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GROUNDHOG DAY!. Alan Murray. Agenda. RC circuit, AC signal using trigonometry ( J ?) using phasors ( K ?) using complex numbers ( L ?). worked examples complex numbers. T=1/f. V S0. V S. Phase. Notation. This is all we need . I. V R. V S. V C. RC Phasor ( K ?).

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GROUNDHOG DAY!

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## GROUNDHOG DAY!

Alan Murray

### Agenda

• RC circuit, AC signal

• using trigonometry(J?)

• using phasors(K?)

• using complex numbers(L?)

• worked examplescomplex numbers

Alan Murray – University of Edinburgh

T=1/f

VS0

VS

Phase

### Notation

Alan Murray – University of Edinburgh

### This is all we need ...

Alan Murray – University of Edinburgh

I

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

I

VR

VS

VC

### RC Phasor (K?)

Alan Murray – University of Edinburgh

I

VR

VS

VC

### RC Phasor (K?)

Alan Murray – University of Edinburgh

I

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

I

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 blanks

Draw 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 the phasor diagram

From ejωt analysis

## Comparison

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

(Notation)

(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 ejωt

Plug 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

Worked examples lecture

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