Question

A transmitter and receiver are connected using a cascaded pair of transmission lines. At the operating frequency, line 1 has a measured loss of $0.1\mathrm{dB}/\mathrm{m}$, and line 2 is rated at $0.2\mathrm{dB}/\mathrm{m}$. The link is composed of $40\mathrm{m}$ of line 1 joined to $25\mathrm{m}$ of line 2 . At the joint, a splice loss of $2\mathrm{dB}$ is measured. If the transmitted power is $100\mathrm{mW}$, what is the received power?

Short answer

Question: Calculate the received power at the end of the transmission system when the transmitted power is 100mW, and the transmission system consists of two lines in cascade with given losses and lengths. The splice loss at the joint between the two lines is 2dB. Line 1 has a loss of 0.1 dB/m and a length of 40m. Line 2 has a loss of 0.2 dB/m and a length of 25m. Answer: The received power at the end of the transmission system is approximately 7.943 mW.

Step-by-step solution

Calculate the loss in each transmission line due to its length

We have the following losses and lengths for each transmission line: - Line 1: Loss of $0.1\text{dB/m}$ for a length of $40\text{m}$ - Line 2: Loss of $0.2\text{dB/m}$ for a length of $25\text{m}$ So the loss in each transmission line can be calculated by multiplying the given loss per meter by the respective lengths: Loss in Line 1: $0.1\text{dB/m}\times 40\text{m}=4\text{dB}$ Loss in Line 2: $0.2\text{dB/m}\times 25\text{m}=5\text{dB}$

Sum up the total losses in the transmission system

To find the total loss in the transmission system, add the losses from Line 1, Line 2, and the splice at the joint: Total Loss = Loss in Line 1 + Splice Loss + Loss in Line 2 Total Loss = $4\text{dB}+2\text{dB}+5\text{dB}=11\text{dB}$

Calculate the received power

To calculate the received power, we will use the formula for power loss in decibels: $P_{\text{received}}=P_{\text{transmitted}}\times {10}^{-\frac{L}{10}}$ Where $P_{\text{received}}$ is the received power in milliwatts, $P_{\text{transmitted}}$ is the transmitted power, and $L$ is the total loss in decibels. Plug in the total loss and transmitted power into the formula: $P_\textrm{received} = 100\textrm{mW} \times 10^{-\frac{11}{10}} = 100\textrm{mW} \times 10^{-1.1} = 100\textrm{mW} \times 0.07943 = 7.943 \textrm{mW}$ So the received power is approximately $7.943\text{mW}$.

Key concepts

Cascaded Transmission Lines

When transmission lines are combined in a series, they are referred to as **cascaded transmission lines**. Imagine them as a series of pipes where water flows; just like water pressure diminishes over distance, the signal loses strength as it travels. Each line has its own unique characteristics, including its loss factor, which is commonly expressed in decibels per meter (dB/m).
In our context, Line 1 has a loss of 0.1 dB/m over 40 meters, leading to a 4 dB loss, while Line 2, with a loss of 0.2 dB/m over 25 meters, incurs a 5 dB loss. Together with the junction (splice) loss of 2 dB, total transmission loss accumulates along these segments.
This combined loss from the cascaded lines essentially dictates how much weaker your signal will be when it finally reaches the receiver. The understanding of cascading is vital for anyone dealing with multiple linked segments as it directly impacts the total system efficiency.

Splice Loss

**Splice loss** refers to the weakening of a signal at a join between two transmission lines. Think of it as a kink in the garden hose where water flow reduces. No matter how perfect the splice, some signal is inevitably lost.
In spicy scenarios, losses can arise from poor alignment or differences in core sizes, not to mention varying material characteristics. In basic terms, even a purely mechanical splice will eat away at the signal, quantified in decibels (dB).
In this particular transmission setup, the splice loss is 2 dB. Understanding splice loss helps engineers minimize joint weaknesses and manage system expectations, ensuring that all factors are considered in the overall transmission line design.

Received Power Calculation

Upon understanding how much signal is lost across the entire transmission link, you can calculate the **received power** at the end of the line. This calculation shows how much of the original power finally makes it to the receiver.
To do this, you first convert the power loss from decibels back into a linear scale using the formula:$$P_{\text{received}}=P_{\text{transmitted}}\times {10}^{-\frac{L}{10}}$$ Where:

• $P_{\text{received}}$ is the power that the receiver actually catches.
• $P_{\text{transmitted}}$ is the starting power sent over the line, in milliwatts (mW).
• $L$ is the calculated total loss in decibels (dB).

For instance, if you start with 100 mW of power and face an overall loss of 11 dB, the received power comes out to about 7.943 mW. Estimating received power equips engineers to ensure signal integrity and reliability across systems.

Decibel-to-Power Conversion

The conversion from **decibels (dB)**, a logarithmic measure, back to power is crucial for analyzing how much signal power makes it through a system.
Decibels are used to simplify the understanding of changes in power levels; for example, a 10 dB decrease corresponds to a power reduction by a factor of 10. This decibel system effectively compresses a wide range of real-world power levels into a smaller, more manageable number scale.
To convert from decibels to a linear power scale, the formula involves raising 10 to the negative power of the decibel loss divided by 10:$$Powerratio={10}^{-\frac{dBloss}{10}}$$This power ratio is then multiplied by the transmitted power to find what arrives at the end of the line. It's indispensable when evaluating efficiency because it outlines if a system has acceptable performance under given losses.