# Microwave Engineering Questions and Answers – Chebyshev Multi-section Matching Transformers

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This set of Microwave Engineering Questions and Answers for Experienced people focuses on “Chebyshev Multi-Section Matching Transformers”.

1. The major disadvantage of binomial multi section transformer is higher bandwidth cannot be achieved.
a) true
b) false

Explanation: In some applications, a flat curve in the operating frequency is a major requirement. This requirement can be satisfied using a binomial transformer. But the disadvantage is that a higher bandwidth can be achieved.

2. Advantage of chebyshev matching transformers over binomial transformers is:
a) higher gain
b) low power losses
c) higher roll-off in the characteristic curve
d) higher bandwidth

Explanation: Chebyshev transformers when designed to operate at a certain frequency called center frequency, the reflection co-efficient is low for a large frequency range implying that they have a higher operating range. This is the major advantage of chebyshev filters.

3. There are passband ripples present in the chebyshev characteristic curve.
a) true
b) false

Explanation: This is a major difference between chebyshev and binomial transformer. Binomial transformers have a flat curve in the passband while chebyshev transformers have ripples in the transformer passband.

4. Chebyshev matching transformers can be universally used for impedance matching in any of the microwave networks.
a) true
b) false

Explanation: Chebyshev transformers have passband ripples in the characteristic curve. In some critical applications, these ripples are not tolerable in the operating bandwidth. Hence, chebyshev transformers cannot be used for all the microwave networks for impedance matching.

5. The 4th order chebyshev polynomial is:
a) 8x4-8x2+1
b) 4x3-4x2+1
c) 4x3-3x
d) none of the mentioned

Explanation: nth order polynomial for a chebyshev polynomial is generated using lower polynomials by the expression Tn (x) = 2xTn-1(x) – Tn-2(x). T2(x) = 2x2-1, T3(x)= 4x3-3x. Substituting the lower level polynomials in the given expression, T4(x) = 8x4-8x2+1.

6. Chebyshev polynomials do not obey the equal-ripple property.
a) true
b) false

Explanation: For -1≤x≤1,│T(x)│≤ 1. In this range, the chebyshev polynomials oscillate between±1. This is the equal ripple property. Chebyshev polynomials obey the equal-ripple property.

7. Chebyshev polynomial can be expressed in trigonometric functions as:
a) Tn(cos θ)=cos nθ
b) Tn(sin θ)= sin nθ
c) Tn(cos θ)=cos nθ.sin nθ
d) none of the mentioned

Explanation: If the chebyshev polynomial variable x is equated to a trigonometric variable cos θ, then the higher order chebyshev polynomials can be defined in terms of the same function with multiples of θ. This can be theoretically proved and function generation becomes simpler.

8. For values of x greater than 1, the chebyshev polynomial in its trigonometric form cannot be determined.
a) true
b) false

Explanation: Since cosine function is defined for values of x between -1 and +1, for x values greater than 1, hyperbolic function is used to define the chebyshev polynomial. Tn(x)=cosh (n cosh-1x).

9. Reflection co-efficient Гn in terms of Zn and Zn+1, successive impedances of successive sections in the matching network are:
a) 0.5 ln (Zn+1/Zn)
b) 0.5 ln (Zn/Zn+1)
c) ln (Zn+1/Zn)
d) ln (Zn/Zn+1)

Explanation: When multiple sections are used in the chebyshev matching network, the reflection co-efficient of the nth matching section, given the impedances at the ends of the section, reflection co-efficient can be obtained using the expression 0.5 ln (Zn+1/Zn).

10. In a 3 section multisection chebyshev matching network, if Z3 = 100Ω, and Z2=50Ω, then the reflection co-efficient Г2 is:
a) 0.154
b) 0.3465
c) 0.564
d) none of the mentioned

Explanation: Гn for ‘n’ section matching chebyshev network is given by Гn=0.5 ln (Zn+1/Zn). substituting the given values in the expression, Г2 is 0.3465.

11. If Г3=0.2 and Z3=50Ω, then the impedance of the next stage in the multi-section transformer is:
a) 100Ω
b) 50Ω
c) 74.6Ω
d) 22.3Ω