For the function F(s) = 3 / (s * (s^2 + 2s + 5)), what is the correct partial-fraction expansion form?
Explanation
When performing a partial-fraction expansion, the form of the numerator for each term depends on the denominator. A simple pole gets a constant numerator (K), while an irreducible quadratic factor (with complex roots) gets a linear numerator (Ks+C).
Other questions
What is the definition of the transfer function, G(s), of a system?
According to the Partial-Fraction Expansion method, what is the correct expansion form for a function F(s) with a repeated real root in the denominator, such as F(s) = 2 / ((s+1)(s+2)^2)?
For a translational mechanical system, which electrical component is analogous to a mass (M) in a series analog (force-velocity analogy)?
What is the process of linearization as described for a nonlinear system?
For the inverting operational amplifier circuit shown in Figure 2.10c, what is the transfer function Vo(s)/Vi(s)?
What is the transfer function G(s) = C(s)/R(s) for a system represented by the differential equation: (d^2c/dt^2) + 6(dc/dt) + 2c = 2(dr/dt) + r ?
When reflecting a rotational mechanical impedance (like inertia J or damping D) from an output shaft to an input shaft through a gear train, what factor is used?
In the context of a DC motor's torque-speed curve, what does 'stall torque' represent?
What are the two properties that define a linear system?
Find the inverse Laplace transform of the function F(s) = (s+2)/(s(s+5)).
What is the impedance, ZM(s), of a translational mass, M, in a mechanical system?
For the series RLC circuit in Figure 2.3, what is the transfer function VC(s)/V(s), where VC(s) is the capacitor voltage and V(s) is the input voltage?
What physical nonlinearity is described as the phenomenon where the input must exceed a certain threshold before the system responds?
In a rotational mechanical system, what is the impedance ZM(s) of a viscous damper with a coefficient of viscous friction D?
What is the primary reason for using a gear train in a rotational mechanical system?
For the two-degrees-of-freedom translational mechanical system in Figure 2.17(a), what is the term representing the forces transmitted between M1 and M2?
What is the Final Value Theorem, as listed in Table 2.2?
How is the equation of motion for a single-degree-of-freedom translational mechanical system like the one in Figure 2.15(a) typically formed?
A DC motor has a stall torque of 100 N-m and a no-load speed of 50 rad/s when the applied armature voltage is 25 V. What is the value of the motor's back emf constant, Kb?
For the noninverting operational amplifier in Figure 2.12, what is the ideal transfer function Vo(s)/Vi(s)?
A transfer function is given by G(s) = 7/(s^2+5s+10). Which differential equation corresponds to this transfer function?
According to the Differentiation Theorem in Table 2.2, what is the Laplace transform of the first derivative of a function, df/dt?
What is the primary advantage of using a transfer function representation over a differential equation, as mentioned in the introduction of Chapter 2?
What is the equivalent inertia, Je, at the input shaft for the gear system shown in Figure 2.32b?
In a parallel analog of a mechanical system (force-current analogy), what electrical component is analogous to a mechanical spring with spring constant K?
Find the transfer function G(s) = X2(s)/F(s) for the two-degrees-of-freedom system in Figure 2.17(a), where the numerator is given by (f_v3*s + K2). What is the denominator, Delta?
What is the key simplifying assumption made when deriving the transfer function for an ideal gear train in Section 2.7?
What is the transfer function G(s) = VL(s)/V(s) for the circuit in Skill-Assessment Exercise 2.6, which has two 1-ohm resistors, three 1-H inductors, and a capacitor implicitly defined by the diagram elements?
How is the transfer function of a nonminimum-phase system, which has a zero in the right half-plane, characterized in its step response?
In the Laplace Transform Solution of a Differential Equation shown in Example 2.3, for the equation d^2y/dt^2 + 12dy/dt + 32y = 32u(t), what are the poles of the system's transfer function?
For the rotational system with gears in Example 2.21, if the input torque T1(t) is applied to the shaft with inertia J1, and the output is the displacement theta2(t) of the shaft with inertia J2, how is the torque reflected to the output shaft?
What is the primary difference between modeling a translational mechanical system and a rotational mechanical system?
In the linearization of the pendulum in the Human Leg case study (Figure 2.52), the term Mg(L/2)sin(theta) is linearized for small angles around theta=0. What is the resulting linear approximation of this torque?
What is the term for the number of linearly independent motions in a mechanical system?
What is the admittance Y(s) of an electrical inductor with inductance L?
According to the Frequency Shift Theorem (Item 4 in Table 2.2), if the Laplace transform of f(t) is F(s), what is the Laplace transform of e^(-at)f(t)?
In the Antenna Control case study (Section 2.12), the transfer function for the power amplifier is given as 100/(s+100). What does this imply about the amplifier's behavior?
When using nodal analysis to write equations for an electrical network, what quantity is summed at each node?
In the DC motor model in Section 2.8, the armature inductance, La, is often neglected. Why is this assumption typically made?
What is the key difference in the procedure for finding the transfer function of a multiple-loop electrical circuit versus a single-loop circuit?
Based on Example 2.16, which shows a mass-spring-damper system, what is the transfer function X(s)/F(s)?
What is the defining characteristic of a system's poles, according to the definition in Section 2.2?
For the nonminimum-phase circuit in Example 4.9, the transfer function is found to be -(s-10)/(s+10). For a unit step input, the response c(t) is what function?
In the context of the equations of motion for multi-degree-of-freedom systems written by inspection (e.g., Equation 2.120), what do the off-diagonal terms in the impedance matrix represent?
A transfer function for a DC motor is given by Theta_m(s)/Ea(s) = (Kt/(Ra*Jm)) / (s * (s + (1/Jm)*(Dm + (Kt*Kb)/Ra))). If Kt=0.5 N-m/A, Ra=2 ohms, Jm=0.1 kg-m^2, Dm=0.2 N-m-s/rad, and Kb=0.5 V-s/rad, what is the simplified transfer function?
If you linearize the function f(x) = 5cos(x) about the point x = pi/2, what is the resulting linear equation?
What is the primary role of the numerator of a transfer function in determining the system's time response?
When is pole-zero cancellation considered a valid approximation for simplifying a higher-order system?
Find the transfer function Vo(s)/Vi(s) for the inverting op-amp circuit in Figure 2.11, where Z1 is a parallel 5.6 uF capacitor and 360 kOhm resistor, and Z2 is a series 0.1 uF capacitor and 220 kOhm resistor.