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SIGNAL AND SYSTEM

1.1. INTRODUCTION

In electrical and electronics engineering, signal is defined as an electrical waveform with some quantifiable properties such as voltage, current, power and frequency. Examples of signals are audio signals, video signals, AC signals and DC signals. A system is a combination and interactions of some independent entities that is designed to perform a function or a number of functions. An example of a system is an audio amplifier system that amplifies low power audio signal (input) and produce high power audio signal at its output.

1.2. SIGNAL

Typical signal is an Alternating Current (AC) signal. An AC signal is an electric current whose magnitude and direction vary cyclically (Figure 1.1). An AC voltage v can be described mathematically as a function of time by the following equation:

,

where

is the peak voltage (unit: volt), Or amplitude of the signal

is the angular frequency, specifies how many oscillations occur in a unit time interval (unit: radians per second)

The angular frequency is related to the physical frequency, , which represents

the number of oscillations per second (unit = hertz), by the equation .

Figure 1.1: AC waveform

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Vpeak

T

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The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak.:

Vp-p = 2 Vpeak

The relationship between voltage and power is:

where represents a load resistance

Rather than using instantaneous power, , it is more practical to use a time averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a root mean square (RMS) value, written as , because

For a sinusoidal voltage:

1.3. SYSTEM

A number of electronics components can be assembled together to form an electronic system that can be used to process or manipulate electrical signals.

Figure 1.2: Audio amplifier

Figure 1.3: Transistor pre-amplifier circuit diagram______________________________________________________________________________

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DECIBELS

2.1. INTRODUCTION

In electronic it is normal practice to concern ourselves with the gain or attenuation of a circuit. This is achieved by expressing values of gain or attenuation as ratios. The use of ratios could tell us at a glance what the overall effect of a circuit is. Let us consider the amplifier circuit shown below (Fig. 2.1.)

Pin = 1mW Po = 1 W

Figure: 2.1

Since this circuit has the power gain of 1000, an input signal emerges 1000 times greater at the output. However, the attenuation circuit shown below (Fig 2.2), gives an output that is 100 times lower than the input.

Pin = 100 mW Po = 1 mW

Figure: 2.2

Hence, the values involved with circuits are large and unwieldy which inturn difficult to manipulate. In practical circuits, amplifiers and attenuators are often connected in cascade (series) to form a complete system as shown in Figure 2.3

Ap3 Ap4 Ap5 Ap1 Ap2 1000 100 0.1 20 0.2

Figure: 2.3

The Overall system gain is determined by multiplying all the sub-system gain together.

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Another method of expressing the ratios which is more meaningful is to introduce the use of logarithmic ratios. The logarithmic ratio between two power levels is expressed in bels (B) after Alexander Graham Bell.

ApPoutPin

log10 B. (1.1 )

The bel is very large unit so for convenience the decibel (dB) is commonly used.

This gives the following equation:

ApPoutPin

10 10log dB. (1.2 )

In other words, the decibel is a logarithmic ratio between two power levels.

Let us calculate the power gain in dB form the previous examples :

a).

Pin = 1mW Pout = 1 W.

b). Pin = 100 mW Pout = 1 mW.

c). Ap1 Ap2 Ap3 Ap4 Ap5 30 dB -6.98 dB 20 dB 13 dB -10 dB

Note: When gain values are quoted in dB the overall system gain is simply a matter of adding together all the gains and subtracting the losses.

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2.2. ADVANTAGES OF EXPRESSING POWER GAIN 1N LOGARITHMIC UNITS

a) It makes the manipulation of large or small numbers much easierb) The minus sign of dB (i.e. - dB) indicates power lossc) The positive sign of dB (i.e. dB) indicates power gaind) Reducing the possibility of errorse) The multiplication and division of ratios becomes addition and

subtractionf) The overall system gain can be obtained by adding the gains and

subtracting the losses

2.3. VOLTAGES and CURRENT RATIOS EXPRESSED DECIBELS

The decibel expression is not only restricted to Power gain of any circuit but it could be expressed in terms of Voltage and Current gain. Figure 2.4. illustrates that in some cases only input and output voltage and current levels are quoted.

Iin Iout

Vin Rin Rout Vout

Figure: 2.4

Then the Power (P) can be calculated by :

P = VI

or P = I2R or P = VR

2

From this, it is easy to see that for any circuit:

P V Iin inx in

or PV

Rinin

in

2

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and

P V Iout out x out

or PVRout

out

out

2

For maximum power transfer from one device to another,the output impedance and input impedance must be the same. (Rout = Rin)

Hence

APPp

in

out

10 log dB

A

VR

VR

p

outout

inin

10

2

2log

Since Rin = Rout, they cancel to give:

AVVp

out

in

102

log dB.

AVVp

out

in

20 log dB. (1.3 )

Likewise:

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UniKL BMI TOPIC 1___________________________________________________________________________________Since Rin = Rout, they cancel to give:

Apout

in

I

I

10

2

log dB.

Apout

in

I

I

20log dB. (1.4 )

However, it is quite acceptable to express Voltage and Current ratios in dB, provided it is made clear that they are not power ratios.

Which are:

Voltage gain, AvVV

out

in

20log dB.

Current gain, AiI

Iout

in

20log dB.

2.4. REFERENCE LEVELS

If the input and output powers are known, then a gain or loss in dB can easily be found

since the decibel is simply a means of comparing the two power level; but the dB scale

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UniKL BMI TOPIC 1___________________________________________________________________________________doesn't define where you're starting from or what your 'zero' is. A reference level is

sometimes useful for the following reasons:

a) It gives an indication for the practical realistic of a circuit. i.e. meaningless to say

that an amplifier has a power gain of 60dB unless some reference power is stated or

understood. A power gain of 60dB represent a numerical gain of 1 X 106, Therefore if the

input to such amplifier was the output would be 1W, fine!!! But if the input was 1W the

output would be MW, hardly likely!!! A stated reference level helps to remove any

misunderstanding

b) If the reference level is known, a change in signal level can be readily understood.

Once a reference power is used any power level can be considered with respect to this

reference level:

Power dB

This reference level is indicated by a subscript to the abbreviation for decibel.

i) dBm : indicating that our scale is relative to 1 milliWatt of power.

dBm : reference level = 1 mW.

Power dBm

ii) dBW: indicating that our scale is relative to 1 Watt of power.

dBW: reference level = 1 W.

Power dBW.

Note:dBm & dBW are different from dB. dBm & dBW represents absolute power, whereas the decibel is used to represent gain or attenuation of an audio amplifier.

Exercise: 1

1. Express in dBm the following power levels:

a) 1.5 W b) 1W c) 1mW

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UniKL BMI TOPIC 1___________________________________________________________________________________2. Calculate the power gain in dB of the circuit in Figure: 1.

Pin = 27mW Po = 1.3 W

Figure :1

3. Refer to the Figure: 2, calculate:

a). The overall system gain in dB.

b). The signal power level at point X.

c). The system output signal power.

X

i/p 5dB 7dB -3dB 2 dB o/p = 1mW

TUTORIAL 1: SIGNAL AND SYSTEMS

Q1. (a) The input to a device is 10 KW at a voltage of 1KV. The output power is 500W while the output impedance is 100.

Find :(i) the power gain in decibels

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(ii) the voltage gain in decibels

(b) An amplifier rated at 40W output is connected to a 10 speaker.Calculate:

(i) the input power for full output if the power gain is 25dB(ii) the input voltage for rated output if the amplifier voltage

gain is 40dB. (assuming Rin = Rout)

Q2. A network has a loss of 12dB, an input resistance of 2000 and an output resistance of 10000. The input and output terminals are matched to the source and load resistance. Calculate:

(a) The current flowing in the load when the current into the network is 3mA.

(b) The input voltage if the load voltage is 15V.

Q3. In the block diagram shown below, the system is match at 300. Find:(a) The overall gain from A to B.(b) The overall gain from A to C.(c) The output voltages, VB and VC if Vin = 20mV.

30dB B

-12 dB -25 dB

A 9dB 8 dB

16 dB 27dB C

NOISE

3.1 INTRODUCTION

In an electronic system noise is defined as any unwanted signals that tend to interfere with the proper / wanted signal. The present of noise worsen the performance of any system...crackles and pops can spoil our enjoyment of a radio broadcast.

Noise is present even when there is no signal. Noise can be classified into:(a) external noise (b) internal noise

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3.2 EXTERNAL NOISE

Is an unwanted signal which occurred outside the system. It enters the system and processed along with the wanted signal

Source / types of external noise:

(a) Man made noise(b) Natural noise:

i) Atmospheric noise - lightning, thunderstormii) Solar noise - radiation from suniii) Extra terrestrial noise - radiation from other stars in the galaxy

(a) MAN MADE NOISE

1. Mains “hum”

Electric hum, mains hum, or power line hum is an audible oscillation at the frequency of the mains alternating current, which is usually 50 or 60 hertz. The sound often has heavy harmonic content and very annoying.

The most common cause of electric hum is magnetostriction, wherein ferromagnetic materials change shape minutely when exposed to magnetic fields. Magnetostrictive electric hum is most often noticed around large linear transformers, particularly when the

transformers are handling large amounts of current.Any mains operated electronic system incorporate a power supply that produce a suitable d.c. operating voltage.The a.c. mains has a frequency of 50 Hz and this 50Hz signal can be picked up before and after rectification to appear at the output of the systems as annoying 50Hz or 100Hz.

Precautions Mains pick up due mostly to electromagnetic induction occurring between the mains wires and the circuit wiring. To reduce this, the following steps can be taken.

(a) Ensure that the main leads are positioned as far as possible from the low voltage side of the unit

(b) Use screened leads for the input leads to the system(c) The electronic circuit can be physically form the mains side by the use of metal

cover or plate that is earthed(d) All earths should be taken to a common point

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UniKL BMI TOPIC 1___________________________________________________________________________________(e) Low voltage d.c. supply lines can be decoupled to earth using capacitors that have

a very low reactance at mains frequency and so will remove an 50 Hz or 100 Hz signal that may be present

(2) Switching or contact noise

Sparks are major sources of noise. Try the following :While your TV is on. Operates a mains light several times and note the interference caused. This shows that whenever a flow of current is interrupted, a burst of electromagnetic radiation occurred that contains multiple frequencies up to hundreds of megahertz. Should these frequencies enter a system, interference will result.

Equipments that likely to generate switching noise include:

(a) Electric motor ... (sparking brushes)(b) Car ignition system ...(contact breakers)(c) Fluorescent lights...(starting mechanism)(d) Solenoid valve on control equipment...inductor filter that prevents the high

frequencies radiating.

Precaution

If noise is generated by switching operation, the equipment producing the noise should have suppressors wired across the switch contacts or brushes.

3) Crosstalk

It is convenient to send information using pair conductors as in telephone and data transmission systemsFor economical reasons multicore cables are used with a number of pairs sharing a common cable. Crosstalk is the signal pickup that occurs between pairs in the cable.

PrecautionPairs that are individually screened within the common cable, multiway ribbon cables that alternate earth conductors in the ribbon, and multiway cables that used twisted pair construction. In addition to these methods, it is usual to reduce the cross - coupling between pairs by keeping signal amplitudes low.

(b) NATURAL NOISE

Atmospheric, Solar, Extra terrestrial noise.Consists of the electrostatic generated by atmospheric conditions, i.e. electrical and magnetic storms. There is also galactic noise from the stars and a certain amount of earth noise is radiated by the planet on which we live.

Precaution

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UniKL BMI TOPIC 1___________________________________________________________________________________These noises are ever present and are only likely to cause interference on systems that use aerials. Minimization of the effects of noise of this type consists of using screened or coaxial cables together with careful aerial design and positioning

3.3 INTERNAL NOISE

This is noise that is introduced by the system itself.

All electronic circuits will introduce noise; it is generated by the very components and wires themselves.Some causes of internal noise are discussed below.

a). Thermal noise

Also called Johnson noise... caused by random motion of electrons in electrical conductors and components.As temperature rises above 0K (Kelvin) thermal agitation occurs and electrons tear free from the atoms outer orbits. These free electrons wander randomly throughout the atomic structure. Any electron movement constitutes a flow of current, so an EMF or voltage will be set up across the conductor. This voltage is thermally generated and will be taking place causing a small but significant noise voltage to be present in all components and conductors.

The r.m.s value of this noise voltage can be calculated using Johnson’s equation below:

Vn kTBR 4 Volts. (1.5 )

where Vn = rms value of noise voltage. k = Boltzmans constant, 1.38 x 10 -23 JK-1

T = Absolute temperature in Kelvin. ( T = C + 273 ).

B = Bandwidth in Hertz.( B = fH - fL ).

R = Resistance in ohms.

You can see that amount of noise that produced in a component is determined not only by the temperature, but by the bandwidth, i.e. the frequency (or range of frequencies) over which it operates and the components resistance.

Example 1.

A 10 kilohms resistor at room temperature operates at 1 MHz

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UniKL BMI TOPIC 1___________________________________________________________________________________Calculate the thermal noise generatedNow consider the same resistor operating at 2 MHz. Calculate thermal noise generated. What can you conclude?

Example 2

An electrical component has a resistance of 200 kilohms. If it is operating at frequency of 5 MHz, calculate the thermal noise generated at

(a) 25C (b) 50 C

From this the need to keep the temperature of circuits as low as possible is apparent, particularly when you realize that the temperature is governed not only ambient conditions but also by the current flowing in a circuit.

Thermal noise power

We are interested in the actual noise power that is transferred form a source into a load. Consider the diagram in Figure:3.1 below:

In

Rs = Rin R

Vn

Input System

Figure:3.1 Noise generator

Maximum power transfer occurs when source and load are matched. In this case Rs = Rin = R. This will also be the condition for maximum noise power

Noise current, InVn

Rin R

But Rs = Rin = R

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UniKL BMI TOPIC 1___________________________________________________________________________________So

InVn

R

2

If the noise power transferred into the input of the system = Pnthen

Pn In xR 2

so PnVn

RxR

2

2

so PnVnR

xR4 2

2

now PnVn

R

4

2

but Vn kTBR 4

i.e. Vn kTBR2 4

Thus PnkTBR

R4

4

Hence Pn kTB watts. ( 1.6 )

b). Shot noise

This is noise that occurs in transistor and active devices due to arrival and departure of change carries at junction within the device.

The rms noise current (In) can be calculated using

In = 2eIB Amps. (1.7 )

where e = charge on an electron, 1.6 x 10-19 C I = DC current

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B = Bandwidth

Example:Calculate the noise current produced by a diode operating at 15kHz when passing a current of :

(a) 250mA (b) 500 mA

This indicates that In is directly proportional to the dc current and so the amount of shot noise can be determined by the device operating current.

Although shot noise can be a source of major irritation for sensitive electronics circuits there is sometimes a need for a noisy component, eg when building a specific sound generator like music synthesize, a noise diode would be used for the production of drum and cymbal sounds.

c). Flicker noise

This is a low frequency noise that occurs in semiconductors due to the random way electron - hole pairs are generated and recombined.Flicker noise power is inversely proportional to frequency with power level virtually insignificant above about 15 kHz. This noise is most apparent in audio frequency systems.

d). Partition noise

This noise occurs in semiconductors due to fluctuations in the way current divides at the junction, i.e. in a BJT where 1E = 1B + IC, the emitter current IE divides to provide the base and collector currents IB and IC. This division of current varies giving rise to partition noise.

Precautions against internal noiseFrom this brief study of the causes of internal noise it can be seen that there is very little the designer can do guard against it. Since it is generated by the components themselves, internal noise can best be minimized by selecting special purpose made low noise components for sensitive circuits and ensuring that a low operating temperature is maintained.

3.4 NOISE COLOUR

Even though noise is a random signal, it can have characteristic statistical properties. Spectral density (power distribution in the frequency spectrum) is such a property, which can be used to distinguish different types of noise. This classification by spectral density is given "color" terminology, with different types named after different colors

White noise.

White noise is a signal, named by analogy to white light, with equal energy per cycle (hertz). This produces a flat frequency spectrum in linear space. In other words, the signal

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UniKL BMI TOPIC 1___________________________________________________________________________________has equal power in any band of a given bandwidth (power spectral density).White noise contains noise of all frequencies. It has a flat or level power / frequency spectrum.

Noise power

Frequency

Figure:3.2

This means that at any frequency over a given range, there is the same amount of noise power present. Thermal noise, shot noise, and partition noise are all examples of white noise.

Pink noise

This is a noise with a power that is inversely proportional to frequency. Sometimes called I/F noise. Its power / frequency spectrum is as shown below:

Noise Power

Frequency

Figure:3.3

As the frequency increase the noise power reduces. Flicker noise is an example of pink noise.

3.5 THE SIGNAL TO NOISE RATIO

Noise is always present and will exist at the input to the system

It will also be produced by the system itself and so appear in greater quantities at the system output.

The significance of the noise power can only be judged by comparing it to the wanted signal power. This is achieved using the signal - to - noise ratio.

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S/N ratio =

= signal powernoise power

Signal to noise ratio in decibels.

S/N = dB (1.8 )

A good quality hi-fi system would required a signal to noise ratio of about 60 dB or greater in order to provide a high quality music output.

3.6 NOISE FIGURE

Sometimes called the noise factor, F is used to define how much noise in a system itself contributed to the overall noise level at the output.

Ideally the noise factor of any system would be unity, indicating that the system introduces no noise at all, but it is IMPOSSIBLE.

The noise figure, F can be determine as the ratio between the input and output signal to noise ratio

F = input S/N ratio output S/N ratio

(1.9 )Or in decibel, F = input S/N ratio in dB - output S/N ratio in dB.

Output Noise Calculation

CASE 1:

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Nout = Nin Ap + Nint

Where

Nin = Input noise or External noise

Nint = Internal noise

Nout = Output noise

CASE 2:

Nout = (Nin+Nint) Ap

Exercises: 2

1. An amplifier has an input signal of 100 mW with 50 W of noise present. The amplifier has power gain, Ap = 20 dB.

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Nout

Nint

Nin

CircuitAp

Nout

Nint

Nin

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Determine the output noise power if 2mW of internal noise is introduced by the amplifier itself.

Nin Ap = 20 dB Nout

Nint

2. An amplifier with a power gain of 20 dB has an input signal of 100 mW with 50W of noise present. The amplifier itself generates additional noise power of 2 mW referred to its input terminals. Calculate the output noise power of this system.

3. Amplifier is operating in the frequency range between fL = 100 Hz to fH = 520,000 Hz. Its electrical component has a resistance of 150 k at temperature of 35 C. Calculate the thermal noise voltage generated by this system

3. The input signal power to an amplifier is 120 mW with 60 W of noise present. Calculate the input Signal to Noise Ratio, SNRinput in dB.

5. The same amplifier has an output SNR of 28 dB. Calculate the output noise power if the output signal power 2.5 W.

6. The input signal to a 40dB amplifier is 25 W. The amplifier itself produces additional 8 nW of internal noise. If the input signal to noise ratio is 50dB, calculate:

a) The input noise power to the amplifierb) The total output noise powerc) The output signal powerd) The output signal to noise ratioe) The noise factor of the amplifier

4 Frequency Response

4.1 Introduction

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UniKL BMI TOPIC 1___________________________________________________________________________________Frequency response of an amplifier refers to the frequency range in which the amplifier will operate satisfactory. This range of frequencies can be called the mid-range or bandwidth, Bw.

At frequencies above and below the midrange, capacitance and any inductance will affect the gain of the amplifier.

At low frequencies• the coupling and bypass capacitors will lower the gain.

At high frequencies• stray capacitances (internal device capacitances) associated with the

active device will lower the gain.

4.2 BODE PLOT

A Bode plot is a graph indicates the frequency response of an amplifier.The horizontal axis indicates the frequency (in log scale) and the vertical indicates the gain (in numerical or in dB).

Figure: 1 Gain versus frequency: (a) RC coupled amplifier; (b) direct coupled amplifiers

4.3 Cut-off Frequencies

The mid-range frequency of an amplifier is called the Bandwidth, Bw of the amplifier.The Bandwidth is defined by the Lower, f1 and Upper Cutoff frequency, f2.

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i.e Bw = f2 – f1

Cutoff frequencies are at which the gain has dropped by:

0.5 powermax, 0.707 voltagemax 0.707 gainmax -3dB from gainmax (gain in dB)

What do these 3 dB points really mean for power and voltage gain?

This represents fall in power gain to 50% of its-mid band value, hence the name half power point.

For a voltage or current gain a loss 3 dB means:

Representing a fall in voltage gain to 70.7% of its mid band value.

4.4 Low Frequency Response – BJT AmplifierAt low frequencies Coupling capacitors (Cs, CC) and Bypass capacitors (CE) will have capacitive reactance (XC) that affect the circuit impedances.

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4.5 High-Frequency Response – BJT Amplifiers

Miller Effect Capacitance

Any P-N junction can develop capacitance. In a CE BJT amplifier this capacitance

becomes noticeable between: the Base-Collector junction at high frequencies.

It is called the Miller Capacitance. It effects the input and output circuits.

Capacitances that will affect the high-frequency response:

• Cbe, Cbc, Cce – junction capacitances

• Cwi, Cwo – wiring capacitances

Example:

1. For the frequency response of a circuit given in Figure 1, determine:

i. The maximum voltage gain, Av max in dB

ii. The bandwidth______________________________________________________________________________

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Figure: 1

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