Question

The characteristic impedance of a certain lossless transmission line is \(72 \Omega\). If \(L=0.5 \mu \mathrm{H} / \mathrm{m}\), find \((a) C ;(b) v_{p} ;(c) \beta\) if \(f=80 \mathrm{MHz} .(d)\) The line is terminated with a load of \(60 \Omega\). Find \(\Gamma\) and \(s\).

Short answer

Answer: The capacitance per unit length (C) is 9.645 pF/m, the phase velocity (vp) is 2.293 x 10^8 m/s, the phase constant (β) is 3.026 rad/m, the reflection coefficient (Γ) is -0.0909, and the VSWR (s) is 1.2.

Step-by-step solution

Calculate capacitance C

Using the given characteristic impedance (Z0) and inductance per unit length (L), we can find the capacitance per unit length (C) using the formula Z0 = sqrt(L/C). Rearranging the formula to solve for C, we get: C = L / Z0^2 Plugging in the given values, L = 0.5 μH/m and Z0 = 72 Ω, we get: C = (0.5 * 10**(-6) H/m) / (72 Ω)^2 = 9.645E-12 F/m So, the capacitance per unit length (C) is approximately 9.645 pF/m.

Calculate phase velocity vp

Now, we can find the phase velocity (vp) using the formula vp = 1/√(LC). Plugging in the values of L and C from the previous steps, we get: vp = 1/√((0.5 * 10**(-6) H/m) * (9.645E-12 F/m)) = 2.293E8 m/s So, the phase velocity (vp) is approximately 2.293 x 10^8 m/s.

Calculate the phase constant β

We can find the phase constant (β) using the formula β = 2πf√(LC), where f is the given frequency (80 MHz). Plugging in the values, we get: β = 2π * (80 * 10**6 Hz) * sqrt((0.5 * 10**(-6) H/m) * (9.645E-12 F/m)) ≈ 3.026 rad/m So the phase constant (β) is approximately 3.026 rad/m.

Calculate the reflection coefficient Γ

We can find the reflection coefficient (Γ) using the formula Γ = (ZL - Z0) / (ZL + Z0), where ZL is the load impedance (60 Ω). Plugging in the values, we get: Γ = (60 Ω - 72 Ω) / (60 Ω + 72 Ω) = -0.0909 So, the reflection coefficient (Γ) is approximately -0.0909.

Calculate the VSWR s

We can find the VSWR (s) using the formula s = (1 + |Γ|) / (1 - |Γ|), where |Γ| is the absolute value of the reflection coefficient. Plugging in the value of |Γ|, we get: s = (1 + 0.0909) / (1 - 0.0909) ≈ 1.2 So, the VSWR (s) is approximately 1.2. To summarize, the capacitance per unit length (C) is 9.645 pF/m, the phase velocity (vp) is 2.293 x 10^8 m/s, the phase constant (β) is 3.026 rad/m, the reflection coefficient (Γ) is -0.0909, and the VSWR (s) is 1.2.

Key concepts

Capacitance per Unit Length

In the context of transmission lines, the capacitance per unit length is a measure of how much electric charge the line can store per unit length when a voltage is applied across its conductors. This parameter is pivotal in determining the speed at which signals travel along the line and how they interact with different impedances. Calculation of this parameter involves the characteristic impedance and inductance per unit length. In our example, using the formula for characteristic impedance and the given inductance, we find a capacitance of approximately 9.645 picofarads per meter (pF/m). This low value suggests that the line has the ability to store a small amount of charge for every meter which is typical for many transmission lines.

Phase Velocity

The phase velocity of a signal on a transmission line represents the speed at which the phase of any single frequency component of the signal propagates. It is intimately connected to the physical properties of the line such as its capacitance and inductance per unit length. The phase velocity can be determined using the relationship v_p = 1/√(LC), signifying that it is inversely proportional to the square root of the product of capacitance and inductance per unit length. In this example, the calculated phase velocity is approximately 2.293 x 10^8 meters per second (m/s), which is slightly less than the speed of light. This speed will affect how quickly information can travel down the line and is an essential parameter for designing high-speed communication systems.

Phase Constant

The phase constant, denoted by β (beta), describes the rate of phase change along the transmission line per unit length, typically in radians per meter (rad/m). It is a critical factor in the phase shift experienced by signals as they travel and is found by accounting for frequency and the line's inductance and capacitance per unit length. The phase constant is computed using the formula β = 2πf√(LC), where f is the frequency of the signal. For an 80 MHz signal, our calculated β is approximately 3.026 rad/m, which means that the phase of the signal will change by about 3.026 radians for every meter the signal travels along the line. This shift must be considered during the design and analysis of transmission line circuits, especially in radio frequency (RF) applications.

Reflection Coefficient

The reflection coefficient, often symbolized by Γ (Gamma), quantifies the reflection of a signal at a discontinuity, such as when a transmission line is terminated with an impedance different from its characteristic impedance. It's calculated by the formula Γ = (Z_L - Z_0) / (Z_L + Z_0), where Z_L is the load impedance, and Z_0 is the characteristic impedance. A negative reflection coefficient, as seen in our example with a value of approximately -0.0909, indicates that the wave is reflected with a phase inversion. Understanding the reflection coefficient is central to mitigating signal loss and ensuring that the maximum power is delivered to the load, avoiding issues like signal degradation or power loss.

Voltage Standing Wave Ratio (VSWR)

The Voltage Standing Wave Ratio (VSWR) is a measure used to describe how effectively a transmission line is matched to its load and is linked to the reflection coefficient. It is represented by the formula s = (1 + |Γ|) / (1 - |Γ|), where |Γ| is the magnitude of the reflection coefficient. This concept reflects the ratio of the maximum to minimum voltage in the standing wave pattern created by the interference of forward and reflected waves on the line. A VSWR of 1.2, as we have calculated here, is fairly close to the ideal value of 1, indicating that there's minimal mismatch and that the line is well-terminated. Low VSWR values are desirable, as they suggest efficient transmission of power to the load with minimal reflections causing losses.