Flashcards in AC Circuits Deck (26):

1

## What is the relationship between E and B in an electromagnetic wave?

### E = Bc

2

## What is the energy density for a magnetic field in an electromagnetic wave?

### um = 1/2 1/μo B²

3

## What is the energy density for an electric field in an electromagnetic wave?

### ue = 1/2 εo E²

4

## What is the speed of an electromagnetic wave?

### c = √[1/μoεo]

5

## What is the relationship between energy density in the electric and magnetic fields that make up an electromagnetic wave?

###
ue = 1/2 εo E² = 1/2 εo (Bc)² = 1/2 εo c²B² =

1/2 εo (1/√[μoεo])² B² = 1/2 1/μo B² = um

-so the energy densities are equal

6

##
Wave Intensity for an Electromagnetic Waves

Energy

###
-consider a power source emitting waves

-lets consider a part of a wave front from this source, this part has surface area A and is travelling at speed v

-the wave front is initially at position 1 and then a small time Δt later it is at position 2

-in the time interval Δt, the wave front has moved a distance v Δt

-the small amount of volume traversed by the wavefront in going from position 1 to 2 is Av Δt

-the wave has brought energy to this volume, if the wave is electromagnetic then in this volume there will now be oscillating electric and magnetic fields that weren't there before

-the amount of energy in this small volume is:

ΔE = uav Av Δt

-where uav is the average energy density of the wave within that volume

7

##
Wave Intensity for an Electromagnetic Waves

Power

###
-we have that the energy in a small volume traversed by part of a wavefront of area A is given by:

ΔE = uav Av Δt

-the power entering this volume is given by:

P = ΔE/ Δt = uav Av Δt / Δt = uav A v

8

##
Wave Intensity for an Electromagnetic Waves

Intensity - General Form

###
-we have that the power entering a volume traversed by an area A of a wavefront in a small time Δt is given by:

P = uav A v

-the intensity of a wave is by definition the amount of power per area:

I = P/A = uav A v / A = uav v

9

##
Wave Intensity for Electromagnetic Waves

Intensity - Electromagnetic Waves

###
I = uav * v

-this is a general result for a wave, for an electromagnetic wave specifically, v=c and total energy density u is given by u = ue + um

-but we know that ue = um, so u = 2um = 2(1/2 1/μo B²)

u = 1/μo B²

-|B fluctuates with time, and we need the average energy density uav

uav = \B² /μo , where \B² indicates the mean value of B²

-we can then write this in terms of the root mean squared value

uav = (Brms * Brms) / μo

-rewrite in terms of B and E

uav = (Erms/c * Brms) / μo

-sub in for intensity

I = uav v = (Erms/c * Brms) / μo * c = Erms*Brms/ μo

10

##
Wave Intensity for Electromagnetic Waves

Poynting Vector

###
-for the intensity of an electromagnetic wave, we have:

I = Erms*Brms/ μo

-recall the Poynting vector:

|S = |Ex|B /μo , where |S is in the direction of propagation of the wave

-find the magnitude of |S

S = | |S | = | |Ex|B | / μo = |E||B|sin90/μo = EB/μo

-we can write this only in terms of B:

S = (Bc)B / μo = B²/μo * c

-since S depends on B and B fluctuates with time, S also fluctuates with time, calculate the average value of S, Sav:

Sav = \S = \B²/μo * c = uav * c = I

-therefore the average magnitude of |S is equal to the intensity of the wave

11

##
Oscillating Voltage

Notation

###
-in AC circuit theory complex numbers are used to express the voltages and currents across and through components in a circuit:

-an oscillating voltage such as

V = Vo cos(ωt)

-this can be written as the real part of a complex number;

V = Re{ Vo exp(j ωt) }

-where j²=-1 to prevent confusion of i with I (current)

-we can then write voltages and currents as complex numbers:

~V = Vo exp (jωt) and ~I = Io exp[j (ωt-𝛿)]

-where Vo and Io are real numbers, ω is angular frequency and 𝛿 is a phase difference

-the measured voltages and currents are then determined from V = Re{~V} and I = Re{~I}

12

## Dealing With Components in AC Circuits

###
-in an AC circuit the direction of current is constantly alternating

-imagine taking a snapshot of the circuit so that the direction of current at that exact moment is constant

-apply Kirchhoff's loop rule

-substitute in complex forms of ε and I

-cancel ωt terms

-remove exponentials using Euler's formula:

exp(jθ) = cosθ + jsinθ

-separate real and imaginary parts

-solve for 𝛿 using the imaginary part

-sub this value for 𝛿 into the real part

13

## Inductor in an AC Circuit

###
-applying Kirchhoff's loop rule, where εL is the emf across the inductor:

~ε - ~εL = 0

-recall that the emf across an inductor is given by εL=L*d~I/dt and sub in:

~ε = L * d~I/dt

-sub in complex forms of ε and I

εo exp(j ωt) = L * d(Io exp(j(ωt - 𝛿))/dt

εo exp(j ωt) = L Io jω * exp(j(ωt) - 𝛿))

-cancel the ωt terms:

εo = L Io jω * exp(-j𝛿)

-remove exponential using Euler's formula:

εo = L Io jω [cos(-𝛿) + jsin(-𝛿)]

εo = L Io jω [cos(𝛿) - jsin(𝛿)]

-move j into bracket:

εo = L Io ω [jcos(𝛿) + sin(𝛿)]

-equate real and imaginary parts to get two equations:

εo = L Io ω sin(𝛿) and 0 = - L Io ω cos(𝛿)

-solve the imaginary part to find 𝛿

𝛿 = π/2 or 3π/2

-but if 𝛿 = 3π/2, then sin(𝛿) = -1 which doesn't make sense since εo, L, Io, and ω are all positive

-therefore 𝛿 = π/2, sub in to the real equation:

εo = L Io ω sin(π/2)

εo = L Io ω

14

##
Complex Impedance

Inductor

###
~V = ~I ~Z :

-for an inductor we have εo = L Io ω and 𝛿=π/2

εo exp(j ωt) = εo/Lω * exp(j(ωt - π/2)) * Zo * exp(j𝛿)

-divide by εo exp(j ωt) :

1 = 1/Lω * exp(-j * π/2) * Zo * exp(j𝛿)

-rearrange for ~Z :

~Z = Zo exp(j𝛿) = ωL exp(j * π/2)

-thus, Zo = ωL and 𝛿 = π/2

-sub in Euler's Formula:

~Z = Zo exp(j𝛿) = ωL [cos(π/2) + jsin(π/2)]

~Z = ωLj

15

## Capacitor in an AC Circuit

###
-applying Kirchhoff's loop rule, where Vc is the voltage across the capacitor:

~ε - ~Vc = 0

-recall that the voltage across a capacitor is given by V=Q/C and sub in:

~ε = ~Q/C

-since ~I is complex, ~Q is too

~Q = C ~ε

-sub in for ~I :

~I = d~Q/dt = d(C~ε)/dt = C * d~ε/dt

-sub in complex form of ε

~I =C * d(εo exp(j ωt))/dt

-differentiate

~I = C * εo j ω * exp(j ωt)

-replace j = exp(j * π/2) from Euler's Formula:

~I = C εo * exp(j * π/2) * ω * exp(j ωt)

~I = C εo ω * exp(j(ωt + π/2)

-equating this to the general form of ~I

~I = Io exp(j(ωt - 𝛿)) = C εo ω * exp(j(ωt + π/2)

-thus 𝛿 = -π/2 , and:

Io = C εo ω

16

##
Complex Impedance

Capacitor

###
-the definition of complex impedance:

~V = ~I ~Z

-for a capacitor, Io exp(j(ωt - 𝛿)) = Cεoω*exp(j(ωt + π/2) , so

εo exp(j ωt) = Cεoω*exp(j(ωt + π/2) * Zo * exp(j𝛿)

-divide by εo exp(j ωt) :

1 = Cω exp(j * π/2) * Zo * exp(j𝛿)

-rearrange for ~Z :

~Z = Zo * exp(j𝛿) = 1/ωC * exp(-j * π/2)

-thus Zo = 1/ωC and 𝛿 = - π/2

-sub in Euler's Formula:

~Zc = 1/ωC [cos(-π/2) + jsin(-π/2)]

~Zc = - j/ωC

17

## Capacitive Reactance

###
-the complex impedance is given by:

~Zc = - j/ωC

-the capacitive reactance is given by Xc = 1/ωC

18

## Inductive Reactance

###
-for an inductor we have a complex impedance:

~Z = ωLj

-the inductive reactance is XL = ωL

19

## Resistor in an AC Circuit

###
-applying Kirchhoff's loop rule, where Vr is the voltage across the resisitor:

~ε - ~Vr = 0

-recall that the voltage across a resistor is given by Ohm's Law: V = IR and sub in:

~ε = ~I R

-sub in complex forms of ε and I

εo exp(j ωt) = RIo exp(j(ωt - 𝛿)

-cancel the ωt terms:

εo = R Io * exp(-j𝛿)

-remove exponential using Euler's formula:

εo = R Io [cos(-𝛿) + jsin(-𝛿)]

εo = R Io [cos(𝛿) - jsin(𝛿)]

-equate real and imaginary parts to get two equations:

εo = R Io cos(𝛿) and 0 = - R Io sin(𝛿)

-solve the imaginary part to find 𝛿

𝛿 = π or 0

-but if 𝛿 = π, then cos(𝛿) = -1 which doesn't make sense since εo, R, and Io are all positive

-therefore 𝛿 = 0, sub in to the real equation:

εo = R Io cos(0)

εo = R Io

20

##
Complex Impedance

Definition

###
~V = ~I ~Z

-where ~Z = Zo exp(j𝛿)

21

##
Complex Impedance

Resistor

###
-for a resistor, V=IR so the complex impedance given by:

~Vr = ~I ~Zr

-is just ~Zr = R

22

## Complex Voltage Notation

### ~V = Vo exp(jωt)

23

## Complex Current Notation

### ~I = Io exp(j(ωt - 𝛿))

24

## Complex Impedance in Series

### ~Zt = ~Z1 + ~Z2 + ~Z3 + ..

25

## Complex Impedance in Parallel

### 1/~Zt = 1/~Z1 + 1/~Z2 + 1/~Z3 + ...

26