Design Methodologies of LTCC Bandpass Filters, Diplexer, and ...

718 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 2, FEBRUARY 2006
Fig. 3.
filter.
Equivalent circuit of the proposed second-order combline bandpass
Fig. 5. Transformations for the matching circuit in the source and load ports.
(a) Positive J-inverter. (b) Negative J-inverter.
is resonance at the central frequency, which will result
in . The capacitor can be derived as
Fig. 4.
Equivalent circuit of the generalized bandpass filter.
where and are, respectively, the odd- and even-mode
admittances of coupled transmission line, and is the corresponding
electric length.
Adding two circuits shown in Fig. 2, and shunting with two
grounded capacitors at two ports separately yields the complete
equivalent circuit of the bandpass filter, as shown in Fig. 3.
Thus, we can apply the immittance inverter to analyze and design
bandpass filters. The admittance inverter and capacitor
are derived as
Following the equivalent circuit in Fig. 3, the equivalent circuit
of the generalized bandpass filter can also be expressed as
Fig. 4, which includes the admittance inverter. The admittance
inverters, susceptance, and its slope parameter are, respectively,
given by
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
where the ’s are the element values of the prototype lowpass
filter, is the fractional bandwidth, and and are
the impedances of source and load transmission lines, respectively
[19].
(11)
where and are electrical length and angular frequency at
the central frequency, respectively.
The bandpass filter with a transmission zero located at the
angular frequency will make . at the center
frequency and can be derived as
when
when
(12)
(13)
The matching circuits located at source and load ports,
as shown in Fig. 1, can be implemented using the following
methods. Note that the quarter-wavelength transmission line is
the simplest form of inverters and, on the other hand, the inductance
or capacitance -network to substitute the -inverter
is the other method. Since the negative inductance or capacitance
cannot be employed to the source or load impedance, a
transformation is needed, as shown in Fig. 5. These equations
are revealed as
III. FILTER DESIGN WITH TRANSMISSION ZERO
(14)
(15)
(16)
(17)
Here, two examples are presented to explain the location of
the frequency of the transmission zero, which will appear at the
lower or higher skirt of the passband.