l22 ac circuits reactance

8/20/2019 L22 AC Circuits Reactance

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Lecture 21. AC Circuits, Reactance.

Outline:

Power in AC circuits, Amplitude and RMS values.

Phasors / Complex numbers.

Resistors, Capacitors, and Inductors in the AC circuits.

Reactance and Impedance.

1

Conflict final exam: December 7, 500 PM: Last day and time to request

conflict exam for Final Exam (you can request conflict exam if you have another

exam at same time OR have 3 exams in 24 hour period). You must e-mail

Professor Cizewski [email protected] with details of why you are

requesting conflict exam.November 25 (Rutgers Thursday): Required E&M Post-test during lecture

times. You can attend any lecture. Makeup post tests will be in early

December. NO Recitations the week of November 24.

mailto:[email protected]:[email protected]


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Oscillations in L-C Circuits

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L-C circuits: the circuits with TWO elements that can store energy

(ideally, without dissipation). The energy flow back and forth

between L and C results in harmonic oscillations of and .

=

2 =

2

electric field energy

in the capacitormagnetic field energy

in the inductor

Let’s say, at t = 0 the capacitor is fully charged, =0.

2=

2+

2


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Oscillations in L-C Circuits (cont’d)

3

= 0,

≡

,

+

1

= 0

Solution: = +

amplitude angl. frequency

phase at t =0

=1

≡

1

≡

2

= 2

=

2=

2 =

41 + 2

=

= +

=

2=

2 =

41 + 2

/2


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L-C-R Circuits

6

weak damping (“small”R

) strong damping (“large”R

)

= 0,

+

+

1

= 0

Solution for weak damping

≫

4:

=

2 ∗ +

∗ =1

4


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Impedance of AC circuits

13

So far we have considered transient processes in R-C and R-L

circuits: the approach to the stationary (time-independent)

state after some perturbation (switch on / off).

Today we’ll discuss how these circuits behave being connected

to the alternating current (AC) power supply: the circuits driven

by a steady external drive, e.g. the AC voltage source.

We disregard all transient processes and instead consider the

steady-state AC currents: currents and voltages vary with time

as + , but their amplitudes are t-independent .

We’ll describe the response of an L-C-R circuit to a harmonic

drive using the notion of impedance.


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Amplitudes, rms Values, and Power in AC Circuits

14

= cos

= cos +

= cos ∙ cos + =1

2 cos 2 + + cos

≡ =1

2 cos

Currents and voltages are

NOT necessarily in phase,

is the phase shift

between V and I (the

“phase angle”).

=

2 =

2

“120V” wall outlet: = 60, = 2 ∙ 60

= 377/, =

≈ 17

= 120, ≈ 170

Root mean square (rms): the

square root of the average of

the square of the quantity:

=

Power, being expressed in the rms values: = ∙ cos

RLC

Instantaneouspower: =

Instantaneous

values:

cos -

the power factor


7/15

Phasors / Complex Number Representation

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Problem: To find current/voltage in R-L-C circuits,

we need to solve differential equations.

Solution: The use of complex numbers / phasors

allows us to replace linear differential equationswith algebraic ones, and reduce trigonometry to

algebra.

We represent voltages and currents in the R-L-C circuits as the phase vectors

( phasors) on the 2D plane. Quantity: = cos. Corresponding

phasor: a vector of length

rotating counter-clockwise with the angular

frequency . Instantaneous value of is the projection of the phasor onto

the horizontal axis.

If all the quantities oscillate with the same , we can get

rid of the term by using the rotating (merry-go-

around) reference frame.

We’ll consider the steady state of AC circuits, when all amplitudes (the phasor lengths)

are t -independent, and the only time dependence remaining is in the single frequency

sinusoidal oscillation of voltages and currents. The angle between different phasors

represents their relative (t -independent) phase.

Instantaneous

value

phasor


8/15

Complex Numbers, Phasors

16

≡ +

= +

∗ ≡

=

=

=

≡ ∙ ∗

≡ 1

= cos + sin Euler’s

relationship

= 1 1

=

complex conjugate of

+ = + + − +

2= +

+ = + − +

2= +

= + = + =

=

− =

Imaginary

unit:

=

The absolute value (or modulus or magnitude):

Phasor : refer to either + or just . In the latter case, it is understood

to be a shorthand notation, encoding the amplitude and phase of an underlying

sinusoid.


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Complex Numbers, Phasors (cont’d)

17

Addition: + + + = + + +

Multiplication:

Differentiation:

∙ = +

=

The use of complex numbers / phasors allows us to replace linear differential equations

with algebraic ones, and reduce trigonometry to algebra:


10/15

Resistor

18

Power dissipated in a resistor: = ()

=1

2 =

1

2 =

phase 2

AC current through a resistor and AC voltage across the resistor are always in phase.


11/15

Capacitor

19

= 0

=

=

= +

=

1

=

1

=

=

=1

2 = =

2= 0

= ()

= −

phase 2 The power IS NOT dissipated in a capacitor:

it is stored in the capacitor for half a period,

and returned to the circuit for another half.

=

= /2

For a capacitor, voltage

LAGS current by 900

.

Current (reference phasor)

Voltage= cos2 + sin 2 cos

2

+ sin

2

= sin2

0 -1


12/15

Inductor

20

= 0

=

= +

=

+

= =

+

=1

2 = =

2= 0

= ()

= +

time t

phase

2 The power IS NOT dissipated in an inductor:

it is stored in the inductor for half a period,

and returned to the circuit for another half.

=

= /2

In an inductor, current

LAGS voltage by 900

Current (reference phasor)

Voltage

= cos2 + sin2 cos

2

+ sin

2

= sin2


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Reactance

21

= = Resistor

=

Capacitor

Inductor

=

1

=

+

=

=

=

=1

=

=

= −

= = =

= =

Wiki: “reactance is the opposition of a circuit element to a change of electric current

due to that element's inductance or capacitance”. The reactance is measured in Ohms.

Major differences between reactance and resistance: the reactance for L and C

changes with frequency, it reflects (being combined with ±) the phase shift between

V and I, and it dissipates no energy.


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Impedance

25

=

Impedance is a measure of the overall opposition of a circuit to current, in other words:

how much the circuit impedes the flow of current. Both the reactance and resistance are

components of the impedance. The magnitudes of V and I are the rms voltage and

current respectively, and the various reactances behave mathematically just likeresistances, except that they are complex.

= + = + 1

For R, C , and L in series:

= + 1

= ∙

∗ = + 1

= ∙ ∗ = + 1

=

For = 120, 60, = 300Ω, = 2 , = 3:

1

= 377 ∙ 3

1

377 ∙ 2 ∙ 10−= 1131 1326 = 195Ω

=120

300 + 195Ω= 0.335


15/15

26

Next time: Lecture 23. Resonance in AC circuits,

Transformers.

§§ 31.3 - 6

l22 ac circuits reactance

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