MODELING IN THE FREQUENCY DOMAIN

Laplace Transform:

System represented by DE is difficult to model as block diagram. Thus, LT is used. By LT input, output and system can be represented separately.

The LT is defined as

we can use Laplace Transform theorems to assist in transforming between f(t) to F(s) and F(s) to f(t) or we called as Inverse Laplace Transform.

The Transfer Function:

The transfer function is the ratio of the Laplace transform of the output of a system to the Laplace transform of the input. As a example, the transfer function, G(s) for a system representation is C(s)/R(s)

Electric Network Transfer Function:
-Apply transfer function to mathematical modeling of electrical circuit including passive Network and Op-Amp circuit.
-Equivalent circuits for the electric networks that we work with first consist of three passive linear components: resistors, capacitors and inductors.
-summarizes the components and the relationships between voltage and current and between current and charge under zero initial conditions.
-From these relationships, we can write the differential equations for the circuit using Khirchhoff’s laws.
-Then we can take the Laplace transforms of the differential equations and finally solve for the transfer function.

Or we can use transform methods:
1. loop or mesh analysis – Kirchhoff’s voltage law
2. nodal analysis – Kirchhoff’s current law

TRANSLATIONAL MECHANICAL SYSTEM TRANSFER FUNCTIONS:

Mechanical systems, like electrical network, can be have 3 passive linear components. Two of them, the spring and the mass, are energy-storage elements; and one of them, the viscous damper, dissipates energy.

where K, fv and M are called spring constant, coefficient of viscous friction and mass, respectively.

ROTATIONAL MECHANICAL SYSTEM TRANSFER FUNCTIONS:

Rotational mechanical systems are handled the same way as translational mechanical systems, except that torque replaces force and angular replaces translational displacement.
The components along with the relationships between torque and angular velocity, as well as angular displacemanet.
Notice that the symbols for the components look the same as translational symbols, but they are undergoing rotation and no translation.
The values of K, D and J are called spring constant, coefficient of viscous friction and moment of inertia, respectively.

ELectric circuit analogs:

The commonality of systems from the various disciplines by demonstrating that the mechanical systems can be represented by equivalent electric circuits.
We have pointed out the similarity between the equations resulting from Kirchhoff’s laws for electrical systems and the equations of motion of mechanical systems.
An electric circuit that is analogous to a system from another discipline is called an electric circuit analog. Analogs can be obtained by comparing the describing equations, such as the equations of motion of a mechanical system, with either electrical mesh or nodal equations.
When compared with mesh equations, the resulting electric circuit is called a series analog.
When compared with nodal equations, the resulting electric circuit is called a parallel analog.

Control System

Control systems are an integral part of modern society.
Numerous applications are all around us.
The control systems also exist in nature such as the pancreas, which regulates our blood sugar.

Control System Definition


Consists of subsystems and processes (or plants) assembled for the purpose of controlling the output of processes.In other word, a control system provides an output or response for a given input or stimulus.

Advantages of Control System
-We can move large equipment with precision
-We can point huge antennas toward the farthest reaches of universe to pick up faint radio signals

We build control systems for four primary reasons:
-Power amplification
-Remote control
-Convenience of input form
-Compensation for disturbances


Response Characteristics and System Configurations





Response characteristic – input, output, transient response, steady-state response and steady-state error.

Input/stimulus – a desired response
Output – the actual response
Transient response – a gradual change before the steady-state response
Steady-state response – after the transient response, which is its approximation to the desired response
Steady-state error – the differences between input and output

There Two major system configurations of control systems :

Open-Loop Systems:

It consists of subsystems called an input transducer, controller and process or plant.
Input transducer converts the form of the input to that used by the controller. Controller drives a process or plant. Other signals, such as disturbances, are shown added to the controller and process outputs via summing junctions. The open-loop system cannot correct for these disturbances.
Examples – toasters, washing machine (washing process)

Closed-Loop (Feedback Control) Systems:

The disadvantages of open-loop systems may be overcome in closed-loop system as shown in Figure 1.3. An output transducer/ sensor, measures the output response and converts into the form used by controller. The closed-loop systems measured the output response through a feedback path, and comparing that response to the input at the summing junction. If there is any difference between the two response, the system drives the plant, via the actuating signal, to make a correction. If there is no difference, the system does not drive the plant.
Examples – air conditioning, lift, washing machine (water level control)

Voltage Regulators

In electronics, a linear regulator is a voltage regulator based on an active device (such as a bipolar junction transistor, field effect transistor or vacuum tube) operating in its "linear region" (in contrast, a switching regulator is based on a transistor forced to act as an on/off switch) or passive devices like zener diodes operated in their breakdown region. The regulating device is made to act like a variable resistor, continuously adjusting a voltage divider network to maintain a constant output voltage. All linear regulators require an input voltage at least some minimum amount higher than the desired output voltage. That minimum amount is called the drop-out voltage.

The transistor (or other device) is used as one half of a potential divider to control the output voltage, and a feedback circuit compares the output voltage to a reference voltage in order to adjust the input to the transistor, thus keeping the output voltage reasonably constant. This is inefficient: since the transistor is acting like a resistor, it will waste electrical energy by converting it to heat. In fact, the power loss due to heating in the transistor is the current times the voltage dropped across the transistor.

Every voltage regulator consists of four basic elements:
a stable reference voltage
a voltage sampling element
a voltage comparator
a power dissipating control device

There are two basic configurations:

Series regulator: regulating dc voltage by controlling the series current supplied to load. The series regulator works by providing a path from the supply voltage to the load through a variable resistance (the main transistor is in the "top half" of the voltage divider). The power dissipated by the regulating device is equal to the power supply output current times the voltage drop in the regulating device

Shunt regulator: regulating dc voltage by shunting away some of the current from the load. The shunt regulator works by providing a path from the supply voltage to ground through a variable resistance (the main transistor is in the "bottom half" of the voltage divider). The current through the shunt regulator is diverted away from the load and flows uselessly to ground, making this form even less efficient than the series regulator. It is, however, simpler, sometimes consisting of just a voltage-reference diode, and is used in very low-powered circuits where the wasted current is too small to be of concern. This form is very common for voltage reference circuits

SIGNAL

A signal is a set of data or information, defined as a function of one or more variables, which conveys information on the nature of a physical phenomenon.
For example, a common form of human communication takes place through the use of speech signals.
Another example of human communication is visual in nature, with signals taking the form of images of people or objects around us.
Examples: Electrocardiogram (ECG) and electroencephalogram (EEG) waves that determine heart and brain conditions.

Example of signal
•Speech signals – speech recognition / communication
•Biomedical signals – brain, heart signals
•Audio signals – CD player, engine sound
•Video and image – medical image, digital camera, face recognition
•Radar signals – range and bearing of distance target

Power amplifier

Generally, an amplifier is any device that will convert one signal (often with a very small amount of energy, a few milliwatt) into a another signal (often with a larger amount of energy e.g. several hundred watts).
In popular use, the term today usually refers to an electronic amplifier, often as in audio applications. The relationship of the input to the output of an amplifier — usually expressed as a function of the input frequency — is called the transfer function of the amplifier, and the magnitude of the transfer function is termed the gain. A related device that emphasizes conversion of signals of one type to another (for example, a light signal in photons to a DC signal in amperes) is a transducer, or a sensor. However, a transducer does not amplify power.

Amplifier Classes:
Most mobile amplifiers use complementary transistor pairs to drive the speakers. In this configuration there is a transistor (or group of transistors) which conducts current from the positive power supply voltage for the positive half of the audio waveform and a different transistor (or group of transistors) which conducts current from the negative power supply voltage for the negative half of the waveform. There are some amplifiers which use the same transistor(s) to drive both the positive and the negative halves of the waveform.
NOTE:Amplifiers in classes A, B, and AB operate their output transistors in a 'linear' mode. Class 'D' amplifiers operate their outputs in 'switch' mode.


Class A
100% of the input signal is used (conduction angle Θ = 360° or 2π). Where efficiency is not a consideration, most small signal linear amplifiers are designed as Class A, which means that the output devices are always in the conduction region. Class A amplifiers are typically more linear and less complex than other types, but are very inefficient. This type of amplifier is most commonly used in small-signal stages or for low-power applications (such as driving headphones).

Class B
50% of the input signal is used (Θ = 180° or π). In Class B, there are two output devices (or sets of output devices), each of which conducts alternately for exactly 180 deg (or half cycle) of the input signal.

Class AB
More than 50% but less than 100% is used. (181° to 359°, π < Θ < 2π). Class AB amplifiers are a compromise between Class A and B, which improves small signal output linearity; conduction angles vary from 180 degrees upwards, selected by the amplifier designer. Usually found in low frequency amplifiers (such as audio and hi-fi) owing to their relatively high efficiency, or other designs where both linearity and efficiency are important (cell phones, cell towers, TV transmitters).

Class AB1 applies to tube or transistor amplifiers in class AB where the grid or base is more negatively biased than it is in class A.

Class AB2 applies to tube or transistor amplifiers in class AB where the grid or base is often more negatively biased than in AB1, and the input signal is often larger. When the drive is high enough to make the grid or the base more positive, the grid or base current will increase. It is possible depending on the level of the signal input for the amplifier to move from class AB1 to AB2.

Class C
Less than 50% is used (0° to 179°, Θ < π). Popular for high power RF amplifiers, Class C is defined by conduction for less than 180° of the input signal. Linearity is not good, but this is of no significance for single frequency power amplifiers. The signal is restored to near sinusoidal shape by a tuned circuit, and efficiency is much higher than A, AB, or B classes of amplification.

Class D
Main article: Switching amplifier
These use switching to achieve a very high power efficiency (more than 90% in modern designs). By allowing each output device to be either fully on or off, losses are minimized. A simple approach such as pulse-width modulation is sometimes still used; however, high-performance switching amplifiers use digital techniques, such as sigma-delta modulation, to achieve superior performance. Formerly used only for subwoofers due to their limited bandwidth and relatively high distortion, the evolution of semiconductor devices has made possible the development of high fidelity, full audio range Class D amplifiers, with S/N and distortion levels similar to their linear counterparts.

Other classes
There are several other amplifier classes, although they are mainly variations of the previous classes. For example, Class H and Class G amplifiers are marked by variation of the supply rails (in discrete steps or in a continuous fashion, respectively) following the input signal. Wasted heat on the output devices can be reduced as excess voltage is kept to a minimum. The amplifier that is fed with these rails itself can be of any class. These kinds of amplifiers are more complex, and are mainly used for specialized applications, such as very high-power units. Also, Class E and Class F amplifiers are commonly described in literature for radio frequencies applications where efficiency of the traditional classes deviate substantially from their ideal values. These classes use harmonic tuning of their output networks to achieve higher efficiency and can be considered a subset of Class C due to their conduction angle characteristics.