V.%20Alternating%20Currents

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V. Alternating Currents

• Voltages and currents may vary in time.

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V1 Alternating Voltages and Currents
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Main Topics

• Introduction into Alternating Currents.
• Mean Values
• Harmonic Currents.
• Phase Shift

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Introduction into Alternating Currents I

• Alternating currents are generally currents which
  vary in time and time to time even change
  polarity i.e. the charges flow in opposite
  directions in the course of time.
• Usually by AC a subgroup of currents is meant,
  which is periodic and harmonic. But also some
  other shapes e.g. a square or saw-tooth are of
  practical importance.

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Introduction into Alternating Currents II

• We shall first define some general mean
  quantities which describe important AC
  properties.
• Later we shall concentrate on the harmonic AC.
  They are important since
• they are being widely produced and used.
• every function can be expressed using an integral
  or series of harmonic functions thereby it
  inherits some of its properties.

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The Mean Value I

• The mean value ltfgt of an time-dependent function
  f(t) is a constant quantity, which has during
  some time ? the same integral (effect) as the
  time-dependent function.
• For instance a mean current is a DC current which
  would transport the same charge over some time ?
  as would the alternating current.

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The Root Mean Square I

• When dealing with AC qualities one more mean is
  needed If there is a time-dependent current
  flowing through a resistor the thermal energy
  loses will be at any instant proportional to the
  square of the current (the resistor doesnt care
  about the direction of the current, it always
  heats).

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The Root Mean Square II

• The root mean square frms of an time-dependent
  function f(t) is a constant quantity, which has
  during some time ? the same thermal effect as
  the time-dependent function.
• Lets for instance feed a bulb by some
  time-dependent current I(t). Then if DC current
  of the value Irms flows through this bulb it
  would shine with the same brightness.

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Harmonic AC I

• From practical as well as theoretical point of
  view harmonic alternating currents and voltages
  play very important role. These are quantities
  the time dependence of which can be described as
  a goniometric functions sin(?), cos(?) exp(i?)
  of time e.g.
• V(t)V0sin(?t ?)
• I(t)I0sin(?t ?)

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Harmonic AC II

• The parameters V0 and I0 are called the peak
  values and from the properties of goniometric
  functions, it is clear that V(t) and I(t) vary
  sinusoidally between the values V0 and V0 or I0
  and I0.
• From now on we shall mean by alternating voltages
  or currents the harmonic ones.

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Harmonic AC III

• The AC voltage can be generated e.g. by the
  electromagnetic induction when rotating a coil of
  area A with N turns in uniform magnetic field B.
  In this case only the angle between the axis of
  the coil and the field changes. Lets suppose the
  dependence
• ?(t) ?t
• where ? 2?f is the angle frequency and f is the
  frequency of the rotation.

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Harmonic AC IV

• Then the flux through the whole coil is
• ?m NABcos?(t)
• And the EMF
• Vemf(t) -d?m/dt ?NABsin(?t)
• This is an AC voltage with the peak value of V0
  ?NAB. If this voltage is applied to a resistor R
  an AC current with the peak value of I0 ?NAB/R
  will flow through it.

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Harmonic AC V

• Lets note some important facts
• ?m(t) and Vemf(t) are phase-shifted by 90 or
  ?/2. When ?m(t) is zero Vemf(t) has a maximum. It
  is of course because the change of ?m(t) is the
  largest.
• V0 depends on ?.

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Harmonic AC V

• Harmonic voltage is also output from the LC
  circuit, if loses can be neglected.
• If we connect a charged capacitor to a coil,
  Kirchhoffs loop law is valid in any instant
• -L dI/dt Vc 0
• This leads to a differential equation of the
  second order. Its solution are harmonic
  oscilations.

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The Mean Value II

• It can be easily shown that the mean value of
  harmonic voltage as well as current is zero.
• It means that charge is not transported but only
  oscillates and the energy which is transported by
  the current is hidden in these oscillations.

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The Root Mean Square III

• It can be also easily shown that the rms values
  of harmonic voltage as well as current are
  non-zero.
• If the mains AC voltage is 120 V it is the rms
  voltage Vrms 120 V. So a bulb connected to this
  AC voltage or to the DC voltage of 120 V would
  shine with the same brightness. The peak voltage
  is V0 ? 170 V.

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The Phase Shift

• We shall see that in AC circuits we have to allow
  a phase shift between the voltage and current. It
  means they do not reach zero and maximum values
  at the same time.
• The power source offers some time-dependent
  voltage and the appliance controls how charge is
  withdrawn.
• We describe it using the phase shift angle ?
• V(t) V0sin(?t) and I(t) I0sin(?t ?)

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Homework

• Chapter 25 44, 45, 46, 47
• Chapter 29 28, 30, 31

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Things to read and learn

• Chapter 25 6, 7
• Chapter 29 4
• Chapter 30 6
• Chapter 31 1,2
• Try to understand all the details of the scalar
  and vector product of two vectors!
• Try to understand the physical background and
  ideas. Physics is not just inserting numbers into
  formulas!

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The Mean Value I

• ltfgt has the same integral as f(t) over some time
  interval?

Often we are interested in mean of a periodic
function over a long time. Then we choose as
representative time the period ? T.

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The Mean Value II

• ltIgt would transport the same charge as I(t) over
  some time ?

• The result of the integration is, of course, a
  charge since I dQ/dt. When divided by ? it
  gives a mean current over ?

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The Root Mean Square I

• frms has the same thermal effect as f(t) over
  some time interval?

For a long-time rms, we again choose a
representative time interval ? T (or T/2) .

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The Root Mean Square II

• Irms has the same thermal effect as I(t) over
  some time interval?

Brightness of a bulb corresponds to the
temperature i.e. thermal losses.

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The Mean Value III

• Let I(t) I0sin(?t) and representative ? T

Since the value of cos for the boundaries is the
same.
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The Mean Value IV

• If I(t) was rectified it would be I(t)
  I0sin(?t) for 0 lt t lt T/2 and I(t) 0 for T/2 lt
  t lt T

Since now cos(?T/2) cos(0) -2 !

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The Root Mean Square III

• Let I(t) I0sin(?t) and representative ? T

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The Mean Value V
Since now cos(?T/2) cos(0) -2 !

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LC Circuit I

• We use definition of the current I -dQ/dt and
  relation of the charge and voltage on a capacitor
  Vc Q(t)/C

• We take into account that the capacitor is
  discharged by the current. This is homogeneous
  differential equation of the second order. We
  guess the solution.

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LC Circuit II

• Now we get parameters by substituting into the
  equation

• These are un-dumped oscillations.

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LC Circuit III

• The current can be obtained from the definition I
  - dQ/dt

• Its behavior in time is harmonic.

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LC Circuit IV

• The voltage on the capacitor V(t) Q(t)/C

• is also harmonic but note, there is a phase shift
  between the voltage and the current.