Electrical signals and trigonometry
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Electrical Signals and Trigonometry. Alan Murray. V L. I L. Agenda. Trigonometry and waves i.e. any old waves! Amplitude, phase and frequency Representing electrical signals Voltage, current Lead, lag, phase shift i.e. not any old waves “CIVIL” mnemonic to remember which way around. 3.

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Electrical Signals and Trigonometry

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Electrical signals and trigonometry

Electrical Signals and Trigonometry

Alan Murray


Agenda

VL

IL

Agenda

  • Trigonometry and waves

    • i.e. any old waves!

    • Amplitude, phase and frequency

  • Representing electrical signals

    • Voltage, current

    • Lead, lag, phase shift

    • i.e. not any old waves

  • “CIVIL” mnemonic to remember which way around

Alan Murray – University of Edinburgh


Sines and cosines frequency wavelength

3

1

1

2

4

2

Sines and CosinesFrequency/Wavelength

y

x

y = cos(x)

y

x

y = cos(2x)

y

x

y = cos(4x)

Alan Murray – University of Edinburgh


Sines and cosines amplitude and phase

y0

-π/2 = -90°

y0

2y0

Sines and CosinesAmplitude and Phase

y = y0cos(x)

y = y0sin(x)

or

y = y0cos(x-90°)

or

y = y0cos(x-π/2)

y = 2y0sin(x)

or

y = 2y0cos(x-90°)

or

y = 2y0cos(x-π/2)

Alan Murray – University of Edinburgh


Sines and cosines amplitude and phase1

y = y0cos(x)

Ф

180° = π

y = y0cos(x+Ф)

Ф

y = y0cos(x+180°)

or

y = y0cos(x+π)

or

y = -y0cos(x)

180°

Sines and CosinesAmplitude and Phase

Clickertime

Sinusoids simulation

Alan Murray – University of Edinburgh


And in circuits

T=1/f

V0

And in Circuits …

  • V = V0 sin (ωt + Ф)

    • a sinusoidal variation of voltage with time

  • Amplitude = V0

    • Volts

  • Time = t

    • seconds

  • Frequency = f

    • cycles/second or Hz

  • Angular freq.= ω (= 2πf)

    • radians/second

  • Phase = Ф

    • radians or °

    • Ф=0 in this example

  • So [ωt] = [radians/second * seconds] = [radians]

  • [ωt + Ф] = [radians + radians]

    • and thus sin(ωt + Ф) makes sense and works!

V

t

Alan Murray – University of Edinburgh


Resistors are trivial

IR

VR

VR

IR

Resistors are trivial (!) ..

  • VR= V0sin(ωt)

  • IR= I0sin(ωt)

  • V & I are in phase

  • VR = RIR

    • Ohm’s Law

    • R is the “impedance”of a resistor

  • V0sin(ωt) = RI0sin(ωt)

    • Ohm’s Law

  • V0= RI0

    • Ohm’s Law, amplitudes only

Alan Murray – University of Edinburgh


But voltage and current ac in a capacitor

V = V0 sin(ωt)

π/2

I = I0cos(wt) = I0sin(ωt+90°)

... but Voltage and Current (AC) in a Capacitor

I

Vs

V

Animate this slide!

Alan Murray – University of Edinburgh


Capacitors are not trivial

IC

VC

90°

IC

time t

Capacitors are not trivial ..

  • VC= V0sin(ωt)

  • IC= I0cos(ωt) = I0sin(ωt+90°)

  • ICleads VC by 90°

  • VC = ZCIC

    • This is Ohm’s Law

    • ZC replaces R

      • Or XC replaces R – see later!

    • ZC = Capacitor “impedance”

  • V0sin(ωt) = ?I0sin(ωt+90°)

  • Ohm’s Law, amplitudes only

    • V0= ?I0

    • What is the capacitor’sequivalent of resistance …

    • What is “ZC “?

    • And how does ZC describe

      the +90° phase shift?

VC

Alan Murray – University of Edinburgh


Inductors are equally awkward

VL

90°

IL

Inductors are equally awkward

IL

  • VL= V0sin(ωt)

  • IL= I0sin(ωt-90°)

  • ILlags VL by 90°

  • VL = ZLIL

    • This is Ohm’s Law

    • ZL replaces R

    • Or XL replaces R – see later!

    • ZL = Inductor “impedance”

  • V0sin(ωt) = ?I0sin(ωt-90°)

  • Ohm’s Law amplitudes only

    • V0= ?I0

    • What is the inductor’sequivalent of resistance …

    • What is “ZL “?

    • And how does ZL describe

      the -90° phase shift?

VL

Alan Murray – University of Edinburgh


A moment of pedantry

A moment of pedantry …

  • VR = RIR

    • R is clearly a resistance

  • VC = ZCIC

    • ZC is officially an impedance

    • IC leads VC by 90°

    • VC = VC0sin(ωt)

    • IC =ICOsin(ωt+90°)

  • ZC describes the magnitude relationship between V and I

  • ZCmust also describe the 90° phase shift in some magical way

    • more anon …

  • However, if we work with amplitudes only

  • VCO = XCICO

  • Then XC is the capacitor’s reactance

    • XC describes the relationship between the magnitudes of V and I

    • XC does not describe the phase shift Ф

    • That is described by the sin(ωt), sin(ωt+Ф)

  • This will make more sense later …

Alan Murray – University of Edinburgh


Mnemonic civil

CIVIL

in a Capacitor, current I leads Voltage (by 90°)

CIVIL

Voltage leads current I (by 90°) in an Inductor L

mnemonic ... "CIVIL"

Alan Murray – University of Edinburgh


Capacitor reactance x c remember 2 f

here's

the

phase

lag

Capacitor Reactance, XC(remember,ω = 2πf )

DC

AC

  • A capacitor blocks DC signals

    • f = 0, ω = 2πf = 0

  • A capacitor passes AC signals

    • f > 0, ω = 2πf > 0

    • Ohm's Law for amplitudes is V0 = XCI0

  • It turns out that

  • VO = 1 IO for a capacitor (blocks DC, passes AC) ωC

  • so XC = 1 ωC

  • DC … f=0 ω=0

    • XC = ∞ … block

  • AC … f>0 w<∞

    • XC < ∞ … pass

  • If I = I0 sin(ωt), then

  • V = 1 x I0sin(ωt-90°) ωC

Alan Murray – University of Edinburgh


Inductor reactance x l remember 2 f

here's

the

phase

lead

Inductor Reactance, XL(remember,ω = 2πf )

  • An inductor passes DC signals

    • f = 0, ω = 2πf = 0

  • An inductor blocks AC signals

    • f > 0, ω = 2πf > 0

    • Ohm's Law for amplitudes is VL0 = XLILO

  • It turns out that

  • VO = ωL IO for an inductor (blocks AC, passes DC)

  • so XL = ωL

  • DC … f=0 ω=0

    • XC = 0 … pass

  • AC … f→∞ w →∞

    • XC → ∞ … block

  • If I = I0sin(ωt), then

  • V = ωL x I0sin(ωt+90°)

Clickertime

Alan Murray – University of Edinburgh


Fill in the blanks

Fill in the blanks …

Alan Murray – University of Edinburgh


Summary so far if i i 0 sin t

Starting too sound messy, isn’t it?

Summary so far,if I =I0 sin(ωt)

  • VR= R xIOsin(ωt)

  • VC= 1 xIOsin(ωt - 90°)ωC

  • VL= ωL xIOsin(ωt + 90°)

  • All obey Ohm’s Law

    • with different constants between V and I

      • R, 1/ ωC and ωL

    • in capacitors, I leads VC by 90°

    • in inductors, VL leads I by 90°

Alan Murray – University of Edinburgh


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