A source with 50 ohm source impedance drives a 50 ohm T-line that is 1/8 of a wavelength long, terminated in a load Z=50-j25 ohms. calculate TL, VSWR, and the input impedance seen by the source
To calculate the transmission line length (TL), VSWR (Voltage Standing Wave Ratio), and the input impedance seen by the source, we will use the following steps:
1. Determine the wavelength (λ) of the signal on the transmission line.
- We can calculate the wavelength using the formula: λ = (2π) / β, where β is the phase constant.
- In this case, since the transmission line is 1/8 of a wavelength long, we can use the formula: TL = λ / 8
2. Calculate the characteristic impedance (Z0) of the transmission line.
- The characteristic impedance of a transmission line with equal source and load impedances is equal to the square root of the source impedance times the load impedance.
- Z0 = √(Z_source * Z_load)
3. Calculate the reflection coefficient (Γ) using the VSWR.
- VSWR (Voltage Standing Wave Ratio) is equal to (1 + |Γ|) / (1 - |Γ|)
- Rearranging the formula, we can calculate the reflection coefficient as: |Γ| = (VSWR - 1) / (VSWR + 1)
4. Calculate the input impedance (Zin) seen by the source.
- The input impedance is given by the formula: Zin = Z_source * ((Z0 + Z_load*tan(β*TL)) / (Z_source + Z0*tan(β*TL)))
Now, let's calculate each of the values step-by-step:
Step 1: Determine the wavelength (λ)
Given: TL (Transmission Line Length) = 1/8 of a wavelength
We know that TL = λ / 8, so λ = TL * 8
Step 2: Calculate the characteristic impedance (Z0) of the transmission line.
Given: Z_source = 50 ohms
Z_load = 50-j25 ohms
Z0 = √(Z_source * Z_load)
Step 3: Calculate the reflection coefficient (Γ) using the VSWR.
Given: VSWR = (1 + |Γ|) / (1 - |Γ|)
We can rearrange the formula to calculate |Γ|:
|Γ| = (VSWR - 1) / (VSWR + 1)
Step 4: Calculate the input impedance (Zin) seen by the source.
Given: Zin = Z_source * ((Z0 + Z_load*tan(β*TL)) / (Z_source + Z0*tan(β*TL)))
Let's substitute the given values and calculate each step.
To calculate the transmission line (T-line) length (TL), voltage standing wave ratio (VSWR), and the input impedance seen by the source, we can follow these steps:
1. Calculate the T-line length (TL):
The T-line length is given as 1/8 of a wavelength. We can use the formula:
Lambda = c / f
Where:
Lambda is the wavelength,
c is the speed of light (approximately 3 x 10^8 meters per second),
and f is the frequency.
Since we have the T-line length as 1/8 of a wavelength, we can calculate the T-line length by:
TL = Lambda / 8
2. Calculate the VSWR:
The VSWR can be calculated using the formula:
VSWR = (1 + |Γ|) / (1 - |Γ|)
Where:
Γ (Gamma) is the reflection coefficient.
The reflection coefficient can be determined using the formula:
Γ = (ZL - Z0) / (ZL + Z0)
Where:
ZL is the load impedance,
and Z0 is the characteristic impedance of the T-line (in this case, 50 ohms).
Once you have the reflection coefficient, you can substitute it into the VSWR formula to obtain the VSWR value.
3. Calculate the input impedance seen by the source:
The input impedance seen by the source can be calculated using the formula:
Zin = Z0 * (ZL + jZ0 * tan(2π * TL / Lambda)) / (Z0 + jZL * tan(2π * TL / Lambda))
Where:
ZL is the load impedance,
Z0 is the characteristic impedance of the T-line,
TL is the T-line length,
and Lambda is the wavelength.
By substituting the given values into this formula, we can calculate the input impedance seen by the source.
Using these steps, you can calculate the TL, VSWR, and the input impedance seen by the source in this scenario.