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Unit 9: Alternating-current circuits Introduction. Alternating current features. Behaviour of basic dipoles (resistor, inductor, capacitor) to an alternating current. RLC series circuit. Impedance and phase lag. Power on A.C. Resonance. Niagara Falls Nikola Tesla 1856-1943

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  • Unit 9: Alternating-current circuits

    Introduction. Alternating current features.

    Behaviour of basic dipoles (resistor, inductor,

    capacitor) to an alternating current.

    RLC series circuit. Impedance and phase lag.

    Power on A.C.

    Resonance.

    Niagara

    Falls

    Nikola Tesla 1856-1943

  • Coil turning inside a magnetic field B.

    NBSwsenwtdt

    d==

    N S

    S

    B

    t

    wtNBS cos== SBNrr

    %Um

    Symbol:

    Alternating-current generation

  • N1 N2

    V1 ~ V2 ~

    The transformer The current on primary produces a current and then a flux varying on time.

    This flux is completely driven by the ferromagnetic material, producing an inducedelectromotive force on terminals of secondary. The flux on each loop u is the same forprimary and secondary.

    It works due to electromagnetic induction and then it doesnt work on D.C.

    Primary

    winding

    Secondary

    windingdt

    dNV u111

    ==

    dt

    dNV u222

    ==

    1

    2

    1

    2

    2

    2

    1

    1 V

    N

    N

    VN

    V

    N

    V==

    Voltage ratio or ratio

    of transformation

    equals the turns ratio

    On an ideal transformer the power on primary equals the power on secondary

    On an real transformer there are three type of losses:

    On windings: Joule heating

    On core: Magnetic Histeresys

    and Eddy currents

    In order to minimize Eddy currents the core is built with

    sheets isolated between them

    B

    r

    i

  • Period T = 2/ (s)

    Frequency f = 1/T (Hz)

    Angular frequency

    w = 2f (rad/s)

    Phase wt+

    Initial phase (degrees or radians) (phase at t=0)

    Amplitude=Maximum voltage Um (V)

    t

    T

    Um

    u(t) = Um cos(t + )u(t)

    f Europe: 50 Hz

    f North America: 60 Hz

    Sinusoidal alternating-current features

  • t

    u =0

    u(t) = Um cos(t + u)

    u(t)

    Initial phase. Examples.

    t

    u(t)

    u=90 (/2 rad)

    t

    u(t)

    u=-90 (-/2 rad)

    t

    u(t)

    u=-45 (-/4 rad)

  • To simplify the analysis of A.C. circuits, a graphical representation of sinusoidalfunctions called phasor diagram can be used.

    A phasor is a vector whose modulus (length) is proportional to the amplitude ofsinusoidal function it represents.

    The vector rotates counterclockwise at an angular speed equal to . The anglemade up with the horizontal axis is the phase (t+).

    Therefore, depending if we are working with the function sinus or the functioncosinus, this function will be repreented by the vertical projection or thehorizontal projection of the rotating vector.

    t

    T

    Um

    u(t) = Um cos(t + )

    u(t)

    Phasor diagram

    U

    t+

    Um

    +

    +

    )sin(:

    )cos(:Pr

    tUVertical

    tUHorizontalojections

    m

    m

  • As the position of phasor is different for any time considered, the graphicalrepresentations are done on time t=0 and then, the initial phase is the anglebetween vector and horizontal axis. In this way, the phasor is a unique vector(not changing on time) for a given function:

    t

    T

    Um

    u(t) = Um cos(t + )

    u(t)

    Phasor diagram

    U

    Um

    Phasor diagram

  • u(t) = Um cos(t + u )i(t) = Im cos(t+i)

    t

    iu =

    Phase lag between two waves (voltage and intensity)

    Phase lag is defined as

    i=0 u

  • t

    iu =Phase lag between two waves (voltage and intensity)

    i=0u>0

    0>

    t

    i

    U =u

    I

    U =-i

    I

  • Behaviour of basic dipoles. Resistor

    Resistor

    t

    i

    u

    u(t) = R i(t) = RIm cost = Um cost

    i(t) = Im cost

    Ri(t)

    u(t)

    Um = R Im

    = 0

    Tipler, chapter 29.1

    uR = iR

    U

    I

  • Behaviour of basic dipoles. Inductor

    i(t) = Im cost

    Li(t)

    u(t)

    Um = LIm

    = /2

    Tipler, chapter 29.1

    Inductor

    t

    iu

    )2

    tcos(U)2

    tcos(ILtsenILdt

    )t(diL)t(u mmm

    +=+===

    XL = L Inductance ()

    dt

    )t(diLuL =

    U

    I

  • Behaviour of basic dipoles. Capacitor

    Ci(t)

    u(t)

    = - /2

    Tipler, chapter 29.1

    Capacitor

    t

    i

    u

    u(t) = Um cost

    )2

    tcos(I)2

    tcos(CU)t(senCUdt

    )t(Cdu

    dt

    )t(dq)t(i mmm

    +=+====

    =

    C

    IU mmXC = 1/C Capacitance ()

    Cuq =

    U

    I

  • R

    L

    C

    )cos( umL wtLwIu +=

    )cos( umR wtRIu +=

    )cos( um

    C wtCw

    Iu +=

    +=

    =

    2

    mLLm IXU

    =

    =

    0

    mRm IRU

    =

    =

    2

    mCCm IXU

    Behaviour of basic dipoles. Review

    Voltage and intensity go on phase

    Voltage goes ahead intensity 90

    Voltage goes behind intensity 90

  • Sinusoidal functions are interesting because additionof (S.F.) is another S.F.

    On the other hand, Fouriers law says that anyperiodic function can be split in S.F. of differentfrequencies. This is the main reason why behaviourof S.F. is studied.

    Knowing response of basic dipoles to a sinusoidalcurrent, we know response to any (no sinusoidal)current.

    Sinusoidal functions (S.F.) and Fouriers law.

  • L R C

    uLuR uC

    i(t)= Im cos (wt)

    u(t) = uL (t)+ uR (t)+ uC (t)= Um cos (wt+)

    Lets take a circuit with resistor, inductor and capacitor inseries. If a sinusoidal intensity i(t)=Imcos(wt) is flowingthrough such devices, voltage on terminals of circuit will bethe addition of voltages on each device:

    RLC series circuit. Impedance of dipole

    u(t)

    Addition of sinusoidal

    functions is another

    sinusoidal function

  • RLC series circuit. Impedance of dipole

    Um cos (wt+) = LwIm cos (wt +/2)+RIm cos (wt)+(1/Cw)Im cos (wt -/2)

    UL

    I URUC

    UL-UC

    I UR

    U

    Um

    (L-1/C) Im

    RIm

    Um

    ZXRXXRI

    U

    CwLwRIU CL

    m

    mmm =+=+=+=

    222222 1)()((

    tgR

    X

    R

    XX

    R

    CwLw

    tg CL ==

    =

    =

    1Z Is called Impedance of dipole ()

    is phase lag of dipole

    Z and are depending not only on parameters of R, L and C, but also on frequency of applied current.

  • R

    XL=Lw

    ZX

    X0)

  • Power on A.C.

    Power consumed on a device on A.C. can be computed multiplying voltage and

    intensity on each time. This is the Instantaneous power:

    )tsin(I)t(i m =

    )tsin(u)t(u m +==+== )tsin(I)tsin(U)t(i)t(u mm

    t2sinsin2

    IUtsincosIUtsin]sintcoscost[sinIU mm

    2

    mmmm

    +=+=

    Reactive powerActive or

    Real power

    p(t)

    )(tp

    Instantaneous power = Active power + Reactive power

    Frequency of reactive power doubles that of active power

    tIU mm 2sincos t

    IU mm 22

    sinsin

    = +

  • Power on A.C.

    > 0 and < 0 zero on a cycleAlways > 0

    Active power + Reactive power = Instantaneous power

    =+

    t2sinsin2

    IU mm

    )t(p+ =

    Example taking: Im = 1 A Um = 1 V = 1 rad/s = 0,6 rad

    Average value on a cycle:

    cos2

    IUtdtsincosIU

    T

    1dt)t(p

    T

    1PP mm

    T

    0

    2

    mm

    T

    0

    a

    av

    ====

    Power on A.C.

    -0,4

    -0,2

    0

    0,2

    0,4

    0,6

    0,8

    1

    0 2 4 6 8 10 12

    wt (rad)

    Po

    wer

    (w)

    Active pow er

    Power on A.C.

    -0,4

    -0,2

    0

    0,2

    0,4

    0,6

    0,8

    1

    0 2 4 6 8 10 12

    wt (rad)P

    ow

    er

    (w)

    Reactive pow er

    Power on A.C.

    -0,4

    -0,2

    0

    0,2

    0,4

    0,6

    0,8

    1

    0 2 4 6 8 10 12

    wt (rad)

    Po

    wer

    (w)

    Inst. pow er

    cos Power factor

    tsincosIU mm 2

    =T

    Tdtwt

    0

    2

    2)(sin

  • DipoleActive

    power Pa(t)Average

    Paav

    Reactive power Pr(t)

    Average Prav

    P(t) Pav

    R=0

    Cos =1

    Sin =00 0

    L=90

    Cos =0

    Sin =10 0 0 0

    C=-90

    Cos =0

    Sin =-10 0 0 0

    Rms magnitudes. Power for basic dipoles

    Consumed power on basic dipoles (R, L and C):

    )(2sinsin2

    sincos 2 tptIU

    tIU mmmm =

    +

    An A.C. current consumes the same power than a D.C. current having the rms magnitudes

    cosIUcos2

    IUpowerAverage rmsrms

    mm =

    =2

    mrms

    UU =

    2

    mrms

    II =

    tsinIU2

    mm R

    Um

    2

    2

    tsinIU2

    mm R

    Um

    2

    2

    t2sinL2

    U2

    m

    t2sinL2

    U2

    m

    t2sinC2

    U2

    m t2sinC2

    U2

    m

    Tipler, chapter 29.1

  • 22 1 )((Cw

    LwRI

    UZ

    m

    m +==

    RLC series circuit. Resonance

    Drawing Z v.s frequency

    Z v.s. freq

    0

    100

    200

    300

    400

    500

    600

    0 500 1000 1500 2000 2500 3000 3500 4000

    frequency (Hz)Z

    (O

    hm

    )

    Example taking: R = 80 L = 100 mH C = 20 F

    Resonance: f0=707 Hz Z=80

    On resonance, impedance of circuit is minimum, and

    amplitude of intensity reaches a maximum (for a given

    voltage). Intensity and voltage on terminals of RLC

    circuit go then on phase. cos =1 and consumed

    power on RLC dipole is maximum.

    There is a frequency where XL=XC and then the impedance gets its minimum value (Z=R).

    This frequency is called Frequency of resonance (f0) and can be easily computed:

    LCf

    LCCL

    1

    2

    11100

    0

    0

    ===