Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 18 views

    The Characteristic Impedance of A Transmission Line

    Uploaded by

    alfredomatiasrojo
    The document discusses the characteristic impedance of a transmission line. It shows that the ratio of the voltage and current waves propagating on the line is constant and depends only on the line's parameters. This ratio is defined as the characteristic impedance of the transmission line. The characteristic impedance describes how voltages and currents are related on the line and allows transmission lines to be characterized using just two complex parameters: the characteristic impedance and propagation constant.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    The Characteristic Impedance of A Transmission Line

    Uploaded by

    alfredomatiasrojo
    18 views4 pages
    The document discusses the characteristic impedance of a transmission line. It shows that the ratio of the voltage and current waves propagating on the line is constant and depends only on the line's parameters. This ratio is defined as the characteristic impedance of the transmission line. The characteristic impedance describes how voltages and currents are related on the line and allows transmission lines to be characterized using just two complex parameters: the characteristic impedance and propagation constant.

    Original Description:

    Original Title

    The Characteristic Impedance of a Transmission Line

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document discusses the characteristic impedance of a transmission line. It shows that the ratio of the voltage and current waves propagating on the line is constant and depends only on the line's parameters. This ratio is defined as the characteristic impedance of the transmission line. The characteristic impedance describes how voltages and currents are related on the line and allows transmission lines to be characterized using just two complex parameters: the characteristic impedance and propagation constant.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views4 pages

    The Characteristic Impedance of A Transmission Line

    Uploaded by

    alfredomatiasrojo
    The document discusses the characteristic impedance of a transmission line. It shows that the ratio of the voltage and current waves propagating on the line is constant and depends only on the line's parameters. This ratio is defined as the characteristic impedance of the transmission line. The characteristic impedance describes how voltages and currents are related on the line and allows transmission lines to be characterized using just two complex parameters: the characteristic impedance and propagation constant.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    1/20/2005 The Characteristic Impedance of a Transmission Line.

    doc 1/4

    The Characteristic
    Impedance of a
    Transmission Line
    So, from the telegrapher’s differential equations, we know that
    the complex current I(z) and voltage V (z) must have the form:

    V ( z ) = V0+ e −γ z + V0− e +γ z

    I ( z ) = I 0+ e −γ z + I 0− e +γ z

    Let’s insert the expression for V (z) into the first telegrapher’s
    equation, and see what happens !

    dV ( z )
    = − γV0+ e −γ z + γV0− e +γ z = −( R + j ω L) I ( z )
    dz

    Therefore, rearranging, I (z) must be:

    γ
    I( z ) = (V0+ e −γ z − V0− e +γ z )
    R + j ωL

    Jim Stiles The Univ. of Kansas Dept. of EECS


    1/20/2005 The Characteristic Impedance of a Transmission Line.doc 2/4

    Q: But wait ! I thought we already knew


    current I(z). Isn’t it:

    I ( z ) = I 0+ e −γ z + I 0− e + γ z ??

    How can both expressions for I(z) be true??

    A: Easy ! Both expressions for current are equal to each other.

    I ( z ) = I 0+ e −γ z + I 0− e + γ z =
    γ (V0+ e −γ z − V0− e + γ z )
    R + j ωL

    For the above equation to be true for all z, I 0 and V0 must be


    related as:

    ⎛ γ ⎞ + −γ z ⎛ −γ ⎞ − +γ z
    I 0+e −γ z = ⎜ ⎟V0 e and I 0−e + γ z = ⎜ ⎟V0 e
    ⎝ R + j ω L ⎠ ⎝ R + j ω L ⎠

    Or—recalling that V0+e −γ z = V + ( z ) (etc.)—we can express this in


    terms of the two propagating waves:

    ⎛ γ ⎞ + ⎛ −γ ⎞ −
    I + (z ) = ⎜ ⎟V ( z ) and I − (z ) = ⎜ ⎟V (z )
    ⎝ R + j ω L ⎠ ⎝ R + j ω L ⎠

    Now, we note that since:

    γ = (R + j ω L )(G + j ωC )

    Jim Stiles The Univ. of Kansas Dept. of EECS


    1/20/2005 The Characteristic Impedance of a Transmission Line.doc 3/4

    We find that:

    γ (R + j ω L )(G + j ωC ) G + j ωC
    = =
    R + j ωL R + j ωL R + j ωL

    Thus, we come to the startling conclusion that:

    V +
    (z ) R + j ωL −V
    −
    (z ) R + j ωL
    = and =
    I+ (z ) G + j ωC I− (z ) G + j ωC

    Q: What’s so startling about this conclusion?

    A: Note that although the magnitude and phase of each


    propagating wave is a function of transmission line position z
    (e.g., V + ( z ) and I + ( z ) ), the ratio of the voltage and current of
    each wave is independent of position—a constant with respect
    to position z !

    Although V0± and I 0± are determined by boundary conditions


    (i.e., what’s connected to either end of the transmission line),
    the ratio V0± I 0± is determined by the parameters of the
    transmission line only (R, L, G, C).

    Æ This ratio is an important characteristic of a transmission


    line, called its Characteristic Impedance Z0.

    Jim Stiles The Univ. of Kansas Dept. of EECS


    1/20/2005 The Characteristic Impedance of a Transmission Line.doc 4/4

    V0+ −V0− R + j ωL
    Z0  + = − =
    I0 I0 G + j ωC

    We can therefore describe the current and voltage along a


    transmission line as:

    V ( z ) = V0+ e −γ z + V0− e + γ z

    V0+ −γ z V0− + γ z
    I( z ) = e − e
    Z0 Z0

    or equivalently:

    V ( z ) = Z 0 I 0+ e −γ z − Z 0 I 0− e + γ z

    I ( z ) = I 0+ e −γ z + I 0− e +γ z

    Note that instead of characterizing a transmission line with real


    parameters R, G, L, and C, we can (and typically do!) describe a
    transmission line using complex parameters Z0 and γ .

    Jim Stiles The Univ. of Kansas Dept. of EECS

    You might also like