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### GROUNDHOG DAY!

### Comparison

Alan Murray

Agenda

- RC circuit, AC signal
- using trigonometry (J?)
- using phasors (K?)
- using complex numbers (L?)

- worked examples complex numbers

Alan Murray – University of Edinburgh

This is all we need ...

Alan Murray – University of Edinburgh

VR

VS

VC

RC Phasor (K?)- Choose I =IC =IR horizontal
- VR = RI (Ohm’s Law)
VRalso horizontal

i.e. VR and I are in phase

- CIVIL → I leads VC by π/2
- Or VC is -π/2 behind I

- VC points ↓ … rotated by -π/2
- VS = VC + VR

Sonny & Cher sing…"I got you babe" ...

Alan Murray – University of Edinburgh

VS

RC Phasor (K?)I and VR are in phase

VC lags I and VR by π/2

VS is at an angle -Φ in between

Plug in numbers forR,C and ω =2πf to get values for VR, VC, I and Φ

(Leave this as a worked example, once we have the same result from the complex number method and from trigonometry)

VR = RI

Φ

VC = I/ωC

Alan Murray – University of Edinburgh

VR

Φ

VS

VC

RC Complex numbers (L?)Here’s the idea ...- Write all currents/voltages as Cej(ωt+phase)
- I = I0ejωt
- VR = VR0ejωt = ZRI0ejωt
- VC = VC0ejωt = ZCI0ejωt
- VS = VS0ejωt= VR + VC= (ZR + ZC)I0ejωt
- ωt spins the complex “phasors”
- I0 and VR0are real
- VC0is imaginary because of an e-jπ/2 term
- -jπ/2 puts the CIV in CIVIL

- VS0 is complex and will includean e-jΦ term
- -Φ is the phase of VS

- To get real voltages and currents - take real parts
- I = Re(I0ejωt)
- VR = Re(VR0ejωt)
- VC = Re(VC0ejωt)
- VS = Re(VS0ejωt)
- The ejωt terms will cancel.
We will leave them in for now

J

Alan Murray – University of Edinburgh

RC Complex numbers (L?)

- And impedances, ZR, ZC?
- VR = RI, VR and I are in phase

“j x” = “I leads VC by 90°”

The 90° phase shift is dealt with by the mathsautomatically.We no longer have to think about it explicitly.

J

Alan Murray – University of Edinburgh

RC Complex numbers (L?)

- This chooses I0 to be real, = “horizontal in the phasor diagram”

Sonny & Cher sing…"I got you babe" ...

X

X

Alan Murray – University of Edinburgh

And if we had anticipated the cancellation of ejωt …

- This chooses I0 to be real, = “horizontal in the phasor diagram”

Alan Murray – University of Edinburgh

RC Complex numbers (L?)

Insert numbers for |VS0|, R, C, f and thus ω

Then …

Alan Murray – University of Edinburgh

IS

VS

C

R

IC

IR

Fill in the blanksDraw a phasor diagram for

VS, VR, VC, IS, IR and IC

here

Similar expressions forVR and VC

VS= VR= VC

Now use Ohm’s Law to

Write an expression for IR,

complete with its ejωt+phase

Then do the same for IC

And tidy it up a little.

Now write this expression

IC as IC0ejωt+phase, with allthe phase information inthe exponential part.

Alan Murray – University of Edinburgh

From ejωt analysis

Note – in the phasor equations, the sinusoidal nature of the voltages andcurrents and the phase differences between them are not spelt out.

In the complex-number version these are explicitas the ejωt and e-jπ/2 terms respectively

(Impedances)

(Ohm’s Law on the total impedance)

(Ohm, continued … ejωt cancels out)

(More Ohm’s Law on R and C individually,then add VR0 and VC0 to get VS0)

Summary : How to use ejωtPlug in numbers, take real parts, calculate relative phases

NB – We can choose to leave out the ejωtthroughout the calculation, although it must go back in atthe end to retrieve the sinusoidal voltages and currents explicitly.

Alan Murray – University of Edinburgh

Clicker exercise

RC Trigonometry! (J?)Should be easy, shouldn't it?

Sonny & Cher sing…"I got you babe" ...

- I = I0cos(ωt)
- Equivalent to choosing I horizontal

- VR = RI0cos(ωt)
- VR is in phase with I

- VC = I/(ωC) and lags I by π/2 (CIVIL)
- VC = I0cos(ωt-π/2)/(ωC)
… which is the same as …

- VC = I0sin(ωt)/(ωC)
- VS = VR + VC
- VS = I0Rcos(ωt) + I0sin(ωt)/(ωC)
- VS = I0[Rcos(ωt) + sin(ωt)/(ωC)]
- VS = I0R[cos(ωt) + sin(ωt)/(RωC)]

Alan Murray – University of Edinburgh

RC Trigonometry! (J?)Should be easy, shouldn't it?

- VS = I0R[cos(ωt) + sin(ωt)(RωC)-1]
- Now for a very non-intuitive step …
- Set Ф = arctan[(RωC)-1],
- tan(Ф) = (RωC)-1 = sin(Ф) cos(Ф)

- VS = I0R[cos(ωt) + sin(ωt)tan(Ф)]
- VS = I0R[cos(ωt)cos(Ф) + sin(ωt)sin(Ф)] cos(Ф)
- VS = I0R cos(ωt+Ф) cos(Ф)
- fatigue setting in?

Alan Murray – University of Edinburgh

RC Trigonometry! (J?)Should be easy, shouldn't it?

X

X

- VS = I0R cos(ωt+Ф) = VS0 cos(ωt+Ф) cos(Ф)
- ... and after several lines of VERY tedious trigonometry(!) ...
- cos(Ф) = [1+(ωRC)-2]-½
- VS is at an angle of Ф = tan-1(ωRC-1)

Alan Murray – University of Edinburgh

Message?

- Trigonometry is familiar from school, but really messy - even for only two components (R and C)
- it gets MUCH worse VERY rapidly for more

- Phasors are excellent for seeing what is happening, but make for messy algebra.
- again - MUCH worse for 3, 4 or more components
- because phasors take you straight back to trigonometry

- Complex numbers are initially threatening, but make the actual maths MUCH easier, ONCE YOU HAVEACCEPTED THE IDEA.
- Groundhog Day finally ends ...
- Goodnight

Alan Murray – University of Edinburgh

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