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Instrument - ELEC 101 A Signals and Instruments 1 Basics in...

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ELEC 101 Signals and Instruments 1 2009/10 Spring A waveform v(t) is periodic with period T if Periodic Waveforms v(t) = v(t + nT) for all values of t, n: integer Basics in signals Signal Waveform Examples 0 v(t) t Step 0 v(t) t Constant (DC) T 0 v(t) t 0 V m t Square Wave T Triangular Wave 2 10 10 10 2 10 v(t) cos (t T) cos (t ) cos( t ) cos( t) π = ω + = ω + ω = ω + π = ω A a b Some periodic waveforms : ω π = 2 T Example Example V m 0 v(t) 0 T 0 v(t) t Sinusoidal Wave 0 2 f ω = π V m V m V m V m Signals contain information about a variety of things and activities in our physical world. E.g. Weather: Signals represent the air temperature, pressure, wind speed, etc. Electronics: Voltage and Current signals. Signals represent music (audio signal), movie (video signal), data (digital signal), etc. t f(t)
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ELEC 101 Signals and Instruments 2 2009/10 Spring Find t. τ τ τ A e A dt t dv t t = = = / 0 ) 1 ( ) ( t/ n / n v(t) Ae v(n ) B Ae Ae − τ − τ τ = τ = = = n e A B = t ∴ =τ Example 0 A t tangent at t = 0 v(t) = Ae -t/ τ τ / ) ( t Ae t v = Find B. Example 0 v(t) = Ae -t/ τ A e / A n τ B τ A is amplitude, τ is time constant, e = 2.718.. Exponential Waveforms (in transient circuit) A A/e 0 τ time v(t) = Ae -t/ τ c Transient voltage and current in RC and RL circuit. Example t = 0 V C (t) i O (t) (Will be discussed in detail later in Transient Circuits)
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ELEC 101 Signals and Instruments 3 2009/10 Spring 10 2 50 50 Sketch V(t) cos( t )V. = π + ° 0 t ω = ° 50 t ω = − ° V 10 t ω V 0 V 10 220 t ω = ° ms s . Hz f T Hz s / f s / rad 20 02 0 50 1 1 50 50 2 50 2 = = = = = = π ω = π = ω ) ( t V Example d Sinusoidal Waveforms (in AC circuit) 0 ° 90 ° 1 m V (t) V cos( t) = ω 2 m V (t) V cos( t ) = ω + θ −θ° 360 t T θ = × ° t ω T ω V m is amplitude (maximum value) Peak to peak voltage (V p-p ) = 2V m ω is angular frequency in radians per second ( ω = 2 π f ) f is frequency in Hertz (Hz = 1/second) θ is phase angle in radians (2 π radians = 360 ° ) V p-p = 2V m V m V m Peak-to-peak voltage is 20V or 20 V p-p
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ELEC 101 Signals and Instruments 4 2009/10 Spring e Average (or DC) Value The average value of a periodic waveform v (t) with period T V av or V DC = T 1 T 0 dt ) t ( v Find V av of v(t).
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