## Mass-spring-damper rotational system. Problems solved. Catalog 3

The transfer function of a rotational Mass-Spring-Damper System.

In this PDF guide, the Transfer Function of the exercises that are most commonly used in the mass-spring-damper system classes that are in turn part of control systems, signals and systems, analysis of electrical networks with DC motor, is determined. electronic systems in mechatronics, etc. It is a good resource to also learn how to obtain the block diagram of the system, or the representation in state variables. Request via email – WhatsApp. Payment is provided by PayPal, Credit or debit card. Cost: € 15

Below, the statements of problems solved in this guide.

1. Given the System of Figure 22, find the transfer function Θ(s)/T(s).

2. Given the System of Figure 23, find the transfer function Θ(s)/T(s).

3. Given the System of Figure 24, find the transfer functions Θ1(s)/T(s)  and Θ2(s)/T(s).

4. Find the state-space representation of the system of the previous exercise, Figure 24, taking Θ1(t)  as the output and T(t) as the input. Build the block diagram of the system and determine the transfer function Θ1(s)/T(s).

5. Given the System of Figure 26, find the transfer function ΘL(s)/Tm(s).

6. Given the System of Figure 27, find the transfer functions Θ1(s)/Tm(s) and Θ2(s)/Tm(s).

7. Given the System of Figure 28, find the transfer functions Θ1(s)/T(s) and Θ2(s)/T(s).

8. Given the System of Figure 29, find the transfer function Θ2(s)/T(s).

9. Given the System of Figure 30, find the transfer functions  Θ1(s)/T(s) and Θ2(s)/T(s) . Consider: k1=9, k2=3 N-m/rad, b1=8, b2=1 N-m-s/rad, J1=5, J2=3 Kg-m2.

10. Find the state-space representation of the system of the previous exercise, Figure 30, taking Θ2(t) as the output and T(t) as the input. Direct Transform the state-space representation obtained into the transfer function Θ2(s)/T(s). Consider k1=9, k2=3 N-m/rad, b1=8, b2=1 N-m-s/rad, J1=5, J2=3 Kg-m2.

11. Find the state-space representation of the system in Figure 32, taking Θ2(t)  as the output and T(t) as the input. Directly, using Matlab, Transform the state-space representation obtained into the transfer function  Θ2(s)/T(s). Consider k1= k2=1 N-m/rad, b1= b2=1 N-m/rad, J=1 Kg-m2.

12. Given the System of Figure 33, find the transfer functions  Θ1(s)/Tm(s) and Θ2(s)/Tm(s).

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Mass-spring-damper rotational system. Problems solved. Catalog 3

In this PDF guide, the Transfer Function of the exercises that are most commonly used in the mass-spring-damper system classes that are in turn part of control systems, signals and systems, analysis of electrical networks with DC motor, is determined. electronic systems in mechatronics, etc. It is a good resource to also learn how to obtain the block diagram of the system, or the representation in state variables. Request via email – WhatsApp

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## Definition of Electromechanical System

“Electromechanical Systems are those hybrid systems of mechanical and electrical variables.” Applications for electromechanical components cover a broad spectrum, from control systems for robots and star-trackers, to household appliances and hard disk position controls on a computer, or the control of DC motors in air conditioning systems for residential installations.

Detail of copper winding, stack and shaft of a electric permeant magnet motor for home appliances.

Figure 2.1 shows an electromechanical drive system. It consists of a power and energy source, a gate circuit for the converter, electronic converters (rectifier, inverter, electronic power controller), current sensors (shunts, current transformer, Hall sensor), voltage sensor (divider voltage, potential transformer), speed sensors (tachometers) and displacement sensors (encoders), three-phase rotary machines, gearboxes and specific loads (pump, fan, car, etc.). In Figure 2-1 all components, with the exception of gears, are represented by a Transfer Function (output variables as a function of time), while the gearbox is represented by a Characteristic Function (Xout output variable depending on the input variable Xin)

The electric machine is perhaps the best example of an electromechanical device because of the frequency with which it is used in numerous applications of daily life. An electric machine is a device that can convert mechanical energy into electrical energy (a hydroelectric plant, for example), or convert electrical energy into mechanical energy (a motor).

For the study of electromechanical systems from the point of view of control engineering, we have decided to focus our attention on DC motors, especially armature-controlled DC servo motors, as they are components intensively used in emerging industries that combine electromechanical engineering with Telematics, as is the case with Robotics and Drones technology. And because, precisely, these areas, together with that of electric vehicles and industry 4.0, are initiating a paradigm shift in all areas of life.

We are dedicated to developing the mathematical model of an electromechanical system with DC motor, as well as the characteristics of this system when it is part of an open loop or closed loop control system (Servomotors). We also provide numerous examples of how to determine and use the Transfer Function of an electromechanical system to analyze its stability and its response over time (transient and steady state).

And gradually we will cover these industries more specifically, with great potential for innovation and future labor demand.

NEXT:

Sources:

1. Control Systems Engineering, Nise
2. Sistemas de Control Automatico Benjamin C Kuo
3. Modern_Control_Engineering, Ogata 4t
4. Libro Rashid – Power Electronic Handbook p 663-666
5. Getty Images

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## Open-loop control system – Electromechanical.

A control system can be open loop or closed loop. To understand this difference we must pay attention to the effect that the output has on the system control action (Ogata, 1998). If the output influences the control action, the system is closed loop. On the other hand, if the output does not affect the control action, we are in the presence of an open-loop control system.

The controlled variable is the quantity or condition that is measured or controlled. The manipulated variable, or control variable, is the quantity or condition that the controller modifies to affect the value of the controlled variable.

To better understand the concept of an open-loop system, consider the following scheme, which represents a very frequent and basic component in every electromechanical system, a Potentiometer:

In practice, the operation of this system is simple. Needle position B (control variable) depends on angular displacement desplazamiento Θi(t) (system input). The position of the needle determines a voltage Vo (t) (system output, controlled variable) that can have a value between +50 and -50 volts. In this system, the output does not affect the control action, which is the mechanical movement of the hand (controller). Therefore, it is an open loop system, which we can represent by the following block diagram:

If we wanted to configure the system of Figure 2 as a closed loop system, we would have to measure the output, first, and compare it with the reference signal, secondly, so that a Controller executes the controlling action based on the result of this comparison. This process could be represented by the following diagram:

Quite often, the Potentiometer in Figure 1 is the component that activates a DC Motor as shown in the following example:

The system of Figure 4 is another example of an open-loop electromechanical system, which involves a greater number of components, including the use of a DC Motor and a Gearbox that allows to transform a rotational movement into a translational displacement, but in which the output does not influence the controlling action.

The following system, on the other hand, also has a Potentiometer that measures the displacement at the output and this measure influences the control action:

The system of Figure 5 is a closed loop electromechanical system that compares the output voltage c with the input voltage r. This comparison is manifested as a voltage difference ev=r-c that then feeds a Differential Amplifier, which in turn activates a DC Motor that, through a Gear system, moves the Potentiometer c. This process is repeated until ev=0, that is, until r = c. In other words, the system looks for the output to match the input, so this system is called the automatic input follower system, Position Control System or Servosystem.

When a DC Motor is part of a Servo System, it is called Servo Motor. The Position Control System is one of the most used essential mechanisms in engineering, hence its great importance.

Sources:

1. Control Systems Engineering, Nise
2. Sistemas de Control Automatico Benjamin C Kuo
3. Modern_Control_Engineering, Ogata 4t
4. Libro Rashid – Power Electronic Handbook p 663-666

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

WhatsApp:  +34633129287

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Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

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## Control System Stability

The most important problem in linear control systems concerns stability. It is the most important system specification among the three requirements enter into the design of a control system: transient response, stability and steady-state error. That is, under what conditions will a system become unstable? If it is unstable, how should we stabilize the system?

The total response of a system is the sum of the forced and natural responses:

Using this concept, we present the following definition of stability, instability and marginal stability:

• A linear time-invariant system is stable if the natural response approaches to zero as time approaches infinity.
• A linear time-invariant system is unstable if the natural responses grows without bound as time approaches infinity.
• A linear time-invariant system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.

Thus the definition of stability implies that only the forced response remains as the natural response approaches zero. An alternate definition of stability is regards the total response and implies the first definition based upon the natural response:

• A system is stable if every bounded input yields a bounded output. We call this statement a bounded-input, bounded-output (BIBO) definition of stability.

We now realize that if the input is bounded but the output is unbounded, the system is unstable. If the input is unbounded we will see an unbounded total response and we cannot draw any about conclusion about stability.

Physically, an unstable system whose natural response grows without bound can cause damage to the system, to adjacent property, or to human life. Many times systems are designed with limited stops to prevent total runaway.

Negative feedback tends to improve stability. From the study of the system poles we must recall that poles in the left half-plane (lhp) yield either pure exponential decay or damped sinusoidal natural responses. These natural responses decay to zero when time approaches infinity. Thus, if the closed-loop system poles are in the left half of the plane and hence have a negative real part, the system is stable. That is:

• Stable systems have closed-loop transfer functions with poles only in the left half-plane.

Poles in the right half-plane (rhp) yield either pure exponentially increasing or exponentially increasing sinusoidal natural responses, which approche infinity when time approaches infinity. Thus:

• Unstable systems have closed-loop transfer functions with at least one pole in the right half-plane and /or poles of multiplicity greater than 1 on the imaginary axis.

Finally, the system that has imaginary axis poles of multiplicity 1 yields pure sinusoidal oscillations as a natural response. These responses neither increase nor decrease in amplitude. Thus,

• Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity one and poles in the left half-plane.

Figure 6.1a shows a unit step response of a stable system, while Figure 6.1b shows an unstable system.

[1]

### Routh’s Stability Criterion

Routh’s Stability criterion tell us whether or not there are unstable roots in a polynomial equation without actually solving for them. This stability criterion applies to polynomials with only a finite number of terms. When the criterion is applied to a control system, information about absolute stability can be obtained directly from the coefficients of the characteristic equation.

The method requires two steps: 1) Generate a data table called a Routh Table and 2) Interpret the to tell how many closed-loop system poles are in the left half-plane, the right half-plane, and on the jw-axis. The power of the method lies in design rather than analysis. For example, if you have an unknown parameter in the denominator of a transfer function, it is difficult to determine via a calculator the range of this parameter to yield stability. We shall see that The Routh-Hurwitz criterion can yield a closed-form expression for the range of the unknown parameter.

• Generating a basic Routh Table. Considering the equivalent closed-loop transfer function in Figure 6.3. Since we are interested in the system poles, we focus our attention on the denominator. We first create the Routh Table shown in Table 6.1:

Begin by labeling the rows with powers of s from the highest power of the denominator of the closed-loop transfer function to s^0. Next start with the coefficient of the highest power of s in the denominator and list, horizontally in the first row every other coefficient.

In the second row, list horizontally, starting with the next higher power of s, every coefficient that was skipped in the first row. The remain entries are filled as follows:

Each entry is a negative determinant of entries in the previous two rows divided by the entry in the first column directly above the calculated row. The left-hand column of the determinant is always the first column of the previous two rows, and the right-hand column is the elements of the column above and to the right.

Figures 6.4 shows an example of building the Routh Table :

The complete array of coefficients is triangular. Note that in developing the array an entire row maybe divided or multiplied by a positive number in order to simplify the subsequent numerical calculation without altering the stability conclusion.

Consider the following characteristic equation, example 5-13:

The first two rows can be obtained directly from the given polynomial. The second is divided by two but we arrive to the same conclusion:

• Interpreting the basic Routh Table. Simply stated, the Routh-Hurwitz criterion declares that the number of roots of the polynomial that are in the right-half plane is equal to the number of sign changes in the first column.

If the closed-loop transfer function has all poles in the left-hand plane the system is stable. Thus, the system is stable if there are no sign changes in the first column of the Routh Table.

The last case, Example 5-13, is one of an unstable system. In that example the number of sign changes in the first column is equal to two. This means that there are two roots with positive real parts. Table 6.3 is also an unstable system. There, the first change occurs from 1 in the s^2 row to -72 in the s^1 row. The second occurs from -72 in the  s^1 row to 103 in the  s^0 row. Thus, the system has two poles in the right-half plane.

Routh’s stability criterion is of limited usefulness when applying to control system analysis because it does not suggest how to improve relative stability or how to stabilize a unstable system. It is possible, however, to determine the effects of changing one or two parameters of the system by examining the values that causes instability. In the following we consider the problem of determining the stability range of a parameter value. Consider the system of the Figure 5-38. Let us determine the range of K for stability.

The characteristic equation is:

And the Routh Table:

For stability, K must be positive and all coefficient in the first column must be positive. Therefore:

When K=14/9 the system becomes oscillatory and, mathematically, the oscillation is sustained at constant amplitude.

### Routh-Hurwitz Criterion Special Cases

Two special cases can occur: (1) The Routh table sometimes will have a zero only in the first column of a row, (2) The Routh table sometimes will have an entire row that consists of zeros

• Zero only in the first column. If the first elemento of a row is zero, division by zero will be required to form the next row. To avoid this phenomenon, an epsilon ε is assigned to replace the zero in the first column. The value is then allowed to approach zero from either the positive or the negative side, after which the signs of the entries in the first column can be determined. To see the application of this, let us look the follow example: determine the stability of the closed-loop transfer function T(s):

The solution is shown in table 6.4:

We must begin by assembling the Routh table down to the row where a zero appears only in the first column (the s^3 row). Next, replace the zero by a small number ε complete the table. To begin the interpretation we must first assume a sign, positive or negative for the quantity  ε. Table 6.5  shows the first column of table 6.4 along with the resulting signs for choices of  ε positive and ε negative.

If is chosen ε positive Table 6.5 shows a sign change from the s^3 row to the s^2 row, and there will be another sign change from the s^2 row to the s^1 row. Hence the system is unstable and has two poles in the right half-plane. Alternatively, we could chose ε negative.  Table 6.5 then shows a sign change from the s^4 row to the s^3 row. Another sign change would occur from the s^3 row to the s^2 row. Our result would be exactly the same as that for a positive choice for ε. Thus, the system is unstable.

• Entire Row is zero. We now look at the second special case. Sometimes while making a Routh table, we can find that an entire row consists of zeros because there is a even polynomial that is a factor of the original polynomial. This case must be handled differently from the previous case. Next example shows how to construct and interpret the Routh table wne an entire row of zeros is present.

Determine the number of right half-plane poles in the closed-loop transfer function T(s):

Start by forming the Routh table for the denominator. We get Table 6.7:

At the second we multiply by 1/7 for convenience. We stop at the third row since the entire row consist of zeros and use the following procedure. First we return to the row immediately above the row of zeros and form an auxiliary polynomial using the entries in that row as coefficients. The polynomial will start with the power of s in the label column and continue by skipping every other power of s. Thus, the polynomial formed for this example is:

Next we differentiate the polynomial with respect to s and we obtain:

Finally we use the coefficients of this last equation to replace the row of zeros. Again, for convenience the third row is multiplied by ¼ after replacing the zeros. The remainder of the table is formed in a straightforward manner by following the standard form shown in Table 6.2

We get Table 6.7. It shows that all entries in the first column are positive. Hence, there are no right half-plane poles and the system is stable.

Source:

1. Control Systems Engineering, Nise pp 301-320
2. Sistemas de Control Automatico Benjamin C Kuo pp
3. Modern_Control_Engineering, Ogata 4t pp 288,

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

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The Block Diagram – Control Engineering

Block Diagram of Electromechanical Systems – DC Motor

Transient-response Specifications

## Block Diagram of Electromechanical Systems – DC Motor

`Introduction`

Electromechanical Systems are hybrids of electrical and mechanical variables. Applications for electromechanical components range from robot control, sun and star trackers, disk-drive position control, DC machines control and central Air-Conditioning systems for residences.

The structure of an Electromechanical Drive System is given in Figure 2.1. It consist of energy/power source, reference values for the quantities to be controlled, electronic controller, gating circuit for converter, electronic converter (rectifier, inverter, power electronic controller), current sensors (shunts, current transformer, Hall sensor), voltage sensor (voltage divider, potential transformer), speed sensors (tachometers) and displacement sensors (encoders), rotating three-phase machines, mechanical gearbox, and the application-specific load (pump, fan, automobile). In Figure 2-1 all but the mechanical gear are represented by a Transfer Functions (output variables as a function of time). Meanwhile, the mechanical gear is represented by the Transfer Characteristic (output variable Xout as a function of the input variable Xin).

[1]

The armature-controlled dc servomotor is perhaps the most important component found in robotic applications. But in general, in electromechanical systems, we can use induction motors and synchronous motors as well, the two most common three-phase AC Machines. Here, we just derive the Dynamic of the DC Motor in order to represent it through block diagrams. So we will have to pass from schematic to block diagram as shown in Figure 2.35:

[2]

Block Diagram from Transfer Function

Block Diagram from State-Space Representation

`The Block Diagram for a DC Motor`

To derive the block diagram representation for a separately excited DC machine, firstly we must derive the Dynamic of the System, the differential equations that govern the DC machine. After that, we use the Laplace Transform to build the block diagram. The motor can be controlled by field or by armature. In this case, let’s suppose that a stationary permanent magnet or a stationary electromagnet generates a constant magnetic flux Ф called the Fixed Field according to Figure 2.35. As a result, the motor is controlled by a voltage ea applied to the armature terminals. The armature is a rotating circuit through which a current ia, flows. When the armature passes through Ф at right angles, it feels a force F=BLia where B is the magnetic field strength and L is the length of the conductor. The resulting torque Tm turns the rotor, the rotating member of the motor. For linear analysis it is supposed that this torque or par is proportional to the flux Ф and the armature current ia, and from here, we obtain the first important relation:

[3]

As Ф is constant, KmФ = Ki, so the previous equation is as follows,

Where Ki is The Constant of Proportionality, also called the motor torque constant (or par constant) and is one the parameter given by the manufacturers when selling a motor. Ki also frequently called KT comes in N-m/A.

(Note: when the motor is controlled by the current in the field, in order to have a linear system the current of the armature must be considered to be constant and the motor torque is given by Tm= KmiF)

There is another phenomenon that occurs in the motor: A conductor moving at right angles to a magnetic field generates a voltage Vb at the terminals of the conductor. Since the current-carrying armature is rotating in a magnetic field, its voltage is proportional to speed. In this way we obtain the second important equation:

[2]

We call Vb the Back Electromotive Force (or back emf); Kb is a constant of proportionality called the back emf constant, another given parameter. It is also important to notice that θm is the angular displacement and the angular velocity of the motor is ωm:

Although the motor is itself an open-loop system, later we will see that The Back EFM Vb, generates a feedback-loop inside the motor, acting as an “electric friction” that tends to improve the stability of the motor.

After this and applying the Kirchhoff’s law for voltages, we find the relation between the armature current ia, the applied armature voltage ea, and the back efm Vb (or eb). Clearing conveniently we obtain:

Where La and Ra represent the inductance and the resistance of the armature respectively. Vb = eb. (La and Ra are given parameters of the motor)

Now, applying Newton’s Law for Rotational Mechanical Systems and clearing conveniently:

Where TL represents the load, JM is the inertia of the rotor, and Bm viscous friction coefficient (JM and Bm are given parameters of the motor)

Actually, the completed Dynamic of the DC Motor in open-loop operation is the following set of differential equations:

[3]

To get a block diagram of this dynamic we have to apply Laplace Transform to this set of equations while focused on the value we are concerned as the output: θm. So, we now that

After applying Laplace:

θm(s) = Ωm(s)/S =(Ωm(s))*(1/S)

We can represent this simple relationship between θm(s) and Ωm(s) using a Block Diagram by recalling the fact that every block has a Transfer Function inside. In this case, the transfer function is:

θm(s)/Ωm(s) = 1/S

The block diagram to represent this system is:

We see now that we need to know what is Ωm(s). To know that we use

We clear for dθm/dt and apply Laplace. We obtain:

Ωm(s)=(Tm(s) – TL(s))/(Jm(s) + Bm(s))

As a dog following its tail, we continue this procedure using each of the rest of the equations just one time, till all the variables are cleared, except the inputs TL(s) and Ea(s).

To get Tm(s) we use:

And we get:

Tm(s)= KiIa(S)

Finally, from:

We get:

Ia(S)= (Ea(s)-Eb(s))/(SLa+Ra)

Eb(s)=KbΩm(s)

Now, using, the interconnecting, θm(s), Ωm(s), Tm(s), Ia(S), Ea(s), Eb(s) and TL(s), with functional blocks and its transfer functions, summing points and pickoff points, we get The Block Diagram for a DC Motor operating as an open-loop:

[3]

Here we can corroborate what we pointed out before, that the back electromotive force, proportional to Ωm (s), represented in the diagram as Eb (s), generates a feedback loop that tends to stabilize the system.

The most commonly used electromechanical system is shown in Figure 2.15., operating at about constant speed without feedback control. Some of the more commonly occurring drive systems are presented using linear transfer function within each and every block of these diagrams as in Figure 2.15. Consequently, we usually need the transfer function of the motor and its load to represent it just through one block.

` The Transfer Function for a DC Motor and Load`

Let’s consider the configuration shown in Figure 2.37:

[2]

The relation between the input Ea(s) and the output θm(s) is given by:

(Note: To see how this equation is derived, please see page 81 of bibliography resource [2])

If we assume that the armature inductance La is small compared to the armature resistance Ra, which is usual for the motor DC, we obtain:

The desired Transfer Function θm(s)/Ea(s) is found to be:

[2]

Where Jm is the equivalent inertia after load inertia JL is reflected back to the armature, and Dm is the equivalent damping after load damping DL is reflected back to the armature, and:

Usually, to find the factors Kt/Ra and Kb, the user counts with the following relations and a Torque-Speed Curve of the motor provided by the manufacturer, as it is shown in the following example:

[2]

As a consequence, we can now represent the motor and its load through one block in the diagram:

`Closed-loop operation.`

The open-loop operation is acceptable for a lot of drive application where a constant speed (a chainsaw) or position (an elevator) is sufficient. However, if a variable speed is needed (conveyor belt) or the position must be accurate (an Antenna), a closed-loop control, with negative feedback, must be chosen. But, in a closed-loop operation, others devices are needed to assess or transform signals: sensors, transducers and amplifiers. A typical closed-loop operation is the position control schema for an Antenna Azimuth shown as follows:

Which equivalent block diagram is as follows:

[2]

`The potentiometer. `

A potentiometer is an electromechanical transducer that converts mechanical energy into electrical energy. The input of the device is a form of mechanical displacement that can be translational or rotational. When a voltage is applied across the fixed terminals of the potentiometer, the output voltage, which is measured between the variable terminal and the ground, is proportional to the input shift. Figure 4-29 shows the schema for a rotational potentiometer.

[3]

When the potentiometer housing is connected to the reference, the output voltage e(t) will be proportional to the position of the axis θc(t) in the case of a rotatory movement. So,

where Ks is the constant of proportionality. So E(s)/θc(s) = Ks will be the transfer function for a block representing this relationship

Potentiometers are often used to feedback the output position in motor control systems. Here it works as a mechanical sensor. Doing so, they allow comparing the actual position of the load with the reference position. The comparison generates an error signal that is then amplified to move the motor until it reaches the correct position. Hence its name, servomotor, slaves of the reference position.

`The Tachometer. `

Like potentiometers, tachometers are electromechanical devices that convert mechanical energy into electrical energy. It works essentially as a voltage generator, with the voltage output proportional to the angular velocity of the input shaft. Figure 4-33 reflects the common use of a tachometer in a speed control system:

[3]

The Dynamic of the tachometer can be represented by:

Where et(t) is the output voltage, θ(t) is the displacement of the motor in radians, ω(t) is the speed of the rotor in rad/s, and Kt is the constant of the tachometer. Later, Et(s)/Ω(s) = Kt. In terms of: the displacement of the motor:

`The gear train. `

Gear trains are used very frequently in electromechanical systems in order to reduce speed, amplify torque or to achieve the more efficient power transfer by matching the driving member with a given load. Consider the gear train of Figure 2-30 (for a more exhaustive analysis please refer to page 66 of the bibliography [5]):

[5]

Generally, in practice, the most commonly used procedure is to reflect the inertia and the damping of the load arrow to the motor shaft. The result is as follows:

Where J1eq is the equivalent inertia seen by the motor arrow and b1eq is the equivalent damping coefficient seen by the motor arrow. In the block diagrams, it is customary to represent the gear train by a block with a proportional transfer function, a constant Kg (which stands for Kgears) equivalent to the n1/n2.

`Example:`
1. Before continuing, let’s see an example of how to apply the theory studied to a fairly common case: Obtain mathematical model of the position control system of the Figure. Obtain your block diagram and the transfer function between the angle of the load and the reference angle θc(s)/θc(s).

Solution:

To see the whole answer see:

Solved Example 1- Electromechanical system transfer function

`DC Motor/Amplifier System Block Diagram`

The CD Motor is always driven by a power amplifier that acts as an energy source. For this reason, it is more practical to present the Torque-Speed Curve of the combinación DC Motor/Amplifier. The Figure 4-51 shows a block diagram for the DC motor-Amplifier arrangement. And Figure 4-52 the Torque-Speed Curve:

[3]

`The Proportional amplifier`

A good example of a proportional amplifier is an Operational Amplifier with negative resistive feedback and inverting configuration such as shown in Figure 2.7a.

[1]

Operational amplifiers, often called Op Amps, are often used to amplify signals in sensor circuits. The Transfer Function of the Operational Amplifier is shown in Figure 2.7b under the name of G2:

Other configurations of this class are shown in Table 3-1 and Table 4-1 with their respective transfer functions:

Table 3-1

[4]

Table 4-1

[3]

`Power Electronic`

Performance of servomotors used for robotics applications highly depends on electric power amplifiers and control electronics, broadly termed Power Electronic. The actuators in robotic applications, mostly DC motors, must be controlled precisely so that desired motions of arms and legs may be attained. This requires a power amplifier to drive de desired level of voltage (or current indirectly) to the motor armature. The use of a linear amplifier, as the operational amplifier discussed in the previous section, is power-inefficient and impractical since it entails a large amount of power loss. An alternative is to control the voltage via ON-OFF switching. Pulse Width Modulation or PWM for short is the most commonly used method for varying the average voltage to the DC motor (For a detailed review of PWM see Motor Drives, p 663, bibliography [7])

Briefly, a typical motor drive system is expected to have some of the systems block diagram indicated in Figure 27.1. The load may be a conveyor system, a traction system, the rolls of a mill drive, the cutting tool of a numerically controlled machine tool, the compressor of an air conditioner, a ship propulsion system, a control valve for a boiler, a robotic arm, and so on.

[7]

The Power Electronic converter block in the previous diagram, for PWM control with inner current loop, may use diodes, MOSFETs, GTOs or IGBTs. Servo drive systems normally use the full four-quadrant converter of Figure 27.9, which allows bidirectional drives and regenerative braking capabilities.

[7]

The PWM is a technique to control an effective armature voltage by using the ON-OFF switching alone. Figure 2.3.3 illustrates the PWM signal:

[6]

PWM varies the ratio of the time length of the complete ON state to the complete OFF state. A single cycle of ON and OFF states is called the PWM period, whereas the percentage of the ON stage in a single period is called duty rate. The first PWM signal of Figure 2.3.3, is of 60% duty, and the second one is 25%. If the voltage supply is V=10 volts, the average voltage actually transmitted to the DC motor is 6 volts and 2.5 volts respectively. The PWM period is set to be shorter than the time constant associated with the mechanical motion. The PWM frequency is usually between 2 and 20 KHz, whereas the bandwidth of a motion control system is at most 100 Hz. Therefore, the discrete switching does not influence the mechanical motion in most cases.

If the electric time constant Te is much larger than PWM period, the actual current flowing to the motor armature is a smooth curve, as illustrated in Figure 2.3.4:

[6]

`Design of Control Systems - Next Topic`

The fundamental objective of analyzing a control system is to facilitate its design. The Block Diagram is the first step because it represents the mathematical model of the system that we want to analyze and improve. The dynamics of a controlled linear process can be represented by the block diagram of Figure 10-1:

[3]

Most control systems are built so that the output vector y (t) meets certain specifications that define what the system should do and how to do it in the desired way. These specifications are unique for each individual application. Relative stability, steady-state precision (error) and transient response are the most commonly used specifications in the market. The essential problem involves determining the control signal U (t) within a prescribed range so that the specifications required by the customer are met.

After determining the above, the engineer must design a fixed configuration of the system and the place where the controller will be placed in relation to the controlled process, which also involves designing the elements that make up the controller, that is, determining the controller parameters. Because most control efforts involve modifying or compensating the performance characteristics displayed by the system during the transient response analysis and steady state response, the design of a fixed configuration is also called compensation.

In short, the art and science of designing control systems can be summarized in three steps, which leads to our following issues:

1. Determine what should be done and how to do it
1. Transient Response Specifications
2. Stability – Routh Criterion
2. Determine the driver configuration
1. Proportional, Integral and Derivative Control Actions
2. PD Controller
3. PI Controller
4. PID controller
3. Determine the controller parameters

Source

1. Chapter 2, Block Diagram of EM Systems, pp 21, 43(23) (Fuchs E.F., Masoum M.A.S. (2011) Block Diagrams of Electromechanical Systems. In: Power Conversion of Renewable Energy Systems. Springer, Boston, MA)
2. Control Systems Engineering, Nise pp 79, 81
3. Sistemas de Control Automatico Benjamin C Kuo pp 159, 203
4. Modern_Control_Engineering, Ogata 4t pp 103,
5. dinamica_de_sistemas p 66
6. Actuators and Drive System – Robótica
7. Libro Rashid – Power Electronic Handbook p 663-666

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

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 Attention: If what you need is to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem… , …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

Ejemplo 1 – Función de Transferencia de Sistema Electromecánico

The Block Diagram – Control Engineering

Ejemplo 1 – Función Transferencia de Sistema masa-resorte-amortiguador

Transient Response Specifications

Control System Stability

PID – Basic Control System Actions

## The Block Diagram – Control Engineering

A control system may consist of a number of components. To show the function performed by each component, control engineers commonly use a diagram called the block diagram. They are used in various control actions at automated control systems. The first application is in representing physical systems.

A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals. Such a diagram depicts the interrelationship that exists among the various components. Differing from a purely mathematical representation, a block diagram has the advantage of indicating more realistically the signals flow of the actual system.The Figure 1.9b and c show a schematic for an Antenna azimuth position control system, and Figure 1.9d shows its functional block diagram:

[2]

The functional block is a symbol for the mathematical operation on the input signal to the block that produces the output. The Transfer Functions of the components are usually entered in the corresponding blocks, which are connected by arrows to indicate the direction of the signal flow.

Figure 3-2 shows an element of the block diagram (In order to guide the readers to the source, I preferred to use the same reference from the book where I found the information). The dimension of the output signal of the block is the dimension of the input signal multiplied by the dimension of the transfer function in the block.

[1]

The main contribution of the block diagrams lies in the fact that the functional operation of the entire system can be visualized more readily by examining its block diagram than by examining the physical system itself. The block diagram contains information concerning dynamic behavior but it does not include any information about the physical construction of the system. Consequently, many dissimilar or unrelated systems can be represented by the same block diagram.

So, each block can be considered as a subsystem. When multiple subsystems are interconnected it is necessary to add new elements to the block diagram: summing points and pickoff points. The characteristics of each element are shown in Figure 5-2:

[2]

One of the most important components of a control system is that which acts as a union point to the comparison of signals. That is what a summing point do. Examples of the devices involved in these operations are the potentiometer and the differential amplifier. Figure 3-3 shows the diversity of this operations:

[3]

A pickoff point, as shown in Figure 5-2, distributes the input signal to several output points.

We will now examine some common topologies for interconnecting subsystems and derive the single Transfer Function representation. These common topologies will form the basis for reducing more complicated systems to a single block.

Figure 5-3 shows an example of cascade configuration. Intermediate signals values are shown at the output of each subsystem. Each signal is derived from the product of the imput times the Transfer Function.

[2]

The Equivalent Transfer Function Ge(s) is shown in Figure 5-3-b and it is the output Laplace Transform divided by the input Laplace Transform as follows:

## Parallel Form

Figure 5-5 shows an example of parallel subsystems.

[2]

Again, by writing the output of each system we can find the equivalent Transfer Function. Parallel subsystems have a common input and the output form by the algebraic sum of the output from all of the subsystems.The equivalent Transfer Function Ge(s) is the output transform divided by the input transform as follows:

## Feedback Form

Figure 3-4 shows an example of a block diagram of a Feedback system, also known as Closed-Loop System. The output C(s) is fed back to the summing point to compare it with the reference input signal R(s).

[1]

Generally, when the output signal C(s) is fed back to the summing point for comparison with the input, it is necessary to transform its form so that it gets the same form of the input signal. For example, the output could be an assessment of the temperature, thus it has the dimensions of the temperature, but the input could be a level of voltage, so it has a typical dimension of electricity. The control system must implement a transducer to set both signals in the same dimension. The conversion is accomplished by the feedback element whose Transfer Function is H(s):

[1]

For the system shown in Figure 3-5 the output C(s) an input R(s) are related as follows:

The Transfer Function relating C(s) and R(s) is called The Closed-Loop Transfer Function and it relates the closed-loop system Dynamic to the Dynamic of the feedforward elements and feedback elements.

To finish, Table 3-1 shows the algebraic rules for Block Diagrams:

[1]

The next example shows how to obtain the Transfer Function of a closed-loop model by using the rules from Table 3-1 at simplifying Block Diagrams:

[1]

You may be interrested in: Diagrama de bloques – Problemas resueltos – Sistema MRA – Catálogo 8

Sources:

1. Modern_Control_Engineering, Ogata 4t pp 71,116
2. Control Systems Engineering, Nise p236
3. Sistemas de Control Automatico, Kuo p108

Posted by: Larry Francis Obando – Technical Specialist –  Educational Content Writer

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Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

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Contact: Caracas, Quito, Guayaquil, Jaén.

Telephone number: +34 633129287

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