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Chapter 10: Problem 13

The incident voltage wave on a certain lossless transmission line for which \(Z_{0}=50 \Omega\) and \(v_{p}=2 \times 10^{8} \mathrm{~m} / \mathrm{s}\) is \(V^{+}(z, t)=200 \cos (\omega t-\pi z)\) V. \((a)\) Find \(\omega .(b)\) Find \(I^{+}(z, t) .\) The section of line for which \(z>0\) is replaced by a load \(Z_{L}=50+j 30 \Omega\) at \(z=0 .\) Find: \((c) \Gamma_{L} ;(d) V_{s}^{-}(z) ;(e) V_{s}\) at \(z=-2.2 \mathrm{~m}\)

Short Answer

Expert verified
Question: Determine the angular frequency (ω) and incident current wave (I⁺(z, t)) for a lossless transmission line with the given incident voltage wave V⁺(z, t) = 200 cos(ωt - πz) V, characteristic impedance Z₀ = 50 Ω, phase velocity vₚ = 2 * 10^8 m/s, and load impedance Zₗ = 50 + j30 Ω at z = 0. Answer: The angular frequency (ω) is omega, and the incident current wave (I⁺(z, t)) is 4 cos(ωt - πz) A.

Step by step solution

01

Identify the given values

Here's what we know: V⁺(z, t) = 200 cos(ωt - πz) V Z₀ = 50 Ω (characteristic impedance) vₚ = 2 * 10^8 m/s (phase velocity) Zₗ = 50 + j30 Ω (load impedance)
02

Find ω (angular frequency)

The equation for V⁺(z, t) is given as: V⁺(z, t) = 200 cos(ωt - πz) Comparing this with the general equation of a wave V(z, t) = A cos(ωt ± βz), we can identify ω as the coefficient of 't': ω = coefficient of 't' = omega
03

Calculate I⁺(z, t) (incident current wave)

To find I⁺(z, t), we need to divide V⁺(z, t) by the characteristic impedance, Z₀: I⁺(z, t) = V⁺(z, t) / Z₀ Plug in the given values and simplify: I⁺(z, t) = (200 cos(ωt - πz)) / 50 I⁺(z, t) = 4 cos(ωt - πz) A
04

Calculate Γₗ (reflection coefficient at the load)

Use the formula for the reflection coefficient at the load: Γₗ = (Zₗ - Z₀) / (Zₗ + Z₀) Plug in the given values and simplify: Γₗ = ((50 + j30) - 50) / ((50 + j30) + 50) Γₗ = j30 / (100 + j30)
05

Calculate Vₛ⁻(z) (reflected voltage wave)

We can find the reflected voltage wave using the reflection coefficient and incident voltage wave: Vₛ⁻(z) = Γₗ * V⁺(z, t=z/vₚ) Plug in the values and simplify: Vₛ⁻(z) = (j30 / (100 + j30)) * 200 cos(ω(z/vₚ) - πz)
06

Calculate Vₛ at z = -2.2m

To find the voltage at z = -2.2m, we need to add the incident and reflected voltage waves: Vₛ = V⁺(z, t) + Vₛ⁻(z) Plug in the values for V⁺(z, t) and Vₛ⁻(z), and set z = -2.2m: Vₛ = [200 cos(ωt - π * -2.2)] + [(j30 / (100 + j30)) * 200 cos(ω * (-2.2/vₚ) + π * -2.2)] Now, we can compute the final results: (a) ω = omega (b) I⁺(z, t) = 4 cos(ωt - πz) A (c) Γₗ = j30 / (100 + j30) (d) Vₛ⁻(z) = (j30 / (100 + j30)) * 200 cos(ω(z/vₚ) - πz) (e) Vₛ = [200 cos(ωt - π * -2.2)] + [(j30 / (100 + j30)) * 200 cos(ω * (-2.2/vₚ) + π * -2.2)]

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Most popular questions from this chapter

A \(50-\Omega\) lossless line is of length \(1.1 \lambda\). It is terminated by an unknown load impedance. The input end of the \(50-\Omega\) line is attached to the load end of a lossless \(75-\Omega\) line. A VSWR of 4 is measured on the \(75-\Omega\) line, on which the first voltage maximum occurs at a distance of \(0.2 \lambda\) in front of the junction between the two lines. Use the Smith chart to find the unknown load impedance.

Two lossless transmission lines having different characteristic impedances are to be joined end to end. The impedances are \(Z_{01}=100 \Omega\) and \(Z_{03}=25 \Omega\). The operating frequency is \(1 \mathrm{GHz}\). \((a)\) Find the required characteristic impedance, \(Z_{02}\), of a quarter-wave section to be inserted between the two, which will impedance-match the joint, thus allowing total power transmission through the three lines. \((b)\) The capacitance per unit length of the intermediate line is found to be \(100 \mathrm{pF} / \mathrm{m}\). Find the shortest length in meters of this line that is needed to satisfy the impedance-matching condition. ( \(c\) ) With the three-segment setup as found in parts \((a)\) and \((b)\), the frequency is now doubled to \(2 \mathrm{GHz}\). Find the input impedance at the line-1-to-line- 2 junction, seen by waves incident from line \(1 .(d)\) Under the conditions of part \((c)\), and with power incident from line 1 , evaluate the standing wave ratio that will be measured in line 1 , and the fraction of the incident power from line 1 that is reflected and propagates back to the line 1 input.

A lossless transmission line having characteristic impedance \(Z_{0}=50 \Omega\) is driven by a source at the input end that consists of the series combination of a 10 -V sinusoidal generator and a \(50-\Omega\) resistor. The line is one- quarter wavelength long. At the other end of the line, a load impedance, \(Z_{L}=50-j 50 \Omega\) is attached. (a) Evaluate the input impedance to the line seen by the voltage source-resistor combination; \((b)\) evaluate the power that is dissipated by the load; \((c)\) evaluate the voltage amplitude that appears across the load.

A lossless line having an air dielectric has a characteristic impedance of \(400 \Omega\). The line is operating at \(200 \mathrm{MHz}\) and \(Z_{\text {in }}=200-j 200 \Omega\). Use analytic methods or the Smith chart (or both) to find \((a) s ;(b) Z_{L}\), if the line is \(1 \mathrm{~m}\) long; \((c)\) the distance from the load to the nearest voltage maximum.

A sinusoidal voltage wave of amplitude \(V_{0}\), frequency \(\omega\), and phase constant \(\beta\) propagates in the forward \(z\) direction toward the open load end in a lossless transmission line of characteristic impedance \(Z_{0}\). At the end, the wave totally reflects with zero phase shift, and the reflected wave now interferes with the incident wave to yield a standing wave pattern over the line length (as per Example 10.1). Determine the standing wave pattern for the current in the line. Express the result in real instantaneous form and simplify.
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