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MIT 6 013 - Fast Electronics and Transient Behavior on TEM Lines

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Chapter 8: Fast Electronics and Transient Behavior on TEM Lines 8.1 Propagation and reflection of transient signals on TEM transmission lines 8.1.1 Lossless transmission lines The speed of computation and signal processing is limited by the time required for charges to move within and between devices, and by the time required for signals to propagate between elements. If the devices partially reflect incoming signals there can be additional delays while the resulting reverberations fade. Finally, signals may distort as they propagate, smearing pulse shapes and arrival times. These three sources of delay, i.e., propagation plus reverberation, device response times, and signal distortion are discussed in Sections 8.1, 8.2, and 8.3, respectively. These same issues apply to any system combining transmission lines and circuits, such as integrated analog or digital circuits, printed circuit boards, interconnections between circuits or antennas, and electrical power lines. Transmission lines are usually paired parallel conductors that convey signals between devices. They are fundamental to every electronic system, from integrated circuits to large systems. Section 7.1.2 derived from Maxwell’s equations the behavior of transverse electromagnetic (TEM) waves propagating between parallel plate conductors, and Section 7.1.3 showed that the same equations also govern any structure, even a dissipative one, for which the cross-section is constant along its length and that has at least two perfectly conducting elements between which the exciting voltage is applied. Using differential RLC circuit elements, this section below derives the same transmission-line behavior in a form that can readily be extended to transmission lines with resistive wires, as discussed later in Section 8.3.1. Since resistive wires introduce longitudinal electric fields, such lines are no longer pure TEM lines. Equations (7.1.10) and (7.1.11) characterized the voltage v(t,z) and current i(t,z) on TEM structures with inductance L [H m-1] and capacitance C [F m-1] as: dv dz =−L di dt (8.1.1)di dz =−Cdv dt (8.1.2)These expressions were combined to yield the wave equation (7.1.14) for lossless TEM lines: 2 2 2(ddz − LCd dt 2 )v(z,t )= 0 (TEM wave equation) (8.1.3) One general solution to this wave equation is (7.1.16): v(z, t )= v(+z − ct)+ v(−z + ct) (TEM voltage) (8.1.4) - 229 -The inductance L [Henries m-1] of the two conductors arises from the magnetic energy stored per meter of length, and produces a voltage drop dv across each incremental length dz of wire which is proportional to the time derivative of current through it40: dv =−L dz (di dt ) (8.1.7)Any current increase di across the distance dz, defined as di = i(t, z+dz) - i(t,z), would be supplied from charge stored in C [F m-1]: 40 An alternate equivalent circuit would have a second inductor in the lower branch equivalent to that in the upper branch; both would have value Ldz/2, and v(t,z) and i(t,z) would remain the same. which corresponds to the superposition of forward and backward propagating waves moving at velocity c = (LC)-0.5 = (με)-0.5 . The current i(t,z) corresponding to (8.1.4) follows from substitution of (8.1.4) into (8.1.1) or (8.1.2), and differentiation followed by integration: iz( ,t )= Y ⎡v(z − ct )− v(z + ct )o ⎣+ −⎦⎤ (TEM current) (8.1.5)Yo is the characteristic admittance of the line, and the reciprocal of the characteristic impedance Zo: Zo = Y−1 o =(LC )0.5 [Ohms ] (characteristic impedance of lossless TEM line) (8.1.6) The value of Yo follows directly from the steps above. A more intuitive way to derive these equations utilizes an equivalent distributed circuit for the transmission line composed of an infinite number of differential elements with series inductance and parallel capacitance, as illustrated in Figure 8.1.1(a). This model is easily extended to non-TEM lines with resistive wires. (a) (b) dz v(t,z) -+ Cdz Cdz Cdz vs(t) Zo v(t,z)+ Zo, c i(t,z) -+ Zo(c) 0 v(z, to) = vs(to - z/c)/2 vs(t) -+ Zov(t, z=0)-+ Ldz i(t,z) Ldz Ldz -z Figure 8.1.1 Distributed circuit model for lossless TEM transmission lines. - 230 -di =−C dz (dv dt ) (8.1.8)These two equations for dv and di are equivalent to (8.1.1) and (8.1.2), respectively, and lead to the same wave equation and general solutions derived in Section 7.1.2 and summarized above, where arbitrary waveforms propagate down TEM lines in both directions and superimpose to produce the total v(z,t) and i(t,z). Two equivalent solutions exist for this wave equation: (8.1.4) and (8.1.9): v (z,t )= f+(t − zc)+ f−(t + zc) (8.1.9) The validity of (8.1.9) is easily shown by substitution into the wave equation (8.1.3), where again c = (LC)-0.5 . This alternate form is useful when relating line signals to sources or loads for which z is constant, as illustrated below. The first form (8.1.4) in terms of (z - ct) is more convenient when t is constant and z varies. Waves can be launched on TEM lines as suggested in Figure 8.1.1(b). The line is driven by the Thevenin equivalent source vs(t) in series with the source resistance Zo, which is matched to the transmission line in this case. Equations (8.1.4) and (8.1.5) say that if there is no negative traveling wave, then the ratio of the voltage to current for the forward wave on the line must equal Zo = Y-1o. The equivalent circuit for this TEM line is therefore simply a resistor of value Zo, as suggested in Figure 8.1.1(c). If the source resistance is also Zo, then only half the source voltage vs(t) appears across the TEM line terminals at z = 0. Therefore the voltages at the left terminals (z = 0) and on the line v(t,z) are: vt,z ()s(= 0)= v t 2 = v+(t,z = 0) (8.1.10) vt(),z = v+(t − zc )= vs(t − zc )2 (transmitted signal) (8.1.11) where we have used the solution form of (8.1.9). The propagating wave in Figure 8.1.1(b) has half the amplitude of the Thevenin source vs(t) because the source was matched to the line so as to maximize the power transmitted from the given voltage vs(t). Note that (8.1.11) is the same as (8.1.10) except that z/c was subtracted from each. Equality is preserved if all arguments in an equation are shifted the same amount. If the Thevenin source resistance were R, then the voltage-divider equation would yield the terminal and propagating voltage v(t,z): vt(),z = vs(t − zc )⎡⎣Zo (R + Zo )⎤⎦ (8.1.12) This more general expression reduces to (8.1.10)


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MIT 6 013 - Fast Electronics and Transient Behavior on TEM Lines

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