传输线理论

[ II-5 ]
which is the dispersion relationship of a discrete, uniform, ideal transmission line (Note that the discrete ideal line is, effectively, a low-pass filter.).
TRANSMISSION LINE THEORY Thus, the set of differential circuit equations for a discrete, uniform transmission line becomes a huge set of algebraic equations -- viz. Vn +1(ω ) = Vn (ω ) − Z s (ω) In +1 (ω ) In +1 (ω) = In (ω ) − Yp (ω) Vn (ω ) where Z s(ω) = R s + j ω L s and Yp (ω ) = Gp + j ω C p are, respectively, the series impedance and the shunt (parallel) admittance of the transmission line. [ I-4a ] [ I-4b ]
We might characterize these solutions as constant phase solutions in the sense that the solution at a given node along transmission line is identical to the solution at an adjacent node except for constant phase factor. If these constant phase solutions are to be valid solutions of Eqs. [ II-1 ], the phase constant φ(ω ) must satisfy the equation Z s(ω) Yp (ω ) exp[ j n φ (ω)] = exp[ j (n + 1) φ (ω)] + exp[ j ( n − 1) φ(ω )] − 2 exp[ j n φ(ω )] [ II-3 ]
Fortunately, here is an amazingly simple set of solutions for this enormous set of algebraic equations. These solutions may be written in the form Vn (ω ) = {a complex constant} exp[ j n φ(ω)] In (ω) = {another complex constant} exp[ j n φ(ω)] [ II-2a ] [ II-2b ]
TRANSMISSION LINE THEORY φ φ′ + j φ ′′ φ′ j φ′′ φ′ j φ ′′ sin = sin = sin cos + cos sin 2 2 2 2 2 2 =±
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[ II-10b ]
For ease of interpretation, we make the small argument approximation so that φ′ ≈ ± φ ′′ = m ω R sCp 2 ω R sC p 2 ← Phase shift per section ← Attenuation per section [ II-11a ]
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II.
Exact Solutions of Transmission Line Equations:
Our task is to solve Eqs. [ I-4 ]. To that end, we first cast this array of coupled inhomogeneous equations in the form of a set of coupled, homogeneous algebraic equations -- viz. Z s(ω) Yp (ω ) Vn (ω) = Vn +1 (ω ) + Vn −1 (ω) − 2 Vn (ω) Z s(ω) Yp (ω ) In (ω) = In +1 (ω ) + In −1 (ω ) − 2 In (ω) [ II-1a ] [ II-1b ]
[ II-11b ]
R. V. Jones, October 23, 2002
TRANSMISSION LINE THEORY
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III.
Continuous Transmission Lines An Approximate Solution of Transmission Line Equations:
+∞
vn (t) =
−∞
∫ V (ω) exp[ j ω t] dω
n
+∞
and in ( t) =
−∞
∫ I (ω) exp[ j ω t] dω
n
[ I-2]
or in the language of circuit analysis vn (t) = ℜ{Vn (ω) exp[ j ω t]} = Vn (ω) cos(ω t + ϕV ) in ( t) = ℜ {I n (ω) exp[ j ω t]} = In (ω) cos(ω t + ϕ I ) [ I-3a ] [ I-3b ]
2 2
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[ II-4 ]
Thus, we have obtained an extremely important result which we will, hereafter, refer to as the dispersion relationship for a discrete, uniform transmission line -- viz. 2 j sin[φ(ω ) 2] = ± Z s(ω ) Yp (ω) Important Special Cases: 1. The "Ideal" or "Lossless" LC-Transmission Line: If we take Z s(ω) = j ω L s and Yp (ω) = j ω C p , then Eq [ II-5 ] becomes sin(φ(ω ) 2) = ω L s Cp 2 [ II-6 ]
−1
j ω R sCp
[ II-7 ]
We have a problem! What, in heavens name, do we mean by the square root of “ j ” (i.e. the fourth root of −1 )? To interpret what is meant by 1 + j j = so that 2 1 + j j = ± 2
R. V. Jones, October 23, 2002
TRANSMISSION LINE THEORY Canceling the common exp[ j n φ (ω)] factor on both sides of the equation, we obtain Z s(ω) Yp (ω ) = exp[ j φ(ω )] + exp[− j φ(ω)] − 2 = {exp[ j φ(ω ) 2] − exp[− j φ (ω) 2]} = {2 j sin[φ (ω) 2]}
The crucial matter is that the voltage and current vary both in time and space! To obtain a solution, we first deal with the time dependence by making use of the "phasor" concept -i.e. we replace the time dependent variables with their Fourier Transforms
In most instances, we are interested in continuous rather than discrete transmission lines. To obtain a representation of the voltage across and current along a continuous line, we develop a "continuous approximation" of the basic circuit equations by making use of a Taylor expansion for small node spatial separation. Before looking at the most general case, it is useful to first look at the lossless or ideal case. From Eqs. [ I-1 ] we may write the basic circuit equations vn +1 (t) = vn ( t) − L s in +1 ( t) = in (t ) − Cp d i (t) dt n +1 d v (t ) dt n [ III-1a ] [ III-1b ]
TRANSMISSION LINE THEORY
I. The Transmission Line Model:
Consider the following repeating (uniform) sequence of "lumped" circuit elements:
Applying elementary circuit analysis to each node of such a "discrete" transmission line we may write a set of basic circuit equations. vn +1 (t) = vn ( t) − R s in +1 (t ) − L s in +1 ( t) = in (t ) − Gp vn (t ) − Cp d i ( t) dt n +1 d v ( t) dt n [ I-1a ] [ I-1b ]
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{1 − j }
2
ω R sC p 2
[ II-9 ]
which yields, upon equating real and imaginary parts, 1 φ′ φ ′′ sin cosh = ± 2 2 2 1 φ′ φ′′ cos sinh = m 2 2 2 ω R sCp 2 ω R sCp . 2 [ II-10a ]
2
j , note that
[ II-8 ]
Therefore, the dispersive relationship for a "lossy" RC-transmission line -- i.e. Eq. [ II-7 ] becomes
R. V. Jones, October 23, 2002
R. V. Jones, October 23, 2002
TRANSMISSION LINE THEORY
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2. The "Lossy" RC-Transmission Line: If we take Z s(ω) = R s and Yp (ω) = j ω Cp , then Eq [ II-5 ] becomes sin(φ(ω ) 2) = [2 j ]