Module 3: Signals and Spectra

3.0 Introduction An understanding of the different types of signals typically present in electrical systems, and the frequency content of such signals is critical to the design of electromagnetically compatible circuits. Many common signals contain high frequency components which may, under certain circumstances, act as sources of interference. Three types of signals will be examined in this module, which are found in many electrical systems. These include narrowband continuous wave (sinusoidal) signals, repetitive broadband signals (such as digital clock signals), and single event broadband signals (such as spark discharges). The emphasis of this module will be to present material required for an understanding of the frequency spectra of these types of signals. In particular, the relationship between rise time, wave shape and spectral content of a signal will be examined.

3.1 Classification of signals For the purposes of the material presented here, a signal is considered to be either a voltage or current waveform that is described mathematically. Signals may be classified in many ways. A few of the more common classifications are listed below. •

energy signals The instantaneous power dissipated by a voltage v(t)in a resistance R is given by


v(t)


p(t)

2

R

and for a current i(t)


i(t)


p(t)

2

R.

It can be seen that the power in each case is proportional to the squared magnitude of the signal. If these signals are applied to a 1 ohm resistor, then both of the equations above assume the same form. For a general signal f(t)


f (t)


p(t)

2

.

The energy associated with this signal during a time interval from t 1 to t2 is given by


f (t)
t2

Ef

2

dt .

t1

A signal whose energy remains finite over an infinite time interval

3-2

Ef is referred to as an energy signal.

•

 
f (t)
dt <    2

power signals

The average amount of power dissipated by a signal f(t) during an interval of time from t1 to t2 is P



t2

1 (t2



t1 )



 f (t) 

2

dt .

t1

A signal which satisfies the relationship 1 0 < lim T T

T/2

 

 f (t)  dt <  2

T/2

has finite average power, and is referred to as a power signal.

•

deterministic signals

A signal whose behavior is precisely known is referred to as being deterministic. These include sinusoidal signals, and digital clock signals. Usually, such signals can be represented by explicit mathematical expressions.

•

non-deterministic signals

A signal whose behavior is not known, and which can only be described statistically is referred to as being non-deterministic, or random. Digital data signals are often nondeterministic.

•

periodic signals

Repetitive, time-domain signals, such as the clock signals often present in digital devices, are referred to as being periodic. A function f(t) is said to be periodic if it satisfies the relationship f (t)



f ( t ± n T0 )

for n

3-3



1, 2 , 3 , ....



for every time t, where T0 is the period of the function. The fundamental angular frequency, o, of a periodic function is

  2 o

f0

 2T

. 0

Periodic signals are classified as power signals because their average power is finite. Sinusoidal signals represent a class of periodic signals that are commonly used in many analysis techniques. These techniques, such as those involving Fourier series, decompose complicated waveforms into a series of sinusoidal waveforms. A sinusoidal waveform f(t) is usually represented by



f (t)

 

where A is the amplitude of the signal, radians per second ( =2 f ).

•



  )

A cos( t

is the phase, and



is the angular frequency in

non-periodic signals

A non-periodic waveform is one that does not satisfy the criteria for a periodic waveform. Non-periodic signals are referred to as energy signals because their total energy is finite.

3.2 The electromagnetic spectrum





In the previous chapter it was seen that, in a source free region, E and B satisfy homogeneous wave equations. For the case of time harmonic excitation, these equations reduce to homogeneous Helmholtz equations, the solutions of which represent propagating waves. Thus electromagnetic energy is transferred in the form of waves which propagate at velocities that depend on the medium of transmission ( v 1/ µ ), and oscillate at frequencies that depend on the nature of the source. Electromagnetic radiation is typically classified according to frequency or wavelength (although in the range including visible and ultraviolet light, x-rays, and -rays, radiation is sometimes classified according to photon energy). The electromagnetic spectrum, shown in Figure 1, is divided into frequency and wavelength bands. At the low frequency end lie the radio bands. Above this is the microwave region, which occupies the range from about 1 GHz to the lower infrared band, and contains the UHF, SHF, EHF, and millimeter-wave bands. The so-called visible spectrum extends from 4.2 x 10 14 Hz (deep red, 720 nm) to 7.9 x 10 14 Hz (violet, 380 nm). At higher frequencies lie ultraviolet light, x-rays, and -rays. It is important to note that certain sections of the electromagnetic spectrum are reserved by regulatory agencies such as the FCC. Broadcasts and emissions in these bands are subject to laws and regulations established by such agencies.









3-4

Frequency

3 kHz

0 Hz

Activities

× × ×

FM Broadcast

10 m

10 m

3m

UHF

30 cm

SHF

3 cm

EHF

30 kHz

3 mm

10 4 nm

3

×

10 3 nm

3

×

10 2 nm

3

16

10 Hz 17

10 Hz

×

Ultraviolet

TV Broadcast Channels 38-69

15

10 Hz

Ultraviolet

Fixed

608 MHz

Maritime Mobile

470 MHz TV Broadcast Ch. 21-36

Visible

100THz

.3 mm

×

10THz

3.5 MHz 4 MHz

Infrared

400 MHz

1THz

2.85 MHz 3 MHz 3.155 MHz

Visible Range

Amateur

100 GHz

10 m

300 GHz

Infrared

Aeronautical Mobile Miscellaneous

10 GHz

Microwave

Mobile, Fixed, Radiolocation, Amateur

Mobile, Fixed

3-5

300 MHz

10 m

30 m

VHF

100 MHz

1 GHz

30 GHz

2

3

HF

10 MHz

14 kHz

1605 kHz

216 MHz

×

AM Broadcast

136 MHz

174 MHz

3

3

1 MHz Radio Spectrum

118 MHz Aeronautical Mobile

TV Broadcast Channels 7-13

4

3

10 m

MF

9 kHz

Radio Navigation

108 MHz Aeronautical Radionavigation

5

3

10 m

LF

88 MHz FM Radio Broadcast

6

3

3

100 kHz

AM Radio Broadcasting

TV Broadcast Channels 5-6

7

×

10 kHz

×

Audio Range

1 kHz 3 kHz

72 MHz

Radionavigation, Radiolocation, Satellite Navigation, Radio Astronomy, Earth Exploration Satellite, Space Research, Amateur, Fixed, Mobile, etc.

The electromagnetic spectrum.

Very Low Frequency (VLF)

100 Hz

TV Broadcast Channels 2-4

Wavelength Infinite

10 Hz

535 kHz

54 MHz

Frequency

Not Allocated

4.2 GHz 4.4 GHz

300 kHz

Aeronautical Radionavigation

Radionavigation

30 MHz

Land Mobile, Mobile, Fixed

Radiolocation

Figure 1.

Activities 3 GHz

3

10 nm

3 nm

806 MHz

10 -2 nm

×

20

Hz

10

21

Hz

3

10

22

Hz

3

10

23

Hz

10

24

Hz

10

25

Hz

10 -3 nm

×

3

×

-4

×

-5

×

-6

×

-7

×

300 kHz

10 -1 nm

×

30 MHz

3

Cosmic-Ray

Aeronautical Radionavigation

3 GHz

Gamma-Ray

190 kHz

3

10

Cosmic-Ray

Radionavigation, Radio Astronomy, Fixed, Miscellaneous

300 GHz

X-Ray

19

10 Hz

Gamma-Ray

1215 MHz

10 Hz

Fixed, Mobile, Miscellaneous

Aeronautical Radionavigation

18

902 MHz 960 MHz

X-Ray

Land Mobile

-8

3

3

3

10 nm

10 nm

10 nm

10 nm

10 nm

Microwave frequency band designations Old Ka K K Ku X X C C S S L UHF

New K K J J J I H G F E D C

Frequency Range (GHz) 26.5 - 40 20 - 26.5 18 - 20 12.4 - 18 10 - 12.4 8 - 10 6-8 4-6 3-4 2-3 1-2 0.5 - 1

3.3 Series expansions and basis functions Complex signals which are periodic can be represented as linear combinations of simpler signals known as basis functions. Thus a periodic signal f(t) with period T may be represented f (t)

     ! c n

cn 0

0 0(t)

"

n (t)

c1 1(t)

"

c2 2(t)

"

...

where the functions n(t) are periodic, having the same period as f(t), and the coefficients cn are referred to as expansion coefficients. The best choice of basis functions depends on the signal f(t) that is to be represented.

•

orthogonality of basis functions

Regardless of the type of basis function selected, the process of determining the expansion coefficients cn is greatly simplified if the basis functions possess the property

#$ %

t1

T

t1

n(t)

'% & (t) dt ( ) 0

m

m

for m

*

for m

+

(

n n

where * indicates the complex conjugate. A set of functions n(t) having this property is said to be orthogonal. If both sides of the signal expansion above are multiplied by m (t) , and then integrated over time interval T, it is seen that

3-6

+',

t1

- . + , (t) f (t) dt /021 3 T

t1

cn

m

n

t1

8

0

cm

9

5 6'7 6 4 (t) T

m

n(t) dt

t1

m

or

8

cn

9

1

t1

5 6'7 4 (t) f (t)dt . T

n

n t 1

This type of signal expansion, using a series of basis functions to represent a more complex signal, is useful to mathematicians and engineers because the response of a linear system to a complex periodic signal can be determined by finding the linear superposition of the responses to much simpler inputs. This representation is useful to EMC engineers for a slightly different reason, however. If the basis functions are chosen correctly, this series expansion corresponds to a complex signal composed of many individual single frequency signals. Thus a square wave, like a digital clock signal, can be thought of as being composed of many components of different frequencies, each oscillating at some integer multiple of a fundamental frequency f0, any one of which may potentially “escape” and act as a source of interference. All that is left is to select basis functions which will form a series that properly represents a particular signal.

3.4 Fourier series The most common series representation of periodic signals is a trigonometric Fourier series. This type of series uses sinusoidal basis functions, such that

:

;

1

cos(n