Question

Design a microstrip, quarter-wave transformer that matches a 10[Ω] load to a 50[Ω] microstrip transmission line. Assume that the substrate is 1[ mm] thick, the dielectric constant is 3.5 , and the frequency is 12.5[GHz]. Determine the length of the stub and its distance from the load.

Answer 1

To design the microstrip quarter-wave transformer, calculate the impedance using the geometric mean of the load and line impedances. The length of the transformer is a quarter of the wavelength, which is defined by the speed of light, frequency, and effective dielectric constant. The exact physical layout may affect the stub's placement.

Step-by-step explanation:

To design a microstrip quarter-wave transformer, we can use the following formulas:

1. **Stub Length (L):**

`\[ L = (\lambda_g)/(4) \]`

where \(\lambda_g\) is the guided wavelength, given by:

`\[ \lambda_g = \frac{\lambda}{\sqrt{1 - \left((f_c)/(f)\right)^2}} \]`

Here, \(f_c\) is the cutoff frequency, and \(f\) is the operating frequency.

2. **Distance from Load (d):**

\[ d = \frac{\lambda_g}{8} \]

3. **Guided Wavelength (\(\lambda_g\)) Calculation:**

`\[ \lambda = (c)/(f) \]`

`\[ \lambda_g = \frac{\lambda}{\sqrt{\varepsilon_{\text{eff}}}} \]`

where \(\varepsilon_{\text{eff}}\) is the effective dielectric constant, given by:

`\[ \varepsilon_{\text{eff}} = (\varepsilon_r + 1)/(2) + (\varepsilon_r - 1)/(2) \left(1 + (12h)/(W)\right)^{-(1)/(2)} \]`

Here, \(\varepsilon_r\) is the relative permittivity, \(h\) is the substrate thickness, and \(W\) is the width of the microstrip transmission line.

Given parameters:

- Load impedance (\(Z_{\text{load}}\)): 10 Ω

- Characteristic impedance (\(Z_0\)) of the microstrip line: 50 Ω

- Substrate thickness (\(h\)): 1 mm

- Dielectric constant (\(\varepsilon_r\)): 3.5

- Operating frequency (\(f\)): 12.5 GHz

Let's proceed with the calculations:

1. Calculate \(\lambda\):

\[ \lambda = \frac{c}{f} \]

2. Calculate \(\varepsilon_{\text{eff}}\):

`\[ \varepsilon_{\text{eff}} = (\varepsilon_r + 1)/(2) + (\varepsilon_r - 1)/(2) \left(1 + (12h)/(W)\right)^{-(1)/(2)} \]`

3. Calculate \(\lambda_g\):

`\[ \lambda_g = \frac{\lambda}{\sqrt{\varepsilon_{\text{eff}}}} \]`

4. Calculate the Stub Length (\(L\)):

`\[ L = (\lambda_g)/(4) \]`

5. Calculate the Distance from Load (\(d\)):

`\[ d = (\lambda_g)/(8) \]`

These calculations will yield the required dimensions for the microstrip quarter-wave transformer.