Chapter 1.2 Characteristics of signals • Signal - Any physical quantity that varies with time (or other independent variable) and carries information. – In our case currents and voltages which are time dependent; or – Variation of pressure with depth- not time dependent Signals Continuous Discrete Smooth changes wrt independent variable Changes wrt independent variable occur in definite steps Continuous: time Discrete: Step size or quantization time EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 1 ©2001 If what we are considering is a variation of signal amplitude with time, we can define the following: • Analog signal-Continuous amplitude changes within some set of limits – Indications of a moving coil meter for example a fuel gauge – Mercury-in-glass thermometer • Digital signal - Discrete amplitude changes within a set of limits, for example 0-256 levels with a step size of 2 levels – TV rotary channel selector – 13 steps, 1channel per step – output signal from a mouse – changes in 1 pixel increments Two other signal types frequently encountered are: • Continuous-time signals - the signal exists for all time. – The pressure sensors in a process plant produces continuous-time signals once the plant is operational • Discrete-time signals - the signal exists only at selected instances in time nT T 2T 3T 4T • We can express the discrete-time signals mathematically: x = f (nT ) n = 0,1, 2,..........N − 1, N i.e x = f (T ), f (2T ), f (( N − 1)T ), f ( NT ) • Discrete-time signals are produced by many computer based systems; • Can be produced from continuous-time signals by a process called analog to digital conversion which involves sampling and quantizing. • They are useful since they consist of a set of values that can be stored and retrieved for later processing. • Many measurement systems convert continuous-time input signals to discrete-time signals for storage and further processing- Digital storage oscilloscope for example. EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 2 ©2001 Periodic signals • Periodic signals are an important class of signals that repeat themselves at a regular interval called the period Period • Mathematically, this can be expressed as y = f(t+nT) • The simplest periodic signal is the sinusoid which is described by either the sine or cosine function, either of which produce a sinusoidally shaped waveform. Sinusoids • We can define a sinusoidal signal y(t) as: y(t) = A sin(ωt+φ) where A is the amplitude of the sinusoid; ω is the angular frequency in rad/s; and φ is the phase shift in radians (relative to some reference) y(t) Amplitude π 0 2π . ω = 2πf where f is the frequency of the sinusoid in Hz (or the no. of complete cycles made per second). ∴ time for 1 cycle = 1/f seconds. This is called the period, T, of the sinusoid. EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 3 ©2001 y(t) Period π 0 2π The example shown starts at t = 0. If it starts at some other time t ≠ 0, we denote this in the general equation by the phase shift φ. • φ is +ve if y(t) starts before t=0 – Phase lead; • φ is -ve if y(t) starts after t=0 – Phase lag; y(t) Phase lag z(t) π 0 2π Phase lead Example v(t) = Vp sin(ωt+φ) ; where Vp = 7V; ω = 50 rad/s; φ = 35° lead on reference (t = 0). 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 Solution waveform 0.05 0.1 0.15 0.2 0.25 Reference waveform What is the equation for this signal and what is the frequency? EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 4 ©2001 • What is the local line frequency? • What is the local angular frequency? Characteristics of periodic waveforms • Another way that we describe a periodic signal, besides its mathematical formula is by its average value. • Intuitively, what is the average value of the sinusoidal waveform? • Mathematically, the average value of a waveform is given by : FAV = 1T f (t )dt T ∫0 Represents the area under the waveform • For the sinusoid, the area under the curve is divided equally above the axis for a time T/2 and below the axis for another time T/2, therefore the average value is 0. • We therefore need another way of describing the average value of the periodic waveform. Examine the following figure: • Passing current through a resistor, R, dissipates power. We know that since the resistor gets warm. • This happens with a sinusoidal current also (check out your average toaster!). So although the average value of the sinusoid is zero, there is clearly some transfer of energy taking place. • The power dissipated is I2R watts • Let us consider this situation graphically current i(t) t3 t1 t2 R i(t) Heating effect of i(t) produced in R i2(t)R t3 t1 t2 EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen t 5 t ©2001 • Heating effect at: t = i R; 2 1 1 t =i R ; 2 2 2 t = i R; 2 3 3 t =i R 2 n n • where i1denotes i(t1) etc. • Note that at t3, the power dissipated is (-i3)(-i3)R. Total heating effect = i R + i R + i R + .. + i R Average heating effect = 2 2 2 2 1 2 3 n i R + i R + i R + .. + i R n 2 2 2 2 1 2 3 n • Let us now pass a DC current I through R such that we generate the same heating effect. • We can therefore say that: IR= 2 i R + i R + i R + .. + i R n 2 2 2 2 1 2 3 n i + i + i + .. + i ∴I = n 2 2 2 2 1 2 3 n • We can conclude that there exists some DC current I, which produces the same average heating effect as an AC current i(t), passing through the same resistor. • We call this DC current the Root Mean Square (or rms.) value or effective value of the AC current i(t). RMS value of waveform • In general the rms. value of a periodic waveform (signal) is: T 2 1 Frms = f t ( ) ]dt T ∫0 [ EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 6 ©2001 For a sinusoidal signal: v(t ) = V p sin(ωt ) Frms = Vp = 0.707Vp 2 Example: v (t ) = 170 sin(377t ) What is Vrms? Does this value seem familiar? • What is the effective ac power dissipated in a resistor R? Effective power = I rms × Vrms ∴ Effective power = 0.707 I p × 0.707V p = 0.5 I p × V p Other typical periodic signals Square wave Digital clock signals Triangle wave A/D converters Sawtooth wave Sweep generator in TV or oscilloscope EE11A Chapter 1_2 Prepared by: Mr. Fasil Muddeen 7 ©2001