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Electrical Signals and Trigonometry

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Electrical Signals and Trigonometry

Alan Murray

VL

IL

- 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

3

1

1

2

4

2

y

x

y = cos(x)

y

x

y = cos(2x)

y

x

y = cos(4x)

Alan Murray – University of Edinburgh

y0

-π/2 = -90°

y0

2y0

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

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°

=π

Clickertime

Sinusoids simulation

Alan Murray – University of Edinburgh

T=1/f

V0

- 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

IR

VR

VR

IR

- 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

V = V0 sin(ωt)

π/2

I = I0cos(wt) = I0sin(ωt+90°)

I

Vs

V

Animate this slide!

Alan Murray – University of Edinburgh

IC

VC

90°

IC

time t

- 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

VL

90°

IL

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

- 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

CIVIL

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

CIVIL

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

Alan Murray – University of Edinburgh

here's

the

phase

lag

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

here's

the

phase

lead

- 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

Alan Murray – University of Edinburgh

Starting too sound messy, isn’t it?

- 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°

- with different constants between V and I

Alan Murray – University of Edinburgh