or dB/octave. A decade is defined as a 10-fold increase in frequency (any 10:1 frequency range). An octave is defined as a doubling in frequency (any 2:1 frequency range). For example, using the information in Figure 4.8, there would be an attenuation of approximately −8 dB/octave [= 20log(0.2) − 20log(0.5)] in the frequency range 1 ≤ ω/ωn ≤ 2 when ζ = 1.
Now examine the phase response shown in Figure 4.9. As ω/ωnincreases,
the phase angle becomes more negative. That is, the output signal begins to lag the input signal in time, with this lag time increasing with ω/ωn. For
values of ω/ωn< 1, this lag is greater for greater values of ζ. At ω/ωn = 1,
all second-order systems having any value of ζ have a phase lag of −90◦ or
1/4 of a cycle. For ω/ωn > 1, the increase in lag is less for systems with
greater values of ζ.