Question

A sinusoidal voltage source drives the series combination of an impedance, \(Z_{g}=50-j 50 \Omega\), and a lossless transmission line of length \(L\), shorted at the load end. The line characteristic impedance is \(50 \Omega\), and wavelength \(\lambda\) is measured on the line. \((a)\) Determine, in terms of wavelength, the shortest line length that will result in the voltage source driving a total impedance of \(50 \Omega .(b)\) Will other line lengths meet the requirements of part \((a)\) ? If so, what are they?

Short answer

Answer: The shortest length of the transmission line that will result in a total impedance of 50Ω is L = λ/8. The other line lengths that meet the requirements are given by L = (λ/8) + n(λ/2), where n is an integer.

Step-by-step solution

Understand the given information

We are given the generator impedance \(Z_{g}=50-j50\Omega\), and a lossless transmission line with characteristic impedance \(Z_{0}=50\Omega\) and length \(L\). The transmission line is shorted at the load end.

Finding the input impedance of the transmission line

The input impedance of a transmission line can be found using the equation: \[Z_{in}=Z_{0}\frac{Z_{L}+jZ_{0}\tan(\beta L)}{Z_{0}+jZ_{L}\tan(\beta L)}\] However, since the transmission line is shorted at the load end, the load impedance \(Z_{L}=0\). Therefore, the equation simplifies to: \[Z_{in}=jZ_{0}\tan(\beta L)\]

Calculate the total impedance driven by the voltage source

The total impedance driven by the voltage source is given by: \[Z_{total}=Z_{g}+Z_{in}\] We are given that \(Z_{g}=50-j50\Omega\), and \(Z_{in}=j50\tan(\beta L)\). So, the equation will be: \[Z_{total}=50-j50+j50\tan(\beta L)\]

Find the line length for the total impedance to be \(50\Omega\)

We need to find the shortest line length \(L\) such that \(Z_{total}=50\Omega\). To do this, set the imaginary part of the total impedance to zero and solve for \(L\), i.e. \[-50+50\tan(\beta L)=0\] \[tan(\beta L)=1\] Now, \(\beta=\frac{2\pi}{\lambda}\). So, we can write: \[tan\left(\frac{2\pi}{\lambda} L\right)=1\] The shortest line length which satisfies this condition is when the argument inside the tangent function is \(\frac{\pi}{4}\), i.e. \[\frac{2\pi}{\lambda} L=\frac{\pi}{4}\] Now, we can solve for the shortest length \(L\) in terms of the wavelength, which is: \[L=\frac{\lambda}{8}\]

Determine if other line lengths will meet the requirements

Now, we must check if other values of line length \(L\) will satisfy the conditions. From the tangent function's properties, we know that \(tan(x+\pi)=tan(x)\). So, we need to check if the following length values also meet the requirements: \[L=\frac{\lambda}{8} + n\frac{\lambda}{2}\] where \(n\) is an integer. Yes, other line lengths will meet the requirements, and they are given as: \[L=\frac{\lambda}{8} + n\frac{\lambda}{2}\] where \(n\) is an integer.

Key concepts

Sinusoidal Voltage Source

A sinusoidal voltage source is an electrical source that outputs a voltage which varies sinusoidally with time. You can imagine it like a wave continuously going up and down, creating peaks and valleys in a smooth, repeated pattern. Mathematically, it's represented as:\[V(t) = V_m \cos(\omega t + \phi)\]where:

• \(V(t)\) is the instantaneous voltage at time \(t\).
• \(V_m\) is the maximum (peak) voltage.
• \(\omega\) is the angular frequency (related to how fast the wave oscillates).
• \(\phi\) is the phase angle (determines where in its cycle the wave begins).

Sinusoidal sources are very common in AC (alternating current) circuits, and they are crucial in analyzing transmission lines. They help in understanding the behavior of systems involving complex impedance, ensuring power is transmitted efficiently.

Characteristic Impedance

Characteristic impedance, often denoted by \(Z_0\), is a fundamental concept in transmission line theory. It's the impedance that a transmission line would have if it were infinitely long. This serves as a measure of the intrinsic opposition a transmission line provides against the flow of electrical current.When dealing with transmission lines, it's essential to consider matching the load impedance with the characteristic impedance for maximum power transfer and minimal reflections. In mathematical terms, the characteristic impedance \(Z_0\) can be defined by the formula:\[Z_0 = \sqrt{\frac{L}{C}}\]where:

• \(L\) is the inductance per unit length of the line.
• \(C\) is the capacitance per unit length of the line.

In our exercise, the characteristic impedance is given as \(50 \Omega\), meaning that the transmission line is designed to optimally transfer energy when connected to a load with the same impedance.

Input Impedance

The input impedance \(Z_{in}\) of a transmission line is the impedance 'seen' at the input end of the line. It's important to calculate this to understand how the line interacts with connected circuits. The input impedance is affected by the characteristic impedance, the length of the transmission line, and the load impedance.For a lossless transmission line, like the one in our example, with a short-circuited end \(Z_L = 0\), the input impedance formula simplifies to:\[Z_{in} = jZ_0 \tan(\beta L)\]where \(\beta\) is the phase constant (related to the wavelength) and \(L\) is the line length. In our context, we solve for \(L\) such that the total impedance becomes real and equals the source impedance. This condition ensures no voltage is reflected and all power is delivered to the load efficiently.

Wavelength and Transmission Lines

Wavelength \(\lambda\) is a key factor in understanding the behavior of waves along a transmission line. It's the physical distance over which the waveform repeats, and it significantly impacts the calculation of line lengths.In transmission line theory, wavelength is tied directly to the phase constant \(\beta\), as \(\beta = \frac{2\pi}{\lambda}\). This relation helps determine crucial parameters, like the length of the transmission line for specific impedance conditions. For example, in the exercise, we calculated the line length \(L\) such that the total impedance matches the source impedance by solving:\[\beta L = \frac{\pi}{4}\]This sets the shortest suitable length to \(\frac{\lambda}{8}\) and gives a sequence of possible lengths as \(\frac{\lambda}{8} + n\frac{\lambda}{2}\) where \(n\) is an integer, signifying other lengths that also perform similarly. Understanding wavelength and its relationship with transmission lines aids in designing systems effectively, ensuring they operate as intended.