Numerical 1
Chebyshev Transformer Design: 50 Ω Source to 100 Ω Load at 3 GHz
Consider the design of a three-section Chebyshev transformer used to match a \(50\,\Omega\) source to a \(100\,\Omega\) load at a center frequency of \(3\,\text{GHz}\). The specified fractional bandwidth is \(71\%\), and the maximum allowable passband reflection coefficient is \(0.05\). The Chebyshev design method uses the equal-ripple property of the Chebyshev polynomial to distribute the reflection coefficient across the passband while keeping its magnitude within the specified limit. For a three-section transformer, the third-order Chebyshev polynomial \(T_3(x)\) is used to determine the individual reflection coefficients and, subsequently, the characteristic impedance of each transformer section.
Given Specifications
The design specifications are:
- Source impedance: \(Z_0=50\,\Omega\)
- Load impedance: \(Z_L=100\,\Omega\)
- Center frequency: \(f_0=3\,\text{GHz}\)
- Number of transformer sections: \(N=3\)
- Maximum passband reflection coefficient: \(\Gamma_m=0.05\)
- Specified fractional bandwidth: \(71\%\)
The transformer therefore consists of three transmission-line sections placed between the \(50\,\Omega\) source and \(100\,\Omega\) load. The characteristic impedances of these three sections must be selected so that the overall reflection coefficient follows the desired Chebyshev equal-ripple response.
Reflection Coefficient Representation
For a three-section Chebyshev transformer, the reflection coefficient can be represented as a cosine series and related to the third-order Chebyshev polynomial. The required form is:
\[ \boxed{ \Gamma(\theta) = 2e^{-j3\theta} \left[ \Gamma_0\cos(3\theta) + \Gamma_1\cos(\theta) \right] } \]
The same reflection coefficient is specified using the third-order Chebyshev polynomial as:
\[ \boxed{ \Gamma(\theta) = Ae^{-j3\theta} T_3\left(\sec\theta_m\cos\theta\right) } \]
Here, \(A\) represents the amplitude of the Chebyshev response, \(\theta\) is the electrical length variable, and \(\theta_m\) represents the passband edge angle. Since the maximum allowable passband reflection coefficient is \(0.05\), the amplitude is selected as:
\[ \boxed{ A=\Gamma_m=0.05 } \]
The common phase factor \(e^{-j3\theta}\) describes the phase variation associated with the three transformer sections. The magnitude response is controlled by the Chebyshev polynomial.
Step 1: Calculate the Passband Edge Angle \(\theta_m\)
The passband edge angle determines the boundary between the equal-ripple passband and the region outside the desired passband. For a Chebyshev transformer, the required value can be obtained from:
\[ \boxed{ \sec(\theta_m) = \cosh \left[ \frac{1}{N} \cosh^{-1} \left( \frac{1}{\Gamma_m} \frac{|\ln(Z_L/Z_0)|}{2} \right) \right] } \]
This expression uses the small-mismatch approximation:
\[ \boxed{ \left| \frac{Z_L-Z_0}{Z_L+Z_0} \right| \approx \frac{|\ln(Z_L/Z_0)|}{2} } \]
This approximation is useful for the small-reflection analysis of multisection transformers and connects the impedance ratio directly with the reflection-coefficient formulation.
For the present design:
\[ Z_L=100\,\Omega \]
\[ Z_0=50\,\Omega \]
\[ N=3 \]
\[ \Gamma_m=0.05 \]
Substituting these values:
\[ \sec(\theta_m) = \cosh \left[ \frac{1}{3} \cosh^{-1} \left( \frac{1}{0.05} \frac{|\ln(100/50)|}{2} \right) \right] \]
Since:
\[ \ln\left(\frac{100}{50}\right) = \ln(2) \approx0.693 \]
we obtain:
\[ \sec(\theta_m) = \cosh \left[ \frac{1}{3} \cosh^{-1} \left( 20\times\frac{0.693}{2} \right) \right] \]
Therefore:
\[ \sec(\theta_m) = \cosh \left[ \frac{1}{3} \cosh^{-1}(6.93) \right] \]
Using:
\[ \cosh^{-1}(6.93)\approx2.623 \]
gives:
\[ \sec(\theta_m) = \cosh(0.874) \approx1.408 \]
Hence:
\[ \cos(\theta_m) = \frac{1}{1.408} \approx0.710 \]
Therefore, the passband edge angle is approximately:
\[ \boxed{ \theta_m\approx44.75^\circ } \]
Step 2: Find the Individual Reflection Coefficients
Once \(\theta_m\) is known, the individual reflection coefficients at the impedance discontinuities can be determined by equating the cosine expansion of the transformer reflection coefficient with the third-order Chebyshev polynomial.
For \(N=3\), the required relationship is:
\[ \boxed{ 2 \left[ \Gamma_0\cos(3\theta) + \Gamma_1\cos(\theta) \right] = A T_3 \left( \sec\theta_m\cos\theta \right) } \]
The third-order Chebyshev polynomial is:
\[ \boxed{ T_3(x)=4x^3-3x } \]
Substituting:
\[ x=\sec\theta_m\cos\theta \]
gives:
\[ T_3 = 4\sec^3\theta_m\cos^3\theta - 3\sec\theta_m\cos\theta \]
Using the identity:
\[ \cos(3\theta) = 4\cos^3\theta-3\cos\theta \]
the polynomial can be expressed in terms of \(\cos(3\theta)\) and \(\cos(\theta)\). After collecting the corresponding terms, the coefficients can be equated with the cosine-series representation of the transformer reflection coefficient.
Coefficient \(\Gamma_0\)
For the \(\cos(3\theta)\) term:
\[ 2\Gamma_0 = A\sec^3\theta_m \]
Therefore:
\[ \Gamma_0 = \frac{A}{2}\sec^3\theta_m \]
Using \(A=0.05\) and \(\sec\theta_m=1.408\):
\[ \Gamma_0 = \frac{0.05}{2}(1.408)^3 \]
Since:
\[ (1.408)^3\approx2.79 \]
we obtain:
\[ \Gamma_0 = 0.025(2.79) \approx0.0698 \]
Thus:
\[ \boxed{ \Gamma_0\approx0.0698 } \]
Coefficient \(\Gamma_1\)
For the \(\cos(\theta)\) term:
\[ 2\Gamma_1 = 3A \left( \sec^3\theta_m-\sec\theta_m \right) \]
Therefore:
\[ \Gamma_1 = \frac{3A}{2} \left( \sec^3\theta_m-\sec\theta_m \right) \]
Substituting the numerical values:
\[ \Gamma_1 = \frac{3(0.05)}{2} \left[ (1.408)^3-1.408 \right] \]
\[ \Gamma_1 = 0.075(2.79-1.408) \]
\[ \Gamma_1 = 0.075(1.382) \approx0.1037 \]
Therefore:
\[ \boxed{ \Gamma_1\approx0.1037 } \]
Symmetry of the Transformer
A three-section Chebyshev transformer has a symmetric reflection-coefficient distribution. Therefore, the remaining coefficients follow the symmetry relationship:
\[ \boxed{ \Gamma_2=\Gamma_1=0.1037 } \]
and:
\[ \boxed{ \Gamma_3=\Gamma_0=0.0698 } \]
The sequence of reflection coefficients is therefore:
\[ \boxed{ \Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3 = 0.0698,\;0.1037,\;0.1037,\;0.0698 } \]
Step 3: Calculate the Characteristic Impedances
The reflection coefficient at each junction is related to the characteristic impedances of two adjacent sections. For relatively small impedance discontinuities, the reflection coefficient can be approximated by:
\[ \boxed{ \Gamma_n \approx \frac{1}{2} \ln \left( \frac{Z_{n+1}}{Z_n} \right) } \]
Rearranging:
\[ \boxed{ \ln Z_{n+1} = \ln Z_n+2\Gamma_n } \]
or equivalently:
\[ \boxed{ Z_{n+1} = Z_ne^{2\Gamma_n} } \]
This relationship allows the characteristic impedance of every transformer section to be calculated successively, beginning with the source impedance \(Z_0=50\,\Omega\).
Finding \(Z_1\)
For the first discontinuity:
\[ \ln Z_1 = \ln Z_0+2\Gamma_0 \]
Substituting \(Z_0=50\,\Omega\) and \(\Gamma_0=0.0698\):
\[ \ln Z_1 = \ln50+2(0.0698) \]
Using:
\[ \ln50\approx3.912 \]
we obtain:
\[ \ln Z_1 = 3.912+0.1396 = 4.0516 \]
Therefore:
\[ Z_1=e^{4.0516} \approx57.5\,\Omega \]
Hence:
\[ \boxed{ Z_1\approx57.5\,\Omega } \]
Finding \(Z_2\)
For the second discontinuity:
\[ \ln Z_2 = \ln Z_1+2\Gamma_1 \]
Substituting:
\[ \ln Z_2 = 4.0516+2(0.1037) \]
\[ \ln Z_2 = 4.0516+0.2074 = 4.259 \]
Therefore:
\[ Z_2=e^{4.259} \approx70.7\,\Omega \]
Thus:
\[ \boxed{ Z_2\approx70.7\,\Omega } \]
Finding \(Z_3\)
For the third discontinuity:
\[ \ln Z_3 = \ln Z_2+2\Gamma_2 \]
Since \(\Gamma_2=0.1037\):
\[ \ln Z_3 = 4.259+2(0.1037) \]
\[ \ln Z_3 = 4.259+0.2074 = 4.4664 \]
Therefore:
\[ Z_3=e^{4.4664} \approx87.0\,\Omega \]
Hence:
\[ \boxed{ Z_3\approx87.0\,\Omega } \]
Step 4: Calculate the Fractional Bandwidth
The fractional bandwidth of the Chebyshev transformer is related to the passband edge angle by:
\[ \boxed{ \mathrm{FBW} = \frac{\Delta f}{f_0} = 2-\frac{4\theta_m}{\pi} } \]
Using \(\theta_m=44.75^\circ\), the angle can be used directly with \(180^\circ\) in the denominator:
\[ \mathrm{FBW} = 2-\frac{4(44.75)}{180} \]
Therefore:
\[ \mathrm{FBW} = 2-\frac{179}{180} \]
\[ \mathrm{FBW} \approx2-0.994 \]
\[ \boxed{ \mathrm{FBW}\approx1.01 } \]
Expressed as a percentage:
\[ \boxed{ \mathrm{FBW}\approx101\% } \]
This result represents the bandwidth associated with the Chebyshev parameters calculated above. If the design specification is strictly \(71\%\), then \(\theta_m\) must instead be determined from that specified bandwidth, giving a different Chebyshev design parameter. Therefore, the \(44.75^\circ\) calculation corresponds to the \(0.05\) ripple and the stated impedance ratio under the approximation used here, rather than simultaneously enforcing a \(71\%\) bandwidth.
Final Three Section Transformer Design
Using the calculated reflection coefficients and the small-reflection approximation, the characteristic impedances of the three transformer sections are approximately:
\[ \boxed{ Z_1=57.5\,\Omega } \]
\[ \boxed{ Z_2=70.7\,\Omega } \]
\[ \boxed{ Z_3=87.0\,\Omega } \]
The complete impedance progression from the source to the load is therefore:
\[ \boxed{ 50\,\Omega \rightarrow 57.5\,\Omega \rightarrow 70.7\,\Omega \rightarrow 87.0\,\Omega \rightarrow 100\,\Omega } \]
fig: Three Section Chebyshev Transformer
Final Design Values
The three-section Chebyshev transformer is designed to transform the \(50\,\Omega\) source impedance gradually toward the \(100\,\Omega\) load impedance. The calculated reflection coefficients are \(0.0698\), \(0.1037\), \(0.1037\), and \(0.0698\), producing the required symmetric coefficient distribution for the three-section design. The corresponding characteristic impedances are approximately \(57.5\,\Omega\), \(70.7\,\Omega\), and \(87.0\,\Omega\).
The center frequency of the design is \(3\,\text{GHz}\). The electrical lengths of the transformer sections are selected according to the center-frequency design condition, while the characteristic impedances determine the impedance transformation between the \(50\,\Omega\) source and \(100\,\Omega\) load. The Chebyshev method provides an equal-ripple response in the passband, allowing a controlled maximum reflection coefficient rather than requiring zero reflection at every frequency.