The stepped impedance resonators (SIR), using alternating sections
of high- and low- characteristic impedance lines, are often used in bandpass
or lowpass filter design. Fig.1 shows a 4th order elliptic highpass
filter schematic which has cutoff frequency at 4 GHz. The filter
is synthesized with a filter synthesis program [1], by extracting elements
from a network impedance or admittance function . Four transmission
zeros are located at the stopband and one zero is placed at DC to make
an elliptic highpass filter. The passband bandwidth is determined
by the coupling capacitors and the parasitic elements of the filter networks.
In order to take account into parasitic inductance of the actual interconnection
of the series coupling capacitors, inductors are inserted into the coupling
capacitors in series, in which 0.3nH is used for the simulation [2].
The simulated response of the circuit is displayed in Fig.2.
In particular, the shunt L-C resonators can be approximated with electrically
short high- and low- impedance lines. These distributed elements
can be realized by TEM-mode transmission line such as microstrip, strip,
and coaxial line etc.
Fig.3 shows a conversion example using LC2COAX program based on the
equivalent impedance formula [3]. This program calculates equivalent
inductance, capacitance, step capacitance due to the discontinuity of the
coaxial lines.
Fig.4 displays a semi-lumped highpass filter. Here high-Z coaxial line represents the inductor of the shunt resonator, low-Z line represents the capacitor of the shunt resonator, and parallel plate capacitors are used for the coupling capacitors. However, upper passband has a limitation restricted by the electrical length of the coaxial line. After quarter-wavelength frequency, open-circuited lines behave as short circuits, and vise versa. Due to the periodic impedance of the distributed elements, the passband attenuation is noticed after 16 GHz in this design. In order to increase the passband width, a shorter transmission line is suggested. In addition, outer conductor diameter of the coaxial line should be chosen as small as possible to avoid the unwanted waveguide resonance in the coaxial line at higher frequencies [4].
The simulated response using distributed shunt resonators is shown in
Fig.5. Spurious responses are noticed after 16 GHz, due to the longest
transmission line of the shunt resonators (one eighth wavelength).
However, the highpsss filter response is maintained from low frequencies
up to 4 times cut-off frequency with the stepped impedance lines.
An example of such filtering is provided by RS
Microwave P/N 4137003, a highpass filter which displays the following
specifications:
Fig.1 Elliptic highpass filter schematic including parasitic inductance.
Fig.2 Simulated frequency response
of Fig.1.( Fc = 4GHz)
Fig.3 Conversion from a L-C shunt
resonator to Stepped impedance lines using a LC2COAX conversion program.[1]
Fig.4 Elliptic highpass filter using
coaxial stepped impedance lines.
Fig.5 Simulated frequency response
using coaxial stepped resonators.
Reference:
[1] N. Yildirim, Filpro, METU, Ankara, Turkey.
[2] Richard V. Snyder. "Filters with an almost constant stopband."
2000
MTT-S International Microwave Symposium Digest, Vol. III [MWSYM], pp.1645-1648.
[3] G. Matthaei, L. Young, E.M.T Jones, " Microwave Filters, Impedance-Matching
Network, and Coupling Structures", Artech House 1980.
[4] Ian Hunter, "Theory and Design of Microwave Filters", IEE Press
2001.