Derivation of equations of Transmission Line for all conditions PDF

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SAI VIDYA INSTITUTE OF TECHNOLOGY Microwaves and Antennas Transmission Line Analysis for all Conditions

Prepared by Wing Commander (Dr.) K Srinivas Rao (R) Professor Department of Electronics and Communication Engineering

Sai Vidya Institute of Technology Bengaluru

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SAI VIDYA INSTITUTE OF TECHNOLOGY Transmission Line Analysis for all Conditions Introduction A transmission line is a pair of electrical conductors carrying an electrical signal from one place to another. Coaxial cable and twisted pair cable are examples. The two conductors have inductance per unit length, which we can calculate from their size and shape. They have capacitance per unit length, which we can calculate from the dielectric constant of the insulation. In the early days of cable-making, there would be current leaking through the insulation, but in modern cables, such leakage is negligible. The electrical resistance of the conductors, however, is significant because it increases with frequency. The magnetic fields generated by high-frequency currents drive those currents to the outer edge of the conductor that carries them, so the higher the frequency, the thinner the layer of metal available to carry the current, and the higher the effective resistance of the cable. In this discussion, we derive and demonstrate the equations that govern the propagation of waves down a transmission line, and show how the frequency-dependent resistance of these cables gives rise to attenuation and distortion of high-frequency signals in all various conditions.

Wave Equations A perfect transmission line will carry an electrical signal from one place to another in a fixed time, regardless of the rate at which the voltage changes. If we apply a signal V(t) to one end of the transmission line, where t is time, the signal at the other end will be V(t − τ), where τ is a constant. We can model a real transmission line with a distributed inductance, capacitance, and resistance. We would like to calculate τ, and so determine the circumstances under which τ will be constant. The following drawing shows a small element of a transmission line.

Figure: Transmission Line Element. Here we have C, L, and R as the capacitance, inductance, and resistance per unit length of line. Voltage with distance and time is V(x,t), and current is I(x,t).

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SAI VIDYA INSTITUTE OF TECHNOLOGY Our element is a short length, δx, of cable. Distance along the cable is x. Although the capacitance, inductance, and resistance of a transmission line are distributed and mingled with one another, we lump them into three separate components in our infinitesimal element. As δx→0, our lumped model becomes a distributed model. Consider the voltage across the inductor and resistor. At position x and time t a current I(x, t) passes through both of them in series.

Equation 1: Voltage Across Inductive Element.

The rate of change of voltage with x at a particular time is a function of the rate of change of current with time and the current itself.

Equation 2: Current Into Capacitive Element.

The rate of change of current with x at a particular time is proportional to the rate of change of voltage with time.

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SAI VIDYA INSTITUTE OF TECHNOLOGY Let us differentiate (1) with respect to x and (2) with respect to t. We use the resulting equations to eliminate terms in I.

Equation 3: Transmission Line Equation.

We arrive at a partial differential equation in V. If we assume R is zero, we are left with the second derivative in x being proportional to the second derivative in t. These are the conditions under which a sinusoidal wave will propagate without distortion or attenuation. Consider a sinusoid of frequency f = ω/2π, as shown below.

Equation 4: Propagating Sine Wave.

The sinusoidal wave has the unique property that its derivatives have the same shape as the original. There is some scaling of the amplitude of the waveform as we differentiate, and it is this scaling that constrains the solution to our transmission line equation. If we set t = √(LC)x, we see the movement of the positive zero-crossing of the sinusoid (the value of sine when its angle is zero). We have dx/dt = 1/√(LC). The velocity of the sine wave is 1/√(LC). Provided that L and C remain constant with ω, the velocity of all sine waves will be the same. Let V(0, t) denote the voltage at position zero and time t. 4

SAI VIDYA INSTITUTE OF TECHNOLOGY If we represent our input V(0, t) as a sum of sinusoids using a Fourier transform, all these sinusoids will propagate along the transmission line at the same speed, so that their sum will remain undistorted as it propagates, and we will have our ideal transmission line: V(x, t) = V(0, t − τ) with τ = x √(LC).

Characteristic Impedance Consider the relationship between voltage and current at the input of our transmission line.

Equation 5: Characteristic Impedance of a Transmission Line.

When we let R = 0, we see that V(x)/I(x) is not a function of t, nor even of x. At any x, V/I = √(L/C). So far as the source of V(0, t) is concerned, the transmission line behaves in exactly the same way as a resistor of value √(L/C). We call this resistance the characteristic impedance of the transmission line. The characteristic impedance of free space, for waves propagating through a vacuum, is 377 Ω. The characteristic impedance of water is 42 Ω. Waves propagate through a vacuum at 299 m/μs, but they propagate through water at only 34 m/μs.

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SAI VIDYA INSTITUTE OF TECHNOLOGY Common Lines The following table gives the values of L and C for various mediums, including free space. The capacitance of free space is the same as the permittivity of free space. The inductance of free space is the same as the permeability of free space.

C (pF/m) L (nH/m) v (m/μs) Z (Ω)

R for f≤ 1kHz (mΩ/m)

RG58/U Coaxial Cable

93.5

273

198

54

53

RG58C/U Coaxial Cable

101

252

198

50

50

RG59B/U Coaxial Cable

72.0

405

185

75

45

CAT-5 Twisted Pair (Solid) 49.2

495

203

100 180

Vacuum

8.85

1260

299

377 0

Water

708

1260

34

42

Medium

0

Table: Wave Velocity and Characteristic Impedance of Various Mediums. Each medium acts as a transmission line. The velocity is v and the characteristic impedance is Z.

The high-frequency resistance of wires is proportional to √ω, due to the skin effect, which we will discuss later. For now, the table gives R for f = ω/2π ≤ 1 kHz. We obtained R for the cables by adding the conductor and shield DC resistance per meter. In a vacuum, there is no heat-generating resistance to the movement of charges so R = 0. We are not confident that we can defend our assignation of R = 0 in water, but that's our best guess. Also in the table is the characteristic impedance of the medium.

General Solution Consider the case when R > 0. We have the second-order differential equation in V given above. Instead of a simple sinusoidal solution, we propose a solution in which the sinusoid amplitude decreases exponentially with distance as it propagates along the cable at a fixed velocity. We have good reason to suggest a solution of this form. No circuit made of capacitors, inductors, and resistors can distort the shape of a sinusoid, so we are confident that a sinusoid propagating down a transmission line will remain a sinusoid. Furthermore, if we consider the case when R is small, the cable beyond an element of transmission line behaves like a pure resistance, Z.

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SAI VIDYA INSTITUTE OF TECHNOLOGY The element's resistance will divide the sinusoidal amplitude by 1/(1+Rδx/Z), which implies exponential attenuation with distance.

Equation 6: General Solution for R > 0. We use j for √−1 and use the complex exponential identity e jθ = cosθ + j sinθ. In our derivations of the general solution, we work with a complex expression for V(t) because doing so greatly abbreviates and clarifies our equations. But our actual applied voltage is the imaginary part of V(t) and the actual currents and voltages farther down the transmission line are the imaginary parts of our complex-valued solutions.

We use γ to denote the wavenumber of the signal propagating down the transmission line. We have γ = 2π/λ, where λ is the wavelength. (The letter k is also popular for denoting the wave number.) The parameter b is the attenuation constant, which defines how the amplitude of our sine wave decreases with distance along the line. In the case where R = 0, we would have b = 0 and γ = ω√(LC). With the complex exponential representation, our derivatives are simpler, as you can see below.

Equation 7: Derivatives of the General Solution.

Now we substitute the Derivatives of the General Solution (7) into the Transmission Line Equation (3). 7

SAI VIDYA INSTITUTE OF TECHNOLOGY We compare the real and imaginary parts of the resulting identity and so obtain two equations in b and γ, which we solve for γ to obtain Equation 8.

Equation 8: Wavenumber of Propagating Sinusoid for R > 0.

The wave velocity, v = ω/γ, is the speed with which a peak in the wave propagates along the transmission line. The wavelength, λ = 2π/γ, is the distance between peaks in the wave at a particular point in time. The following table gives γ, v, and λ under various conditions.

Condition

γ

v

Λ

ω√(LC)

1 / √(LC)

2π / √(LC)

ωL >> R > 0 ω√(LC)

1 / √(LC)

2π / √(LC)

ω→∞

1 / √(LC)

2π / √(LC)

R=0

ω√(LC)

ωL = R > 0 1.1 ω√(LC) 0.91 / √(LC) ωL...