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V. Alternating Currents

V. Alternating Currents. Voltages and currents may vary in time. V–1 Alternating Voltages and Currents. Main Topics. Introduction into Alternating Currents. Mean Values Harmonic Currents. Phase Shift. Introduction into Alternating Currents I.

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V. Alternating Currents

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  1. V. Alternating Currents Voltages and currents may vary in time.

  2. V–1 Alternating Voltages and Currents

  3. Main Topics • Introduction into Alternating Currents. • Mean Values • Harmonic Currents. • Phase Shift

  4. Introduction into Alternating Currents I • Alternating currents are generally currents which varyintime and time to time even changepolarity 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.

  5. Introduction into Alternating Currents II • We shall first define some generalmeanquantities 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.

  6. The Mean Value I • The meanvalue<f> 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.

  7. 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 doesn’t care about the direction of the current, it always heats).

  8. 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 samethermaleffect as the time-dependent function. • Let’s 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 samebrightness.

  9. 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 goniometricfunctions [sin(), cos() exp(i)] oftime e.g.: V(t)=V0sin(t + ) I(t)=I0sin(t + )

  10. 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.

  11. 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. Let’s suppose the dependence: (t) = t • where  = 2f is the anglefrequency and f is the frequency of the rotation.

  12. 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.

  13. Harmonic AC V • Let’s 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 .

  14. Harmonic AC V • Harmonic voltage is also output from the LCcircuit, if loses can be neglected. • If we connect a charged capacitor to a coil, Kirchhoff’s 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.

  15. The Mean Value II • It can be easily shown that the meanvalue of harmonic voltage as well as current is zero. • It means that charge is nottransported but only oscillates and the energy which is transported by the current is hidden in these oscillations.

  16. The Root Mean Square III • It can be also easily shown that the rmsvalues 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.

  17. The Phase Shift • We shall see that in AC circuits we have to allow a phaseshift 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 phaseshiftangle: • V(t) = V0sin(t) and I(t) = I0sin(t + )

  18. Homework • Chapter 25 – 44, 45, 46, 47 • Chapter 29 – 28, 30, 31

  19. 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!

  20. The Mean Value I • <f> 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. ^

  21. The Mean Value II • <I> 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 : ^

  22. 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) . ^

  23. 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. ^

  24. The Mean Value III • Let I(t) = I0sin(t) and representative  = T: Since the value of cos for the boundaries is the same.

  25. The Mean Value IV • If I(t) was rectified it would be I(t) = I0sin(t) for 0 < t < T/2 and I(t) = 0 for T/2 < t < T: Since now cos(T/2) – cos(0) = -2 ! ^

  26. The Root Mean Square III • Let I(t) = I0sin(t) and representative  = T: ^

  27. The Mean Value V Since now cos(T/2) – cos(0) = -2 ! ^

  28. 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.

  29. LC Circuit II • Now we get parameters by substituting into the equation: • These are un-dumped oscillations.

  30. LC Circuit III • The current can be obtained from the definition I = - dQ/dt: • Its behavior in time is harmonic. ^

  31. 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. ^

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