A matching network consists of a length of T-line in series with a capacitor. Determine the length(in wavelength) required of the T-line section and the capacitor value needed (at 1 GHz) to match a 10-j35 ohm load impedance to the 50 ohm line.
To determine the length of the T-line section and the required value of the capacitor in the matching network, we can use the Smith chart and the formulas for impedance transformation.
Step 1: Convert the load impedance and line impedance to their normalized values on the Smith chart.
Given:
Load impedance: Zl = 10 - j35 ohms
Line impedance: Z0 = 50 ohms
The normalized load impedance Znl is calculated as:
Znl = (Zl - Z0) / (Zl + Z0)
Substituting the values, we get:
Znl = (10 - j35 - 50) / (10 - j35 + 50)
Calculating the numerator and denominator separately:
Numerator = 10 - j35 - 50 = -40 - j35
Denominator = 10 - j35 + 50 = 60 - j35
Now, substitute these values back into the formula:
Znl = (-40 - j35) / (60 - j35)
Step 2: Plot the normalized load impedance Znl on the Smith chart.
Using the Smith chart, locate the point corresponding to the calculated Znl value and mark it.
Step 3: Find the desired reflection coefficient.
The reflection coefficient Γ is given by:
Γ = (Zl - Z0) / (Zl + Z0)
Substituting the values, we get:
Γ = (10 - j35 - 50) / (10 - j35 + 50)
Calculating the numerator and denominator separately:
Numerator = 10 - j35 - 50 = -40 - j35
Denominator = 10 - j35 + 50 = 60 - j35
Now, substitute these values back into the formula:
Γ = (-40 - j35) / (60 - j35)
Step 4: Determine the electrical length of the T-line section.
The electrical length of the T-line section in wavelengths (βL) can be found from the Smith chart as the angle between the normalized load impedance (Znl) and the center of the Smith chart.
Measure this angle and convert it into radians.
Step 5: Convert the electrical length of the T-line section into physical length.
The physical length (L) in wavelengths can be calculated using the equation:
L = βL * λ
Where β is the propagation constant and λ is the wavelength at the operating frequency.
Step 6: Calculate the value of the capacitor.
The impedance transformation provided by the T-line can be given as:
Zin = Z0 * ((Zl + jZ0 * tan(βL)) / (Z0 + jZl * tan(βL)))
Simplify this expression by substituting the given values and solving for the required value of the capacitor.
Note: The process outlined above is a general approach to matching networks using the Smith chart and formulas. The steps may vary slightly depending on the specific context and assumptions made.