Question

An absolute measure of power is the \(\mathrm{dBm}\) scale, in which power is specified in decibels relative to one milliwatt. Specifically, \(P(\mathrm{dBm})=10 \log _{10}[P(\mathrm{~mW}) / 1 \mathrm{~mW}]\). Suppose that a receiver is rated as having a sensitivity of \(-20 \mathrm{dBm}\), indicating the mimimum power that it must receive in order to adequately interpret the transmitted electronic data. Suppose this receiver is at the load end of a \(50-\Omega\) transmission line having \(100-\mathrm{m}\) length and loss rating of \(0.09 \mathrm{~dB} / \mathrm{m}\). The receiver impedance is \(75 \Omega\), and so is not matched to the line. What is the minimum required input power to the line in \((a) \mathrm{dBm},(b) \mathrm{mW} ?\)

Short answer

The minimum required input power to the transmission line is approximately: (a) -11.46 dBm (b) 0.677 mW

Step-by-step solution

Calculate the total line loss in dB

To calculate the total line loss in dB, multiply the loss rating of the transmission line (0.09 dB/m) by the length of the transmission line (100m): $$\mathrm{Line~Loss} = 0.09 \mathrm{~dB/m} * 100 \mathrm{~m} = 9 \mathrm{~dB}.$$

Add line loss to the receiver sensitivity

Now, add the line loss to the receiver sensitivity to calculate the required input power in dBm: $$\mathrm{Required~Input~Power~(dBm)} = -20 \mathrm{~dBm} + 9 \mathrm{~dB} = -11 \mathrm{~dBm}.$$

Calculate the impedance mismatch factor

Since the receiver impedance (\(75\Omega\)) is not matched to the transmission line impedance (\(50\Omega\)), we need to calculate an impedance mismatch factor to account for the reflected power due to the mismatch. The reflection coefficient, denoted as \(\Gamma\), can be calculated using the following formula: $$\Gamma = \frac{Z_\text{load} - Z_\text{line}}{Z_\text{load} + Z_\text{line}} = \frac{75\Omega - 50\Omega}{75\Omega + 50\Omega} = 0.2.$$ We can now calculate the mismatch factor (MF) as: $$\mathrm{MF} = 1 - \Gamma^2 = 1 - 0.2^2 = 0.96.$$

Calculate the minimum required input power in dBm

With the mismatch factor calculated, we can now find the minimum required input power in dBm, taking into account for reflection due to impedance mismatch. Divide the previously calculated required input power (in dBm) by the mismatch factor: $$\mathrm{Required~Input~Power~(dBm)}_\text{corrected} = \frac{-11 \mathrm{~dBm}}{0.96} \approx -11.46 \mathrm{~dBm}.$$

Convert the required input power in dBm to mW

Now, use the dBm to mW conversion formula to convert the required input power from dBm to mW: $$P(\mathrm{mW}) = 10^{P(\mathrm{dBm})/10} * 1 \mathrm{mW}.$$ Plugging in the corrected required input power, we get: $$P(\mathrm{mW}) = 10^{-11.46/10} * 1 \mathrm{~mW} \approx 0.677 \mathrm{~mW}.$$

Final Answer

So, the minimum required input power to the transmission line is: (a) \(\approx -11.46 \mathrm{~dBm}\) (b) \(\approx 0.677 \mathrm{~mW}\)

Key concepts

dBm Scale

When working with electrical power measurements, the dBm scale is a unit of measure that provides a convenient way to express power levels. dBm stands for decibels relative to one milliwatt. It's a logarithmic scale where 0 dBm is equal to 1 milliwatt (mW) of power.

Using dBm to express power is beneficial because it can represent very large or small values in a compact form. This makes it easier to deal with numbers that would otherwise be unwieldy. Moreover, when calculating power levels across components in a system, addition and subtraction in dBm can be used instead of multiplication and division of absolute power values, simplifying the math involved.

Transmission Line Loss

A key factor in the design and operation of transmission lines is transmission line loss. It refers to the power that is lost as it travels along the line from the source to the receiver. These losses come in the form of heat and electromagnetic radiation and are often measured in dB per unit length (e.g., dB per meter).

The total loss can be calculated by multiplying the loss per unit length by the length of the transmission line. This information is essential for determining the minimum power required at the input, so the receiver still gets adequate power despite these inherent losses.

Impedance Mismatch

Impedance mismatch is a critical consideration when transmitting signals over a cable or transmission line. Impedance is the opposition that a circuit offers to the flow of alternating current, and it is measured in ohms (). When the impedance of the transmission line doesn't match the device connected to it, some power is reflected back toward the source rather than being absorbed by the load.

This mismatch can lead to inefficiency and reduced signal quality. For maximum power transfer, the impedance of the transmission line should ideally match the load impedance. When they do not match, a correction factor must be applied to ensure that the minimum necessary power reaches the receiver.

Reflection Coefficient

The reflection coefficient, represented by the Greek letter Gamma (Gamma), quantifies the amount of power reflected due to impedance mismatch. This coefficient can range from -1 to 1, where 0 indicates no reflection (complete impedance match) and 1 or -1 indicates total reflection (no power delivered to the load).

The formula to calculate the reflection coefficient is Gamma = (Z_load - Z_line)/(Z_load + Z_line), where Z_load is the load impedance and Z_line is the line impedance. Once Gamma is derived, it can be used to adjust calculations of the power required at the input of a transmission line to ensure the load receives sufficient power.

Decibel-milliwatts Conversion

The conversion between decibel-milliwatts () and milliwatts () is a fundamental operation in working with transmission line power calculations. To convert from to , use the formula: P() = 10^(P()/10) * 1 .

If you need to convert from milliwatts to dBm, apply the formula: P_{dBm} = 10 * log_{10}(P_{mW} / 1 ).

These conversions are critical when assessing the signal levels in a system since equipment specifications, signal strength indicators, and regulatory limits are often provided in dBm.