Electric Circuits II - Philadelphia ... The Resistor The angles ®¸ and...

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Transcript of Electric Circuits II - Philadelphia ... The Resistor The angles ®¸ and...

  • Electric Circuits II Phasor Diagram

    1

    Dr. Firas Obeidat

  • Dr. Firas Obeidat – Philadelphia University

    2

    Phasor diagram for the Passive Circuit Elements

    Let

    In polar form

    But Vm∟θ and Im∟ 𝝓 merely represent the general voltage and current phasors V and I. Thus

    𝒊(𝒕) = 𝑰𝒎𝒄𝒐𝒔⁡(𝝎𝒕 + 𝝓)

    The Resistor

    The angles θ and 𝝓 are equal, so that the current and voltage are always in phase.

    The Inductor

    Let 𝒊(𝒕) = 𝑰𝒎𝒄𝒐 𝒔 𝝎𝒕 + 𝝓 = 𝑰𝒎𝒆 𝒋(𝝎𝒕+𝝓)

  • Dr. Firas Obeidat – Philadelphia University

    3

    Phasor diagram for the Passive Circuit Elements

    We obtain the desired phasor relationship

    Note that the angle of the factor jωL is exactly

    +90◦ and that I must therefore lag V by 90° in

    an inductor.

    The Capacitor

    Let 𝒗(𝒕) = 𝑽𝒎𝒄𝒐𝒔(𝝎𝒕 + 𝝓) = 𝑽𝒎𝒆 𝒋(𝝎𝒕+𝝓)

    𝒊 𝒕 = 𝑪 𝒅𝒗(𝒕)

    𝒅𝒕 𝑰𝒎𝒆

    𝒋𝜽 = 𝒋𝝎𝑪𝑽𝒎𝒆 𝒋𝝓

    𝑰 = 𝑗𝜔𝐶𝑽 𝑽 = 𝟏

    𝑗𝜔𝐶 I

    Note that the angle of the factor 1/jωC is

    exactly -90◦ and that I must therefore lead V

    by 90° in an Capacitor.

  • Dr. Firas Obeidat – Philadelphia University

    4

    Phasor diagram for series RL circuit

    Example: for the circuit shown in figure (a), draw the phasor

    circuit , impedance diagram and voltages phasor diagram.

    V=100∟0, so the phasor circuit is shown in figure (b).

    ZT=ZR+ZL=3Ω+j4Ω =5∟53.13 o.

    Impedance diagram is shown in figure (c).

    𝐼 = 𝑉

    𝑍 𝑇

    = 100∟0𝑜

    5∟53.13o = 20∟−53.13o

    VR=IZR=(20∟-53.13 o A)(3∟0Ω)=60∟-53.13o V.

    VL=IZL=(20∟-53.13 o A)(4∟90Ω)=80∟36.87o V.

    Phasor diagram is shown in figure (d).

    In rectangular form

    VR=60∟-53.13 o =36-j48V.

    VL=80∟36.87 o =64+j48V.

    V=VR+VL=36-j48+64+j48=100+j0V=100∟0 V.

  • Dr. Firas Obeidat – Philadelphia University

    5

    Phasor diagram for series RC circuit

    Example: for the circuit shown in figure (a), draw the phasor

    circuit , impedance diagram and voltages phasor diagram.

    I=5∟0, so the phasor circuit is shown in figure (b).

    ZT=ZR+ZC=6Ω-j8Ω =10∟-53.13 o.

    Impedance diagram is shown in figure (c).

    𝑉 = 𝐼𝑍𝑇= (5∟53.13 o)(10∟−53.13o)=50∟0 V

    VR=IZR=(5∟53.13 o)(6∟0o)=30∟53.13o0 V

    VC=IZC=(5∟53.13 o A)(8∟-90Ω)=40∟-36.87o V.

    Phasor diagram is shown in figure (d).

    In rectangular form

    VR=30∟53.13 o=18+j24 V

    VC=40∟-36.87 o =32-j24V.

    V=VR+VC=18+j24+32-j24=50+j0=50∟0 V.

  • Dr. Firas Obeidat – Philadelphia University

    6

    Phasor diagram for series RLC circuit

    Example: for the circuit shown in figure (a), draw the

    phasor circuit , impedance diagram and voltages phasor

    diagram.

    V=50∟0, so the phasor circuit is shown in figure (b).

    ZT=ZR+ZL+ZC=3Ω+7Ω-j3Ω =3+j4= 5∟53.13 o.

    Impedance diagram is shown in figure (c).

    VR=IZR=(10∟−53.13 o)(3∟0o)=30∟−53.13o0 V

    VC=IZC=(10∟-53.13 o A)(3∟-90Ω)=30∟-143.13o V.

    Phasor diagram is shown in figure (d).

    In rectangular form

    V=VR+VL+VC=18-j24+56+j42-24-j18

    V=50+j0=50∟0 V.

    𝐼 = 𝑉

    𝑍 𝑇

    = 50∟0𝑜

    5∟53.13o = 10∟−53.13o

    VL=IZL=(10∟-53.13 o A)(7∟90Ω)=70∟36.87o V.

    VR=30∟−53.13 o0 V=18-j24 V

    VC=30∟-143.13 o V=-24-j18.

    VL=70∟36.87 o V=56+j42 V.

  • Dr. Firas Obeidat – Philadelphia University

    7

    Phasor diagram for parallel RL circuit

    Example: for the circuit shown in figure (a), draw the

    phasor circuit , impedance diagram and currents

    phasor diagram.

    V=20∟53.13, so the phasor circuit is shown in figure

    (b).

    YT=YR+YL=1/3.33+1/j2.5=0.3-j0.4 =0.5∟-53.13

    Impedance diagram is shown in figure (c).

    Currents Phasor diagram is shown in figure

    (d).

    I=IR+IL=3.6+j4.8+6.4-j4.8=10+j0=10 ∟0.

    𝑍𝑇 = 1

    𝑌 𝑇

    = 1

    0.5∟−53.13 =2 ∟53.13

    𝐼 = 𝑉

    𝑍 𝑇

    = 20∟53.13𝑜

    2∟53.13o = 10∟0o

    𝐼𝑅 = 𝑉

    𝑍 𝑅

    = 20∟53.13𝑜

    3.33∟0o = 6∟53.13o

    𝐼𝐿 = 𝑉

    𝑍 𝐿

    = 20∟53.13𝑜

    2.5∟90o = 8∟−36.87o

  • Dr. Firas Obeidat – Philadelphia University

    8

    Phasor diagram for parallel RC circuit

    Example: for the circuit shown in figure (a), draw the

    phasor circuit , impedance diagram and currents

    phasor diagram.

    I=10∟0, so the phasor circuit is shown in figure (b).

    YT=YR+YC=1/1.67+1/-j2.5=0.6+j0.8 =1∟53.13

    Impedance diagram is shown in figure (c).

    Currents Phasor diagram is shown in figure

    (d).

    𝑍𝑇 = 1

    𝑌 𝑇

    = 1

    1∟53.13=1∟-53.13

    𝑉 = 𝐼𝑍𝑇 = (10∟0 𝑜)(1∟-53.13)= 10∟−53.13o

    𝐼𝑅 = 𝑉

    𝑍 𝑅

    = 10∟−53.13𝑜

    1.67∟0o = 6∟−53.13 o

    𝐼𝐶 = 𝑉

    𝑍 𝐶

    = 10∟−53.13𝑜

    1.25∟−90o= 8∟36.87 o

  • Dr. Firas Obeidat – Philadelphia University

    9

    Phasor diagram for parallel RLC circuit

    Example: for the circuit shown in figure (a),

    draw the phasor circuit , impedance diagram

    and currents phasor diagram.

    V=100∟53.13, so the phasor circuit is shown

    in figure (b).

    YT=YR+YL+YC=1/3.33+1/j1.43+1/-j3.33

    =0.3+j0.4 =0.5∟-53.13

    Impedance diagram is shown in figure (c).

    Currents Phasor diagram is shown in figure

    (d).

    𝑍𝑇 = 1

    𝑌 𝑇

    = 1

    0.5∟−53.13=2∟53.13 o=1.2+j1.6

    𝐼𝑅 = 𝑉

    𝑍 𝑅

    = 100∟53.13𝑜

    3.33∟0o = 30∟53.13 o

    𝐼𝐶 = 𝑉

    𝑍 𝐶

    = 100∟53.13𝑜

    3.33∟−90o= 30∟143.13 o

    𝐼 = 𝑉

    𝑍 𝑇

    = 100∟53.13𝑜

    2∟53.13o = 50∟0o

    𝐼𝐿 = 𝑉

    𝑍 𝐿

    = 100∟53.13𝑜

    1.43∟90o = 70∟−36.87 o

  • Dr. Firas Obeidat – Philadelphia University

    10

    Series-parallel AC circuit

    Example: for the circuit, calculate ZT, Is, VR, VC,

    IL and IC

    𝐈𝐋 = 𝐕𝐋 𝐙𝐋

    = 𝟏𝟏𝟖. 𝟒𝟒∠ − 𝟗. 𝟒𝟔𝐨

    𝟑∠𝟗𝟎𝐨 = 𝟑𝟗. 𝟒𝟖∠ − 𝟗𝟗. 𝟒𝟔𝐨

  • Dr. Firas Obeidat – Philadelphia University

    11

    Series-parallel AC circuit

    Example: for the circuit, calculate Is and Vab

  • Dr. Firas Obeidat – Philadelphia University

    12

    Series-parallel AC circuit

  • Dr. Firas Obeidat – Philadelphia University

    13

    Series-parallel AC circuit

    Example: Determine the current I and

    the voltage V.

  • Dr. Firas Obeidat – Philadelphia University

    14

    Series-parallel AC circuit

    Example: calculate I, I1, I2, I3 and ZT.

  • Dr. Firas Obeidat – Philadelphia University

    15

    Series-parallel AC circuit

  • Dr. Firas Obeidat – Philadelphia University

    16

    Series-parallel AC circuit

    Example: calculate ZT, I, I1, andI2.

  • Dr. Firas Obeidat – Philadelphia University

    17

    Series-parallel AC circuit

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