Chapter 4 Basic Balun Introduction
4.2 Marchand Balun
4.2.2 Chebyshev Model Establishment
The mathematical model should be established before the network synthesis. In general, Butterworth and Chebyshev polynomials are the common choices when we define the mathematical model for filter or balun design. In this section, Chebyshev approximation is used to synthesize the following second-order and third-order baluns.
The exact broadband synthesis theory [29, 30] is a well design method to analyze the network responses using in the network consisting entirely of equal wavelength lines, stubs and coupled lines. These are called commensurate lines. In next chapter, the complete design procedures of the novel fifth-order balun are presented.
Now, we have a short discussion about the Chebyshev model establishment of the second-order and third-order baluns by these following steps:
(1) An ideal balun is a three-port network which can separate the incident power into equal portions and 180-degree phase difference responses. For the two-port condition of exact synthesis method, balun must be reduced to the equivalent two-port network. For example, the two 50Ω output impedances of balun can be viewed as a series connection as a 100Ω load system.
(2) The Z-, Y-, and S-parameter representations are often used to characterize a microwave network with arbitrary number of ports. However, in many practical microwave networks consist of several cascade connection forms. In this case, by
multiplying ABCD-matrix of the individual two-port sub-networks is a convenient way to easily derive the whole network representation. The basic definition for a two-port network in terms of the total voltages and currents is given by
1 2 2 There are some useful transmission (ABCD-) parameters as shown in Table 4.1.
Z
Table 4.1 The ABCD parameters of some useful two-port circuits.
(3) According to Richard’s transformation definition, the distributed network composed of commensurate length of transmission-lines and load resistance can be treated as a lumped R-L-C network by using the complex frequency variable S for network analysis and synthesis. The complete exact synthesis method and Richard’s
transformation will be presented in chapter five.
The second-order model which is the minimum order of Marchand prototype balun consists of only a series open-circuited and a shunt short-circuited stub. A series open-circuited stub can be regarded as a series capacitor in Richard’s domain and a shunt short-circuited one is known as a shunt inductor. In the third-order model, a cascade transmission-line section is added behind the shunt short-circuited one.
From the Richard’s transformation theory, the corresponding transmission-line impedance transformations are introduced as follows [27].
0 , where Z denotes the distributed transmission-line impedance in real frequency 0 domain; L and C are the corresponding lumped elements in complex plane (Richard’s) domain; Z is the corresponding impedance of unit element. The equivalent UE circuits of second-order and third-order balun models are shown in Figure 4.5.
2
(a) Second-order prototype. (b) Third-order prototype.
(4) Due to the unmatched impedance property of the equivalent two-port network, the conversions between S-parameter and ABCD-parameter should be revised to the following equations.
02 01 02 01 To simplify the analysis, the impedance transformation ratio 02
01
Z
Z is supposed to be 2 in this procedure. Finally, the complete mathematical representation of squared magnitude of transmission coefficient parameter denoting as S212 can be derived by several minute and complicated calculations in Richard’s domain. It is a polynomial of the complex frequency variable S. Table 4.2 shows some useful ABCD-parameters in Richard’s domain.
Distubuted Circuit Lumped Circuit
1 1
Table 4.2 The ABCD parameters in Richard’s domain.
The squared magnitude of transmission coefficient of the second-order balun network can be expressed as
2
, where L and C are lumped element values in Richard’s domain. They represent a quarter-wavelength shunt short-circuited and series open-circuited stub in real frequency domain, respectively.
The third-order balun network can be defined as
2
, where Z is the characteristic impedance of unit element in Richard’s domain. It represents a cascade quarter-wavelength transmission-line in real frequency domain.
∆ is the fractional bandwidth (FBW) of the balun; F , 6' F , 4' F and 2' F are the 0' coefficients of S to the power of six, four, square and zero, respectively. They are functions of Z, L and C.
(5) For the reason that the equivalent two-port network of balun can be viewed as a bandpass filter response, so we choose a high-pass Chebyshev prototype in complex domain to model the desired design. The following two Chebyshev prototypes presented for second-order and third-order Marchand balun are shown in Figure 4.6
and 4.7. Specially, due to the special denominator form of ABCD-matrix of unit element, the Chebyshev polynomial is in need of some reasonable revisions. The method of mathematical function establishment is dilated in chapter five.
This is a second-order high-pass Chebyshev polynomial form shown below.
2
21 4 2 2 4
2 2 2
4
1 1
4 4
1 N( C) 1 ( C C)
S S S S S S
T S S
ε ε
= =
+ +
+ − +
(4.19)
, where ε is expressed as ripple-level of Chebyshev response and its related response diagram is shown in Figure 4.6.
(a)
(b)
Figure 4.6 Normalized Chebyshev second-order prototype.
(a) High-pass response. (b) Bandpass response.
In thick-line model, the ripple-level is ε =0.1922 and the normalized cutoff frequency in Richard’s domain is SC = j0.5 . Another thin-line model has
0.0934
ε = and SC = j1.
This is a third-order high-pass Chebyshev polynomial form [30]
2
21 2
1 1 N( )
S = F S
+ (4.20)
2 2 2
2 2
2 2
1 1
( ) 1 1
C C
C C
N m n m n
S S
S S
F S T T U U
S S S S
ε − −
= − − −
(4.21)
, where FN2( )S is revision type of original Chebyshev polynomial. The complete theory will be introduced later and its related response diagram is shown below.
(a)
(b)
Figure 4.7 Normalized Chebyshev third-order prototype.
(a) High-pass response. (b) Bandpass response.
Similar to the values of Second-order prototype, the ripple-level is ε =0.1922 and the normalized cutoff frequency in Richard’s domain is SC = j0.5 in thick-line model. Another thin-line model has ε =0.0934 and SC = j1. In general, the wider the bandwidth is, the worse the return loss is. Based on this simple trade-off concept, it gives a good sense for specific balun design.
The parameters of baluns such as impedance transformation ratio, bandwidth and return loss can be arbitrarily determined. The one or more reasonable numerical solutions will be obtained by solving these mathematical equations and discriminants.
They are the solutions which can exactly describe the presented network responses at all frequencies. Finally, the theoretical values of each quarter-wavelength section are well-defined in real frequency domain by converting the lumped element values.
The basic balun design procedures have been summarized. We will establish the new novel fifth-order Marchand balun prototype as same as the methods presented in next chapter.