Investigating Lossless Transmission Line Phase Constant and Infinite Bandwidth

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One of the frustrating things you can encounter when starting to learn about RF design is transmission line effectWires, which were just interconnects in the good old days of working with low frequency boards, are now complex components described by some dreaded equations! . However, as we learn more about transmission lines, we may eventually come to see them as blessings in disguise and appreciate the complexity of them all. of bandwidth can be provided. This is in stark contrast to our intuition from low frequency circuits.

In this article, we will derive the wave equation for a lossless transmission line and look at its salient feature of infinite bandwidth. But before that, let’s explain what we mean by the “distributed” nature of transmission lines.

Centralized and decentralized regimes

If the circuit dimensions are comparable to the shortest wavelength in the circuit, the wire should be treated as a transmission line. This highlights the demarcation between two modes of operation for circuit components and interconnections: “centralized” and distributed. In the lumped region we deal with lower frequencies and the electrical signal is assumed to travel at infinite speed on the wire. The term bulk refers to the fact that they can be identified separately. capacitor When inductor in a specific area of ​​the circuit.As an example, consider passive bandpass filter Shown in Figure 1.

Figure 1. An example diagram of a passive bandpass filter.

In “Region A” of the diagram above, magnetic energy storage is dominant, so this part of the circuit operates inductively. On the other hand, since electric energy is stored in the electric field in “area B”, this part is modeled as a capacitor. In this example, we can use some lumped capacitors, inductors, etc. to model the behavior of the circuit. The circuit theory and analysis taught in the first semester of university is actually lumped element circuit analysis. In a lumped circuit, Kirchhoff’s Voltage Law (KVL) When Kirchhoff’s Current Law (KCL) can be easily applied to find voltage and current in the circuit.

In contrast, the distributed regime cannot identify separate capacitors and inductors. For example, a distributed uniform lossless wire is modeled as an infinite ladder network of LC sections (Fig. 2).

  Infinite ladder network of LC sections.

Figure 2. Infinite ladder network of LC sections.

This model suggests that each very short length of wire stores energy in both forms. magnetic and electric fieldsAnd these two forms of energy storage are distributed throughout the wire. In this case, the capacitive and inductive parts of the circuit cannot be separated. they are mixed.

Furthermore, in the dispersive domain, the electrical signal travels as waves along the wire. That is, voltage and current are functions of time and position along the wire. As a result, we can say that KVL and KCL do not hold at high frequencies.

In the next section, we will try to derive the phase constant formula in a relatively friendly way. If you are not interested in learning about derivation, you can skip this next section and continue from the Equation Summary section.

Derivation of phase constant formula

A lossless transmission line can be characterized by two important parameters: Impedance Z0 and the phase constant β. of characteristic impedance Specifies the ratio of voltage waves to current waves for infinitely long lines. The phase constant characterizes how the wave varies with position. Z for lossless lines0 is given by Equation 1.

\[Z_0 = \sqrt{\frac{L}{C}}\]

Formula 1.

To derive the equation for β, we need to find the steady-state voltage and current signals that appear in the ladder network model of Figure 3.

Ladder network model.

Figure 3. Ladder network model.

Using the voltage and current parameters given in the first LC section, Kirchhoff’s voltage law produces

\[v(x+ \Delta x, t)-v(x, t)= – L \Delta x \times \frac{d i(x, t)}{dt}\]

Dividing both sides by Δx gives:

\[\frac{v(x+ \Delta x, t)-v(x, t)}{\Delta x}= – L \times \frac{d i(x, t)}{dt}\]

Considering the limit of this equation as Δx approaches 0, the left-hand expression is actually the derivative of v(x, t) with respect to x. Therefore, the above equation can be rewritten as Equation 2.

\[\frac{dv(x, t)}{d x}=- L \times \frac{d i(x, t)}{dt}\]

Formula 2.

For an infinite line, the ratio of voltage to current at any point along the line is equal to Z.0. From Equation 1, we get:

\[i(x, t)=\frac{v(x,t)}{Z_0}=v(x,t)\sqrt{\frac{C}{L}}\]

Substituting i(x, t) in Equation 2 gives Equation 3.

\[\frac{dv(x, t)}{d x}=- \sqrt{LC} \times \frac{d v(x, t)}{dt}\]

Formula 3.

where the sides are with respect to line voltage, while the left side is the derivative of v(x, t) with respect to position and the right side contains the derivative of the function with respect to time. Since we want a steady-state response to a sinusoidal excitation (such as v),s

For this analysis, we can assume that the input is a complex exponential voltage Ae.jωt not vsFind the desired voltage or current signal from

When applying Aejωt The term e to the circuitjωt Displayed in all voltage and current quantities. For example, v(x, t) can be viewed as V(x)e.jωtwhere V(x) is a complex value called the phasor of v(x, t). Since basic circuit theory deals with lumped circuits, phasors are clearly position-independent. However, in transmission line analysis, phasors are expected to be a function of position. replace v(x, t) with V(x)ejωtEquation 3 produces

\[\frac{d}{d x}\Big (V(x)e^{j \omega t} \Big )=- \sqrt{LC} \times \frac{d}{dt}\Big (V(x)e^{j \omega t} \Big )\]

V(x) is not a function of time, but ejωt is not a function of x. So, with a little algebra, the above equation simplifies to

\[\frac{dV(x)}{d x} =- j \omega \sqrt{LC} \times V(x)\]

The solution of this first-order differential equation should be familiar to EE.

\[V(x) =V_0 e^{- j \omega \sqrt{LC}x}\]

where V.0 A constant that can be found from the boundary conditions of the line’s input and output ports.From phasor analysis, the real part of V(x)e isjωt Here is the output you get when you apply v.s

\[v(x,t)=Real \text{ } Part \text{ }of \Big (V(x)e^{j \omega t} \Big ) = V_0 cos(\omega t-\omega \sqrt{LC}x)\]

Defining the phase constant \(\beta = \omega \sqrt{LC}\) gives

\[v(x,t)= V_0 cos(\omega t-\beta x)\]

Formula 4.

This is the same wavefunction we used in the previous article to describe how voltage waves travel along transmission lines.Divide equation 4 by Z0 The forward current wave is given as Equation 5.

\[i(x,t)= \frac{V_0}{Z_0} cos(\omega t-\beta x)\]

Equation 5.

Summary of Lossless Transition Line Equations

In the previous section, we derived the forward voltage and current wave equations. In general, both forward and reflected waves can exist on the line at the same time. For a lossless line, the overall voltage and current waves are of the form:

\[v(x,t)= A cos(\omega t-\beta x) + B cos(\omega t+\beta x)\]

\[i(x,t)=\frac{A}{Z_0} cos(\omega t-\beta x)- \frac{B}{Z_0} cos(\omega t+\beta x)\]

where the characteristic impedance Z0 and the phase constant β is

\[Z_0 = \sqrt{\frac{L}{C}}\]

\[\beta = \omega \sqrt{LC}\]

Dispersion Effects of Power Lines: Desirable or Nuisance?

Due to the wave behavior of high frequency electrical signals, RF designers are obsessed with matching the load impedance to the characteristic impedance of the line. If the impedances are not matched, maximum power cannot be transferred to the load, and large voltage peaks caused by standing waves can damage circuit components and interconnections. However, the dispersion behavior of transmission lines leads to very interesting properties.

Based on the above analysis, exciting the line with a sine wave v givess

\[v(x,t)= V_0 cos(\omega_1 t-\beta x)\]

Substituting β gives:

\[v(x,t)= V_0 cos(\omega_1 t- \omega_1 \sqrt{LC} x)=V_0 cos\Big(\omega_1 ( t- \sqrt{LC} x) \Big)\]

The signal obtained at a given position x is the same as the input, except delayed by $$ x \sqrt{LC}$$. This result is valid at any frequency. The only assumption is that the line is lossless and acts as a transmission line. If the delay is frequency dependent, different frequency components of the input experience different amounts of delay, leading to distortion at the output. For example, applying a pulse to a system with frequency-dependent delays can completely distort the output because the different frequency components arrive at the output with unequal time shifts.

As the equation shows, a lossless transmission line passes all frequency components with the same delay. In other words, the bandwidth of the line is infinite. Increasing L and C increases the delay, but produces a constant delay for all frequency components. We find this feature even more surprising when compared to intuition from lumped low-frequency circuits, where increasing the capacitor value usually reduces the system bandwidth.

In addition to uniform delay, frequency-independent attenuation is also required to achieve systems with infinite bandwidth. In the discussion above, we assumed that the line was lossless and therefore the attenuation was zero at all frequencies.

What about lossy transmission lines?

In a real transmission line, losses include conductor loss (skin effect), dielectric loss, and hysteresis effectThese losses are also frequency dependent. however, lossy transmission line, the line parameters can be adjusted to have uniform attenuation and group delay (at least in principle). You can read more about this in the bookDesign of CMOS high frequency integrated circuitsBy Thomas H. Lee.

Higher order mode propagation

In addition to lossy components, there are other factors that limit the usable bandwidth of transmission lines. As the frequency increases, the wavelength of the signal eventually becomes comparable to the cross-sectional size of the transmission line. In this case, an electromagnetic field configuration is created that is different from what we would normally expect. These modes are known as higher order modes of propagation. The propagation velocities of higher-order modes are different from the propagation velocities of dominant (or principal) modes. Therefore, we usually try to operate the transmission line below the first higher order mode. For example, depending on the size of the conductor and the type of dielectric used, coaxial lines may be specified to operate up to about 18 GHz to avoid higher order mode propagation.

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