For the circuit shown in Figure P6.16, determine a

Determine the equivalent resistance \(R_{e}\) of the circuit shown in Figure P6.1, such that \(v_{s}=R_{e} i\). All the resistors are identical and have the resistance R.

Determine the voltage \(v_{1}\) in terms of the supply voltage \(v_{s}\) for the circuit shown in Figure P6.2.

The Wheatstone bridge, like that shown in Figure P6.3, is used for various measurements. For example, a strain gage sensor utilizes the fact that the resistance of wire changes when deformed. If the sensor is one resistance leg of the bridge, then the deformation can be determined from the voltage \(v_{1}\). Determine the relation between the voltage \(v_{1}\) and the supply voltage \(v_{s}\).

The power supply of the circuit shown in Figure P6.4 supplies a voltage of 9 V. Compute the current i and the power P that must be supplied.

Obtain the model of the voltage \(v_{1}\), given the current \(i_{s}\), for the circuit shown in Figure P6.5.

(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.6. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.7. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.8. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

(a) The circuit shown in Figure P6.9 is a model of a solenoid, such as that used to engage the gear of a car’s starter motor to the engine’s flywheel. The solenoid is constructed by winding wire around an iron core to make an electromagnet. The resistance R is that of the wire, and the inductance L is due to the electromagnetic effect. When the supply voltage \(v_{s}\) is turned on, the resulting current activates the magnet, which moves the starter gear. Obtain the model of the current i given the supply voltage \(v_{s}\). (b) Suppose \(v_{s}(t)=V u_{s}(t)\) and i(0) = 0. Obtain the expression for the response for i(t).

The resistance of a telegraph line is \(R=10 \ \Omega\), and the solenoid inductance is L = 5 H. Assume that when sending a “dash,” a voltage of 12 V is applied while the key is closed for 0.3 s. Obtain the expression for the current i(t) passing through the solenoid. (See Figure 6.2.15)

Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.11.

Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.12.

Obtain the model of the current i, given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.13.

Obtain the model of the voltage \(v_{o}\), given the supply current \(i_{s}\), for the circuit shown in Figure P6.14.

Obtain the model of the currents \(i_{1}, i_{2}\), and \(i_{3}\), given the input voltages \(v_{1}\) and \(v_{2}\), for the circuit shown in Figure P6.15.

Obtain the model of the currents \(i_{1}, i_{2}\), and the voltage \(v_{3}\), given the input voltages \(v_{1}\) and \(v_{2}\), for the circuit shown in Figure P6.16.

For the circuit shown in Figure P6.14, determine a suitable set of state variables, and obtain the state equations.

For the circuit shown in Figure P6.15, determine a suitable set of state variables, and obtain the state equations.

For the circuit shown in Figure P6.16, determine a suitable set of state variables, and obtain the state equations.

Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.20.

Use the impedance method to obtain the transfer function \(I(s) / V_{s}(s)\) for the circuit shown in Figure P6.21.

Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.22.

Use the impedance method to obtain the transfer function \(V_{o}(s) / I_{s}(s)\) for the circuit shown in Figure P6.23.

Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.24.

Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.25.

Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.26.

Draw a block diagram of the circuit shown in Figure P6.15. The inputs are \(v_{1}\) and \(v_{2}\). The output is \(i_{2}\).

Draw a block diagram of the circuit shown in Figure P6.16. The inputs are \(v_{1}\) and \(v_{2}\). The output is \(v_{3}\).

Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.29.

Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.30.

Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.31.

Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.32.

Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.33.

(a) Obtain the transfer function \(\Theta(s) / V_{i}(s)\) for the D’Arsonval meter. (b) Use the final value theorem to obtain the expression for the steady-state value of the angle \(\theta\) if the applied voltage \(v_{i}\) is a step function.

(a) Obtain the transfer function \(\Omega(s) / T_{L}(s)\) for the field-controlled motor of Example 6.5.2. (b) Modify the field-controlled motor model in Example 6.5.2 so that the output is the angular displacement \(\theta\), rather than the speed \(\omega\), where \(\omega=\dot{\theta}\). Obtain the transfer functions \(\Theta(s) / V_{f}(s)\) and \(\Theta(s) / T_{L}(s)\).

Modify the motor model given in Example 6.5.2 to account for a gear pair between the motor shaft and the load. The ratio of motor speed to load speed \(\omega_{L}\) is N. The motor inertia is \(I_{m}\) and the motor damping is \(c_{m}\). The load inertia is \(I_{L}\) and the load damping is \(c_{L}\). The load torque \(T_{L}\) acts directly on the load inertia. Obtain the transfer functions \(\Omega_{L}(s) / V_{f}(s)\) and \(\Omega_{L}(s) / T_{L}(s)\).

The derivation of the field-controlled motor model in Section 6.5 neglected the elasticity of the motor-load shaft. Figure P6.37 shows a model that includes this elasticity, denoted by its equivalent torsional spring constant \(k_{T}\). The motor inertia is \(I_{1}\), and the load inertia is \(I_{2}\). Derive the differential equation model with \(\theta_{2}\) as output and \(v_{f}\) as input.

Figure P6.38 is the circuit diagram of a speed-control system in which the dc motor voltage \(v_{a}\) is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not as sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage \(v_{a}\) is varied by changing the generator input voltage \(v_{f}\). The voltage \(v_{a}\) is related to the generator field current \(i_{f}\) by \(v_{a}=K_{f} i_{f}\). Derive the system model relating the output speed \(\omega\) to the voltage \(v_{f}\), and obtain the transfer function \(\Omega(s) / V_{f}(s)\).

The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c=5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \end{array}\) The manufacturer’s data states that the motor’s maximum speed is 3500 rpm, and the maximum armature current it can withstand without demagnetizing is 40 A. Compute the no-load speed, the no-load current, and the stall torque. Determine whether the motor can be used with an applied voltage of \(v_{a}=15 \mathrm{~V}\).

The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ R_{a}=0.8 \ \Omega \\ L_{a}=3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{array}\) where I includes the inertia of the armature and that of the load. Investigate the effect of the damping constant c on the motor’s characteristic roots and on its response to a step voltage input. Use the following values of c (\(\text { in } \mathrm{N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad}\)): c = 0, c = 0.01, and c = 0.1. For each case, estimate how long the motor’s speed will take to become constant, and discuss whether or not the speed will oscillate before it becomes constant.

The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c=5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a}=4 \times 10^{-3} \mathrm{H} \quad I=5 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{array}\) where c and I include the effect of the load. a. Obtain the step response of \(i_{a}(t)\) and \(\omega(t)\) if the applied voltage is \(v_{a}=10 \mathrm{~V}\). b. Obtain the step response of \(i_{a}(t)\) and \(\omega(t)\) if the load torque is \(T_{L}=0.2 \mathrm{~N} \cdot \mathrm{m}\).

The following measurements were performed on a permanent magnet motor when the applied voltage was \(v_{a}=20 \mathrm{~V}\). The measured stall current was 25 A. The no-load speed was 2400 rpm and the no-load current was 0.6 A. Estimate the values of \(K_{b}, K_{T}, R_{a}\), and c.

A single link of a robot arm is shown in Figure P6.43. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque \(T_{m}\) through spur gears. Suppose that the equivalent inertia felt at the motor shaft is \(0.215 \mathrm{~kg} \cdot \mathrm{m}^{2}\). As the arm rotates, the effect of the arm weight generates an opposing torque that depends on the arm angle, and is therefore nonlinear. For this problem, however, assume that the effect of the opposing torque is a constant \(4.2 \mathrm{~N} \cdot \mathrm{m}\) at the motor shaft. Neglect damping in the system. It is desired to have the motor shaft rotate through \(3 \pi / 4 \ \mathrm{rad}\) in a total time of 2 s, using a trapezoidal speed profile with \(t_{1}=0.3 \mathrm{~s}\) and \(t_{2}=1.7 \mathrm{~s}\). The given motor parameters are \(R_{a}=4 \ \Omega, L_{a}=3 \times 10^{-3} \ \mathrm{H}\), and \(K_{T}=0.3 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}\). Compute the energy consumption per cycle; the maximum required torque, current, and voltage; the rms torque; and the rms current.

A conveyor drive system to produce translation of the load is shown in Figure P6.44. Suppose that the equivalent inertia felt at the motor shaft is \(0.05 \mathrm{~kg} \cdot \mathrm{m}^{2}\), and that the effect of Coulomb friction in the system produces an opposing torque of \(3.6 \mathrm{~N} \cdot \mathrm{m}\) at the motor shaft. Neglect damping in the system. It is desired to have the motor shaft rotate through 11 revolutions in a total time of 3 s, using a trapezoidal speed profile with \(t_{1}=0.5 \mathrm{~s}\) and \(t_{2}=2.5 \mathrm{~s}\). The given motor parameters are \(R_{a}=3 \ \Omega, L_{a}=4 \times 10^{-3} \ \mathrm{H}\), and \(K_{T}=0.4 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}\). Compute the energy consumption per cycle; the maximum required torque, current, and voltage; the rms torque; and the rms current.

Consider the accelerometer model in Section 6.7. Its transfer function can be expressed as \(\frac{Y(s)}{Z(s)}=-\frac{s^{2}}{s^{2}+(c / m) s+k / m}\) Suppose that the input displacement is z(t) = 10 sin 120t mm. Consider two cases, in SI units: (a) k/m = 100 and c/m = 18 and (b) \(k / m=10^{6}\) and c/m = 1800. Obtain the steady-state response y(t) for each case. By comparing the amplitude of y(t) with the amplitudes of z(t) and \(\ddot{z}(t)\), determine which case can be used as a vibrometer (to measure displacement) and which can be used as an accelerometer (to measure acceleration).

An electromagnetic microphone has a construction similar to that of the speaker shown in Figure 6.7.2, except that there is no applied voltage and the sound waves are incoming rather than outgoing. They exert a force \(f_{s}\) on the diaphragm whose mass is m, damping is c, and stiffness is k. Develop a model of the microphone, whose input is \(f_{s}\) and output is the current i in the coil.

Consider the speaker model developed in Example 6.7.1. The model, whose transfer function is given by equation (3) in that example, is third order and therefore we cannot obtain a useful expression for the characteristic roots. Sometimes inductance L and damping c are small enough to be ignored. If L = 0, the model becomes second order. (a) Obtain the transfer function X(s)/V(s) for the case where L = c = 0, and obtain the expressions for the two roots. (b) Compare the results with the third-order case where \(\begin{aligned} m & =0.002 \mathrm{~kg} & & k=4 \times 10^{5} \mathrm{~N} / \mathrm{m} \\ K_{f} & =16 \mathrm{~N} / \mathrm{A} & & K_{b}=13 \mathrm{~V} \cdot \mathrm{s} / \mathrm{m} \\ R & =12 \ \Omega & & L=10^{-3} \ \mathrm{H} \\ c & =0 & & \end{aligned}\)

The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H} \quad I=5 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\) where c and I include the effect of the load. The load torque is zero. Use MATLAB to obtain a plot of the step response of \(i_{a}(t)\) and \(\omega(t)\) if the applied voltage is \(v_{a}=10 \mathrm{~V}\). Determine the peak value of \(i_{a}(t)\).

Consider the motor whose parameters are given in Problem 6.48. Use MATLAB to obtain a plot of the response of \(i_{a}(t)\) and \(\omega(t)\) if the applied voltage is the modified step \(v_{a}(t)=10\left(1-e^{-100 t}\right) \mathrm{V}\). Determine the peak value of \(i_{a}(t)\).

Consider the circuit shown in Figure P6.50. The parameter values are \(R=10^{3} \ \Omega, C=2 \times 10^{-6} \mathrm{~F}\), and \(L=2 \times 10^{-3} \ \mathrm{H}\). The voltage \(v_{1}\) is a step input of magnitude 5 V, and the voltage \(v_{2}\) is sinusoidal with frequency of 60 Hz and an amplitude of 4 V. The initial conditions are zero. Use MATLAB to obtain a plot of the current response \(i_{3}(t)\).

The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =3 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H} \quad I=4 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\) The system uses a gear reducer with a reduction ratio of 3:1. The load inertia is \(10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\), the load torque is 0.04 N · m, and the load damping constant is \(1.8 \times 10^{-3} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad}\). Use MATLAB to obtain a plot of the step response of \(i_{a}(t)\) and \(\omega(t)\) if the applied voltage is \(v_{a}=20 \mathrm{~V}\). Determine the peak value of \(i_{a}(t)\).

The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =0 \quad R_{a}=0.8 \Omega \\ L_{a} & =3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\) where I includes the inertia of the armature and that of the load. The load torque is zero. The applied voltage is a trapezoidal function defined as follows. \(v(t)=\left\{\begin{array}{ll} 60 t & 0 \leq t \leq 0.5 \\ 30 & 0.5<t<2 \\ 60(2.5-t) & 2 \leq t \leq 2.5 \\ 0 & 2.5<t \leq 4 \end{array}\right.\) a. Use MATLAB to obtain of plot of the response of \(i_{a}(t)\) and \(\omega(t)\). b. Compute the energy consumption per cycle; the maximum required torque, current, and voltage; the rms torque; and the rms current.

A single link of a robot arm is shown in Figure P6.43. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque \(T_{m}\) through spur gears. Suppose that the equivalent inertia felt at the motor shaft is \(0.215 \mathrm{~kg} \cdot \mathrm{m}^{2}\). As the arm rotates, the effect of the arm weight generates an opposing torque that depends on the arm angle, and is therefore nonlinear. The effect of the opposing torque at the motor shaft is \(4.2 \sin \theta \ \mathrm{N} \cdot \mathrm{m}\). Neglect damping in the system. It is desired to have the motor shaft rotate through \(3 \pi / 4 \ \mathrm{rad}\) in a total time of 2 s, using a trapezoidal speed profile with \(t_{1}=0.3 \mathrm{~s}\) and \(t_{2}=1.7 \mathrm{~s}\). The given motor parameters are \(R_{a}=4 \ \Omega, L_{a}=3 \times 10^{-3} \ \mathrm{H}\), and \(K_{T}=0.3 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}\). Use MATLAB to obtain of plot of the response of the motor current and the motor speed.

Consider the circuit shown in Figure P6.50. The parameter values are \(R=10^{4} \ \Omega, C=2 \times 10^{-6} \mathrm{~F}\), and \(L=2 \times 10^{-3} \ \mathrm{H}\). The voltage \(v_{1}\) is a single pulse of magnitude 5 V and duration 0.05 s, and the voltage \(v_{2}\) is sinusoidal with frequency of 60 Hz and an amplitude of 4 V. The initial conditions are zero. Use Simulink to obtain a plot of the current response \(i_{3}(t)\).

Consider the circuit shown in Figure P6.55. The parameter values are \(R=2 \times 10^{4} \ \Omega\) and \(C=3 \times 10^{-6} \mathrm{~F}\). The voltage \(v_{s}\) is \(v_{s}(t)=12 u_{s}(t)+3 \sin 120 \pi t \mathrm{~V}\). The initial conditions are zero. Use Simulink to obtain a plot of the responses \(v_{o}(t)\) and \(v_{1}(t)\).

The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H} \quad I=5 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\) where c and I include the effect of the load. The load torque is zero. a. Use Simulink to obtain a plot of the step response of the motor torque and speed if the applied voltage is \(v_{a}=10 \mathrm{~V}\). Determine the peak value of the motor torque. b. Now suppose that the motor torque is limited to one-half the peak value found in part (a). Use Simulink to obtain a plot of the step response of the motor torque and speed if the applied voltage is \(v_{a}=10 \mathrm{~V}\).

The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =0 \quad R_{a}=0.8 \Omega \\ L_{a} & =3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\) where I includes the inertia of the armature and that of the load. The load torque is zero. The applied voltage is a trapezoidal function defined as follows. \(v(t)=\left\{\begin{array}{ll} 60 t & 0 \leq t \leq 0.5 \\ 30 & 0.5<t<2 \\ 60(2.5-t) & 2 \leq t \leq 2.5 \\ 0 & 2.5<t \leq 4 \end{array}\right.\) A trapezoidal profile can be created by adding and subtracting ramp functions starting at different times. Use several Ramp source blocks and Sum blocks in Simulink to create the trapezoidal input. Obtain a plot of the response of \(i_{a}(t)\) and \(\omega(t)\).