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Unit 10: Alternating-current circuits Introduction. Alternating current features. Phasor diagram. Behaviour of basic dipoles (resistor, inductor, capacitor) to an alternating current. RLC series circuit. Impedance and phase lag. Resonance. Filters Niagara Falls Nikola Tesla 1856-1943

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  • Unit 10: Alternating-current circuits

    Introduction. Alternating current features.

    Phasor diagram.

    Behaviour of basic dipoles (resistor, inductor,

    capacitor) to an alternating current.

    RLC series circuit. Impedance and phase lag.

    Resonance. Filters

    Niagara

    Falls

    Nikola Tesla 1856-1943

  • Period T = 2π/ω (s)

    Frequency f = 1/T (Hz)

    Angular frequency

    w = 2πf (rad/s)

    Phase wt+ϕ

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

    Amplitude=Maximum voltage Um (V)

    is the root mean square value. Is that measured by the

    measurement devices on A.C.

    ωt

    T

    ϕ

    Um

    u(t) = Um cos(ωt + ϕ) u(t)

    f Europe: 50 Hz

    f North America: 60 Hz

    Sinusoidal alternating-current features

    2

    m

    rms

    UU =

  • 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 ω. The anglemade up with the horizontal axis is the phase (ωt+φ).

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

    ωt

    T

    ϕ

    Um

    u(t) = Um cos(ωt + ϕ)

    u(t)

    Phasor diagram

    U

    ωt+φ

    Um

    ω

    +

    +

    )sin(:

    )cos(:Pr

    ϕω

    ϕω

    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 φ is the anglebetween vector and horizontal axis. In this way, the phasor is a unique vector(not changing on time) for a given function:

    ωt

    T

    ϕ

    Um

    u(t) = Um cos(ωt + ϕ)

    u(t)

    Phasor diagram

    U

    φ

    Um

    Phasor diagram

  • ωtϕϕϕϕu =0

    u(t) = Um cos(ωt + ϕu)

    u(t)

    Initial phase. Examples.

    ωt

    u(t)

    ϕϕϕϕu=90º (ππππ/2 rad)

    ωt

    u(t)

    ϕϕϕϕu=-90º (-ππππ/2 rad)ωt

    u(t)

    ϕϕϕϕu=-45º (-ππππ/4 rad)

    UU

    U U

  • u(t) = Um cos(ωt + ϕu )i(t) = Im cos(ωt+ϕi)

    ϕ ωt

    iu ϕϕϕ −=

    Phase lag between two waves (voltage and intensity)

    Phase lag is defined as

    ϕϕϕϕi=0 ϕϕϕϕu

  • ϕ ωt

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

    ϕϕϕϕi=0ϕϕϕϕu>0

    0>ϕ

    ϕ ωt

    ϕϕϕϕiϕ

    Uφ =φu

    I

    Uφ =φi

    I

  • Behaviour of basic dipoles. Resistor

    Resistor

    ωt

    i

    u

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

    i(t) = Im cosωt

    Ri(t)

    u(t)

    Um = R Im

    ϕ = 0

    Tipler, chapter 29.1

    uR = iR

    U

    I

  • Behaviour of basic dipoles. Inductor

    i(t) = Im cosωt

    Li(t)

    u(t)

    Um = LωIm

    ϕ = π/2

    Tipler, chapter 29.1

    Inductor

    ωt

    iu

    XL = Lω Inductive reactance (Ω)

    U

    I

    =Ldi(t)

    u Ldt

    m m m

    di(t)u(t) L L I sen t L I cos( t ) U cos( t )

    dt

    π πω ω ω ω ω= = − = + = +

    2 2

  • Behaviour of basic dipoles. Capacitor

    Ci(t)

    u(t)

    φ = - π/2

    Tipler, chapter 29.1

    Capacitor

    ωt

    i

    u

    u(t) = Um cosωt

    XC = 1/Cω Capacitive Reactance (Ω)

    Cuq =

    UI

    Cd(u )i(t) Cdt

    =

    m m m

    dq(t) Cdu(t)i(t) CU sen t CU cos( t ) I cos( t )

    dt dt

    π πω ω ω ω ω= = = − = + = +

    2 2

    mm

    IU

    Cω=

  • R

    L

    C

    )cos( umL wtLwIu ϕ+=

    )cos( umR wtRIu ϕ+=

    )cos( um

    C wtCw

    Iu ϕ+=

    +=

    =

    2πϕ

    mLLm IXU

    =

    =

    mRm IRU

    −=

    =

    2πϕ

    mCCm IXU

    Behaviour of basic dipoles. Review

    Voltage and intensity go on phase

    Voltage goes ahead intensity 90º

    Voltage goes behind intensity 90º

  • L R C

    uLuR uC

    i(t)= Im cos (wt)

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

    Let’s take a circuit with resistor, inductor and capacitor in series. If asinusoidal intensity i(t)=Imcos(wt) is flowing through such devices,voltage on terminals of circuit will be the addition of voltages on eachdevice:

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

    UL

    I URUC

    UL-UC

    I UR

    U

    ϕ Um

    (Lω-1/Cω) Im

    RIm

    ϕ Um

    ZXRXXRI

    U

    CwLwRIUI

    CwLwRIU CL

    m

    mmmmmm =+=−+=−+=−+=

    222222222)()

    1(())

    1(()(

    ϕϕ tgR

    X

    R

    XX

    R

    CwLw

    tg CL ==−

    =

    =

    1 Z Is called Impedance of dipole (Ω)

    ϕ is phase lag of dipole

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

    ϕ is ranging between - and 2

    π2

    π

    U = UL + UR + UC

  • R

    ZX

    ϕϕϕϕ

    X0)

  • 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 Ω L = 100 mH C = 20 μF

    Resonance: f0=707 Hz Z=80 Ω

    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.

    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

    ωω

    ω ===

  • RLC series circuit as a Bandpass filter

    C

    Ru(t)

    L

    uR(t)

    Input

    Outp

    ut

    22

    mmmRouput

    )C

    1L(R

    RU

    Z

    URRIUU

    ωω −+

    ====

    22m

    R

    input

    output

    )C

    1L(R

    R

    U

    U

    U

    U

    ωω −+

    ==

    2

    1

    U

    U

    21 f,fm

    R =

    Bandwith [f1 , f2]

    1 LQ

    R C=

    The tunning circuit of a

    radio is a Bandpass filter

  • RLC series circuit as a Highpass filter

    Input

    Outp

    ut

    1

    1

    2

    L

    m f

    U

    U=Bandpass [f1 , ∞]

    1 LQ

    R C=

    2 21( )

    m m

    ouput L m

    U L UU U L I L

    ZR L

    C

    ωω ω

    ωω

    = = = =

    + −

    2 21( )

    ouput L

    input m

    U U L

    U UR L

    C

    ω

    ωω

    = =

    + −

  • RLC series circuit as a Lowpass filter

    Input

    Outp

    ut

    1

    1

    2

    C

    m f

    U

    U=Bandpass [∞, f1]

    1 LQ

    R C=

    2 2

    1 1

    1( )

    m m

    ouput C m

    U UU U I

    C C ZC R L

    C

    ω ωω ω

    ω

    = = = =

    + −

    2 2

    1

    1( )

    ouput C

    input m

    U U

    U UC R L

    Cω ω

    ω

    = =

    + −