Solved Numerical 1
A lossless transmission line with a characteristic impedance of 300 Ω is fed by a generator having an internal impedance of 100 Ω. The transmission line is 100 m long and terminated with a resistive load of 200 Ω.
Determine:
- Reflection Loss
- Transmission Loss
- Return Loss
Given Data
- Generator Impedance, Zg = 100 Ω
- Characteristic Impedance, Z0 = 300 Ω
- Transmission Line Length, l = 100 m
- Load Impedance, ZL = 200 Ω
Step 1: Calculate the Load Reflection Coefficient
The reflection coefficient at the load is given by:
$ \Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0} $
Substituting the given values:
$ \Gamma_L=\frac{200-300}{200+300} $
$ \Gamma_L=\frac{-100}{500} $
$ \Gamma_L=-\frac{1}{5} $
Therefore:
$ |\Gamma_L|=0.2 $
Step 2: Calculate Reflection Loss
Reflection loss measures the power that fails to enter the load because of impedance mismatch.
The formula is:
$ \text{Reflection Loss} = 10\log_{10} \left( \frac{1}{1-|\Gamma|^2} \right) $
Substituting the reflection coefficient:
$ = 10\log_{10} \left( \frac{1}{1-(0.2)^2} \right) $
$ = 10\log_{10} \left( \frac{1}{1-\frac{1}{25}} \right) $
$ = 10\log_{10} \left( \frac{25}{24} \right) $
$ = 0.1773\text{ dB} $
Reflection Loss = 0.1773 dB
Step 3: Calculate Transmission Loss
Transmission loss consists of attenuation loss plus reflection loss.
$ \text{Transmission Loss} = \text{Attenuation Loss} + \text{Reflection Loss} $
Since the transmission line is lossless:
$ \text{Attenuation Loss}=0 $
Therefore:
$ \text{Transmission Loss} = 0 + 0.1773 $
$ = 0.1773\text{ dB} $
Transmission Loss = 0.1773 dB
Step 4: Calculate Return Loss
Return loss indicates how much power is reflected back toward the source because of impedance mismatch.
The formula is:
$ \text{Return Loss} = -20\log_{10} |\Gamma| $
Substituting:
$ = -20\log_{10} \left( \frac{1}{5} \right) $
$ = 13.98\text{ dB} $
Return Loss = 13.98 dB
Final Answers
- Reflection Loss = 0.1773 dB
- Transmission Loss = 0.1773 dB
- Return Loss = 13.98 dB
Even though the transmission line is lossless, the mismatch between the characteristic impedance (300 Ω) and the load impedance (200 Ω) causes part of the incident wave to be reflected. This reflected power creates reflection loss and return loss. Since there is no attenuation along the line, the transmission loss is equal to the reflection loss.