Chebyshev Polynomial transformer
Chebyshev Transformer Design: Polynomial, Equal Ripple Response and Derivation
Before designing a multisection Chebyshev transformer, it is important to understand the mathematical behavior of Chebyshev polynomials. These polynomials form the mathematical basis of the Chebyshev transformer because their equal ripple characteristic can be used to control the maximum reflection coefficient over a specified frequency band. Instead of attempting to make the reflection coefficient continuously decrease from its maximum value, the Chebyshev approach allows a controlled ripple in the passband while keeping the magnitude of reflection below a specified maximum value. This property makes the Chebyshev transformer particularly useful when a wider bandwidth is required with a specified limit on the allowable reflection.
Chebyshev Polynomial Review
Chebyshev polynomials are a sequence of polynomials represented by \(T_N(x)\), where \(N\) denotes the order of the polynomial. In transformer design, the Chebyshev polynomials of the first kind are used because they provide the required equal ripple behavior. The order of the polynomial is directly related to the number of sections used in the transformer. Higher order polynomials provide greater design flexibility and can be used to obtain a different relationship between bandwidth and the maximum allowable reflection coefficient.
The first few Chebyshev polynomials of the first kind are:
\[ \begin{aligned} T_0(x) &= 1 \\ T_1(x) &= x \\ T_2(x) &= 2x^2-1 \\ T_3(x) &= 4x^3-3x \\ T_4(x) &= 8x^4-8x^2+1 \end{aligned} \]
Higher order polynomials can be generated using the recurrence relationship:
\[ \boxed{ T_N(x)=2xT_{N-1}(x)-T_{N-2}(x) } \]
This recurrence relation is useful because it allows the required polynomial to be constructed without independently deriving every higher order expression. For a transformer having \(N\) sections, the corresponding \(N\)-th order Chebyshev polynomial is used to describe the desired frequency dependence of the reflection coefficient.
Equal Ripple Property of Chebyshev Polynomials
The most important characteristic of the Chebyshev polynomial for transformer design is its equal ripple property. Within the interval \(-1\leq x\leq1\), the polynomial remains bounded between \(-1\) and \(+1\). This behavior can be understood by representing \(x\) in terms of an angle:
\[ x=\cos\theta \]
With this substitution, the Chebyshev polynomial becomes:
\[ \boxed{ T_N(\cos\theta)=\cos(N\theta) } \]
Because the cosine function always lies between \(-1\) and \(+1\), the Chebyshev polynomial also satisfies:
\[ -1\leq T_N(x)\leq1 \qquad \text{for} \qquad -1\leq x\leq1 \]
The polynomial therefore oscillates between equal positive and negative peak values. These oscillations are called equal ripples. In a Chebyshev transformer, this mathematical property is transferred to the reflection coefficient. The reflection coefficient is allowed to vary within a controlled range in the passband, but its magnitude does not exceed the specified maximum value.
This is the fundamental difference between an equal-ripple Chebyshev design and a design that attempts to minimize reflection uniformly at every frequency. The Chebyshev approach deliberately distributes the reflection error across the passband so that the maximum error can be controlled.
Behavior Outside the Equal Ripple Region
The behavior of the Chebyshev polynomial changes when \(|x|>1\). In this region, the magnitude of \(T_N(x)\) increases rapidly as \(x\) moves away from the interval \([-1,1]\). This property is useful in transformer design because the frequency region outside the selected passband can be associated with the rapidly increasing portion of the Chebyshev polynomial.
For \(|x|>1\), the polynomial can be expressed using the hyperbolic cosine function. For \(x>1\):
\[ \boxed{ T_N(x)=\cosh\left[N\cosh^{-1}(x)\right] } \]
Therefore, the polynomial is bounded and oscillatory within the equal-ripple region but grows rapidly outside that region. This behavior provides the mathematical mechanism for defining a controlled passband and the corresponding reflection response of the transformer.
Mapping the Transformer Passband to the Chebyshev Polynomial
To apply the Chebyshev polynomial to a multisection transformer, the electrical length \(\theta\) is mapped onto the polynomial variable \(x\). The passband is defined between the lower electrical boundary \(\theta_m\) and the upper boundary \(\pi-\theta_m\). The required transformation is:
\[ \boxed{ x=\frac{\cos\theta}{\cos\theta_m} } \]
At the lower edge of the passband, \(\theta=\theta_m\), so:
\[ x= \frac{\cos\theta_m}{\cos\theta_m} =1 \]
At the upper edge of the passband, \(\theta=\pi-\theta_m\), and therefore:
\[ x= \frac{\cos(\pi-\theta_m)} {\cos\theta_m} = \frac{-\cos\theta_m}{\cos\theta_m} =-1 \]
Thus, the complete transformer passband is mapped onto the equal-ripple interval:
\[ -1\leq x\leq1 \]
This mapping is important because it allows the equal-ripple property of the Chebyshev polynomial to control the reflection coefficient over the desired transformer bandwidth.
Chebyshev Transformer Design
The objective of a Chebyshev multisection transformer is to match a load impedance \(Z_L\) to a source or transmission-line impedance \(Z_0\) over a specified frequency range while keeping the magnitude of the reflection coefficient below a specified maximum value. The transformer consists of multiple transmission-line sections, each having a selected characteristic impedance. The impedance of these sections is chosen so that the overall reflection coefficient follows a Chebyshev equal-ripple response.
For an \(N\)-section transformer, the desired reflection coefficient can be represented in the form:
\[ \boxed{ \Gamma(\theta) = A e^{-jN\theta} T_N\left(\sec\theta_m\cos\theta\right) } \]
where \(N\) is the number of transformer sections, \(A\) determines the maximum reflection coefficient in the passband, and \(\theta_m\) determines the passband edge. For the specified maximum passband reflection coefficient \(\Gamma_m\), the constant is selected as:
\[ \boxed{ A=\Gamma_m } \]
The factor \(e^{-jN\theta}\) represents the phase variation associated with the electrical length of the transformer sections, while the Chebyshev polynomial determines the required equal-ripple magnitude characteristic.
Finding the Passband Edge \(\theta_m\)
The next step is to determine the electrical angle \(\theta_m\) corresponding to the desired maximum passband reflection coefficient. At the passband extrema, the magnitude of the Chebyshev polynomial reaches unity within the equal-ripple region. Therefore, the maximum reflection magnitude in the passband is:
\[ \boxed{ |\Gamma|_{\max}=\Gamma_m } \]
At the center frequency or zero electrical length, the reflection coefficient associated with the direct impedance mismatch between the load and the main transmission line is:
\[ \boxed{ \Gamma(0) = \frac{Z_L-Z_0}{Z_L+Z_0} } \]
Using the Chebyshev response and the relationship between the mismatch at \(\theta=0\) and the passband ripple level, the passband edge is obtained from:
\[ \boxed{ \sec\theta_m = \cosh \left[ \frac{1}{N} \cosh^{-1} \left( \frac{1}{\Gamma_m} \left| \frac{Z_L-Z_0}{Z_L+Z_0} \right| \right) \right] } \]
This equation shows how the passband edge depends on the number of transformer sections, the maximum allowable reflection coefficient, and the impedance mismatch between \(Z_L\) and \(Z_0\). Increasing the number of sections generally provides greater control over the response, while reducing the allowable ripple requires a different electrical passband width for a given transformer order.
Fractional Bandwidth of the Chebyshev Transformer
Once \(\theta_m\) has been determined, the fractional bandwidth can be obtained from the relationship between the electrical angle and frequency. For the standard equal-ripple transformer formulation, the fractional bandwidth is:
\[ \boxed{ \mathrm{FBW} = \frac{\Delta f}{f_0} = 2-\frac{4\theta_m}{\pi} } \]
where \(\Delta f\) is the bandwidth and \(f_0\) is the center frequency. This expression connects the electrical passband defined by \(\theta_m\) with the frequency bandwidth of the transformer.
Finding the Individual Reflection Coefficients
After determining \(\theta_m\), the next step is to determine the individual reflection coefficients associated with the impedance discontinuities between adjacent transformer sections. These coefficients are obtained by expanding the Chebyshev polynomial into cosine terms and comparing the resulting coefficients with the frequency-domain expression for the transformer reflection coefficient.
For a three-section transformer, the third-order Chebyshev polynomial is:
\[ \boxed{ T_3(x)=4x^3-3x } \]
Substituting:
\[ x=\sec\theta_m\cos\theta \]
gives:
\[ T_3(\sec\theta_m\cos\theta) = 4\sec^3\theta_m\cos^3\theta - 3\sec\theta_m\cos\theta \]
Using the trigonometric identity:
\[ \cos3\theta=4\cos^3\theta-3\cos\theta \]
the polynomial can be expressed in terms of cosine harmonics. Comparing the resulting coefficients with the reflection coefficient representation gives the individual discontinuity coefficients.
For the three-section case, the coefficient relationships can be written as:
\[ \boxed{ 2\Gamma_0 = A\sec^3\theta_m } \]
and:
\[ \boxed{ 2\Gamma_1 = 3A \left( \sec^3\theta_m-\sec\theta_m \right) } \]
For a symmetric transformer, the remaining coefficients satisfy the corresponding symmetry relationships:
\[ \boxed{ \Gamma_2=\Gamma_1 } \]
\[ \boxed{ \Gamma_3=\Gamma_0 } \]
The symmetry of the reflection coefficients leads to a smooth and symmetric impedance progression from the input impedance toward the load impedance.
Converting Reflection Coefficients into Section Impedances
The reflection coefficient at each impedance discontinuity is related to the characteristic impedances of the adjacent sections. For small discontinuities, the relationship can be approximated by:
\[ \boxed{ \Gamma_n \approx \frac{1}{2} \ln \left( \frac{Z_{n+1}}{Z_n} \right) } \]
Rearranging this relationship gives:
\[ \ln \left( \frac{Z_{n+1}}{Z_n} \right) \approx 2\Gamma_n \]
Therefore:
\[ \boxed{ Z_{n+1} = Z_n e^{2\Gamma_n} } \]
This relationship allows the characteristic impedance of each transformer section to be calculated recursively. Starting with the source-side impedance \(Z_0\), the first section is obtained from \(\Gamma_0\):
\[ \boxed{ \ln Z_1 = \ln Z_0+2\Gamma_0 } \]
The next section is obtained using \(\Gamma_1\):
\[ \boxed{ \ln Z_2 = \ln Z_1+2\Gamma_1 } \]
Similarly:
\[ \boxed{ \ln Z_3 = \ln Z_2+2\Gamma_2 } \]
The process is continued until all \(N\) transformer section impedances have been determined. Equivalently, the recursive relationship can be written as:
\[ \boxed{ Z_{n+1}=Z_n e^{2\Gamma_n} } \]
This step converts the mathematically determined reflection coefficients into the physical characteristic impedances required for constructing the multisection transformer.
Three Section Chebyshev Transformer Design Example
Consider the design of a three-section Chebyshev transformer used to match a load impedance of:
\[ \boxed{ Z_L=100\,\Omega } \]
to a transmission line having characteristic impedance:
\[ \boxed{ Z_0=50\,\Omega } \]
Let the number of transformer sections be:
\[ \boxed{ N=3 } \]
and let the maximum allowable passband reflection coefficient be:
\[ \boxed{ \Gamma_m=0.05 } \]
The direct load-to-line reflection coefficient is:
\[ \Gamma(0) = \frac{100-50}{100+50} = \frac{50}{150} = 0.3333 \]
Using the Chebyshev passband relationship for \(N=3\) and \(\Gamma_m=0.05\), the passband edge is obtained as approximately:
\[ \boxed{ \theta_m\approx44.75^\circ } \]
The corresponding fractional bandwidth is approximately:
\[ \boxed{ \mathrm{FBW}\approx101\% } \]
The individual reflection coefficients obtained from the Chebyshev polynomial expansion are approximately:
\[ \boxed{ \Gamma_0=0.0698 } \]
\[ \boxed{ \Gamma_1=0.1037 } \]
For the symmetric three-section transformer:
\[ \boxed{ \Gamma_2=\Gamma_1 } \]
Using the recursive relationship between the reflection coefficients and section impedances, the characteristic impedances are approximately:
\[ \boxed{ Z_1=57.5\,\Omega } \]
\[ \boxed{ Z_2=70.7\,\Omega } \]
\[ \boxed{ Z_3=87.0\,\Omega } \]
Therefore, the impedance progression of the transformer is approximately:
\[ 50\,\Omega \rightarrow 57.5\,\Omega \rightarrow 70.7\,\Omega \rightarrow 87.0\,\Omega \rightarrow 100\,\Omega \]
fig: Final Circuit Diagram
Key Design Points of the Chebyshev Transformer
The Chebyshev transformer uses the equal-ripple property of the Chebyshev polynomial to distribute the reflection coefficient over the desired passband. The normalized polynomial variable is mapped to the electrical angle of the transformer so that the passband corresponds to the interval \(-1\leq x\leq1\). The maximum reflection coefficient is controlled by the specified ripple level \(\Gamma_m\), while the number of sections determines the order of the Chebyshev polynomial used in the design.
The overall design procedure therefore consists of determining the passband edge \(\theta_m\), calculating the fractional bandwidth, expanding the appropriate Chebyshev polynomial to obtain the individual reflection coefficients, and finally converting those reflection coefficients into the characteristic impedances of the transformer sections. The resulting multisection transformer provides a controlled equal-ripple reflection response while transforming the source impedance \(Z_0\) to the load impedance \(Z_L\).