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### Electrical Signals and Trigonometry

Alan Murray

1

1

2

4

2

Sines and CosinesFrequency/Wavelengthy

x

y = cos(x)

y

x

y = cos(2x)

y

x

y = cos(4x)

Alan Murray – University of Edinburgh

-π/2 = -90°

y0

2y0

Sines and CosinesAmplitude and Phasey = 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

Ф

180° = π

y = y0cos(x+Ф)

Ф

y = y0cos(x+180°)

or

y = y0cos(x+π)

or

y = -y0cos(x)

180°

=π

Sines and CosinesAmplitude and PhaseClickertime

Sinusoids simulation

Alan Murray – University of Edinburgh

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

VR

VR

IR

Resistors are trivial (!) ..- VR = V0sin(ωt)
- IR = I0 sin(ωt)
- V & I are in phase
- VR = RIR
- Ohm’s Law
- R is the “impedance”of a resistor
- V0 sin(ωt) = RI0sin(ωt)
- Ohm’s Law
- V0 = RI0
- Ohm’s Law, amplitudes only

Alan Murray – University of Edinburgh

π/2

I = I0cos(wt) = I0sin(ωt+90°)

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

Vs

V

Animate this slide!

Alan Murray – University of Edinburgh

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

90°

IL

Inductors are equally awkwardIL

- 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 …

- 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

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

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

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 …

Alan Murray – University of Edinburgh

Starting too sound messy, isn’t it?

Summary so far,if I =I0 sin(ωt)- VR= R xIO sin(ωt)
- VC= 1 xIO sin(ωt - 90°)ωC
- VL= ωL xIO sin(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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