Chapter 3 Microwave Filter Theorem
3.6 Impedance and Admittance Inverters
It is often to use only series, or only shunt elements when implementing a filter with a particular type of transmission line. It is possible to use impedance or admittance inverters[5]. Such inverters are useful for implementing a band-pass or band-stop filter with narrow (<10%) bandwidth.
Since these inverters form the inverse of the load impedance or admittance, they can be used to transform series-connected elements to shunt-connected elements, or vice versa.
In its simplest form, a J or K inverter can be constructed using a quarter-wave transformer of the appropriate characteristic impedance, as shown in figure 3.6(b).
Figure 3.6 Impedance and admittance inverters (a)Operation of impedance and admittance inverters (b)Impedance as quarter-wave transformation
Series LC resonators and parallel LC resonators are used in band-pass and band-stop prototype filter. The resonance frequency
LC 1
0=
ω of the resonator is not enough to describe the characteristics of the resonator. We define the susceptance slope parameter
j
As for the series LC resonator, we define the reactance slope parameter
k
s
From equation (81), we may get L=C. If we would like to change the relation between L and C, then K will be changed as well.
Figure 3.8 Impedance inverter
Compare the equation (82) and (83), we may get
C1
K = L (84)
The analysis of J converter is similar to that of K converter. We may get
p
If we would like to change the relation between L and C, then J will be changed as well.
L1
J = C (86)
We can make use of this feature to convert the filter to all series circuits or all shunt
circuits. From below figure, we may easily change the component value of the series resonators or the parallel resonators.
Figure 3.9 Impedance and admittance inverters
Figure 3.10 Impedance and admittance inverters
The slope parameter will be changed once K or J is changed. Since the left circuit and the right circuit are equivalent in the below figure, the susceptance of the two circuits will be equivalent. We can get
2 0 2 1
K x K
x =
′ (93)
1 0
1 0
L K L x K x
K = ′⋅ = ′⋅ (94)
Figure 3.11 Impedance scaling of the K inverter
Similarly, the reactance of the left circuit and the right circuit in the below figure are equivalent.
2 0 2 1
J b J
b =
′ (95)
1 0
1 0
C J C b J b
J = ′⋅ = ′⋅ (96)
Figure 3.12 Admittance scaling of the J inverter
If the value of K in the left circuit is not the same as the value of K in the right circuit, we can still get the relationship in figure 3.13.
Figure 3.13 Impedance scaling of the K inverter
1
We can get similar relationship as below for J converter:
1
Quarter-wavelength sections of line between the stubs act as admittance inverters to effectively convert alternative shunt resonators to series resonators. The stub and the transmission line sections are λ 4 long at the center frequency ω0.
For narrow bandwidths the response of such a filter using N stubs is essentially the same as that of a coupled line filter using N+1 sections. The internal impedance of the stub filter is Z , while in the case of the coupled line filter end sections are required to 0 transform the impedance level. This makes the stub filter more compact and easier to design.
Figure 3.14 Band-stop and band-pass filters using shunt transmission line resonators (θ =π 2 at the center frequency) (a)Band-stop filter (b)Band-pass filter
Consider a band-stop filter using N open-circuited stubs, the design equations for the required stub characteristic impedances, Z0n, will be derived in terms of the element values of a low-pass filter prototype through the use of the equivalent circuit. The analysis of band-pass filter using short-circuited stubs follows the same procedure.
An open-circuited stubs can be approximated as a series LC resonator when its length is near 90 . The input impedance of an open-circuited transmission line of characteristic ° impedance Z0n is
cotθ
for frequency in the vicinity of the center frequency ω0. The impedance of a series LC circuit is
( )
resonator parameters:π
ω n
n
Z0 4 0L
= (104)
If we consider the quarter-wave sections of line between the stubs as ideal admittance inverters, the band-stop filter can be represented by the equivalent circuit of figure 3.15(b). The circuit elements of the equivalent circuit can be related to those of the lumped-element band-stop filter prototype of figure 3.15(c).
Figure 3.15 Equivalent circuit for the band-stop filter. (a)Equivalent circuit of open-circuited stub for θ near π 2 (b)Equivalent filter circuit using resonators and admittance inverter (c)Equivalent lumped-element band-stop filter
With reference to figure 3.15(b), the admittance, Y, seen looking toward the L2C2 resonator is
( ) ( )
The admittance at the corresponding point in the circuit of figure 3.15(c) is
( ) ( )
These two results will be equivalent if the following conditions are satisfied:
1
We can get impedances of a band-stop filter is
∆
The characteristic impedances of a band-pass filter with short-circuited stub resonators is cannot be used for equal-ripple design for N even.
Chapter 4 Circuit Implementation 4.1 Chebyshev Low-pass Filter
We can get below parameters for Chebyshev low-pass prototype filter.
0 =1
L (0.0875 dB ripple at the pass-band), and the value of g elements of low-pass prototype filter are as below.
Table 4.1 The elements of low-pass prototype filter of return loss 17dB
g0 g1 g2 g3
1 0.8112 0.6104 1.3290
If we design f0 =2.5GHz, ∆=0.15 , we can get the lumped elements of the band-pass filter as below.
Figure 4.1 Band-pass filter of return loss 17dB, center frequency 2.5GHz, fractional bandwidth 15%
Figure 4.2 Simulated insertion loss and return loss of the band-pass filter in figure 4.1
We can make use of equation (88) to transfer the series LC into the inverters and shunt LC.
Figure 4.3 Impedance and admittance inverters
From equation (96), we can change the impedance of the J inverters. We can also change the load impedance to 50 Ohm.
Figure 4.4 Admittance scaling of the J inverter
The equivalent circuit is as below.
Table 4.2 The elements of low-pass prototype filter of return loss 17dB
RL Z1 Cpbp Lpbp Z2
21.6191 Ohm 45.7788 Ohm 3.5515pF 1.1412nH 84.0868 Ohm
Figure 4.5 Band-pass filter of return loss 17dB, center frequency 2.5GHz, fractional bandwidth 15%
Figure 4.6 Simulated insertion loss and return loss of the band-pass filter in figure 4.5
From figure 3.3(c) and equation (44), λ 4 coupled line can be taken as a transmission
line of impedance ( )
o e
i Z Z
Z 0 0
2
1 −
= . When we design the coupled line with line width w 10= mil and line spacing s=8mil , we can get Z0e =130.686Ω andZ0o =64.9301Ω, and it can be taken as a transmission line of impedance
(
−)
= Ω= 32.8780
2 1
0
0e o
i Z Z
Z if the trace length is λ 4 at the operation frequency. If the load impedance of the coupled line is 50 Ohm, the input impedance is
Ω
=
=
= 21.6192
50 8780 .
32 2
2 0
L
in Z
Z Z .
We can use the equivalent short-circuited stub to implement the shunt LC resonator.
The input admittance of the short-circuited stub can be approximated as
( ) equation (126) and equation (127).
( ) ( )
Figure 4.7 Band-pass filter of return loss 17dB, center frequency 2.5GHz, fractional bandwidth 15%
Figure 4.8 Simulated insertion loss and return loss the band-pass filter in figure 4.7
We use diodes to switch the mode of the filter. Since the forward resistance of the diode
is not 0, then = ≠∞
L
in Z
Z Z
2
0 . Therefore, we may take diode parameters into
consideration and increase impedance to get better performance. Firstly, we consider the status of diode on as in figure 4.9.
Figure 4.9 Band-pass filter of return loss 17dB, center frequency 2.5GHz, fractional bandwidth 15%
Figure 4.10 Simulated insertion loss and return loss of the band-pass filter in figure 4.9
As for diode on, we need to consider the effect of blocking capacitor for even mode.
Figure 4.11 Band-pass filter of return loss 17dB, center frequency 2.5GHz, fractional bandwidth 15%
Figure 4.12 Simulated insertion loss and return loss of the band-pass filter in figure 4.11
4.3 Band-stop Filter
Figure 2.9 shows a band-stop filter. Since the parameters of the band-pass are fixed, we cannot change the band-stop filter design. We just need to find out a set of parameters to achieve the performance. The two pairs of coupled line in the middle only contribute limited performance variance to the band-pass filters. We may change the electrical length of the coupled lines as below. The return loss of the band-pass filter will degrade from 17dB to about 15dB, but we may get better bandwidth of the band-stop filter.
Figure 4.13 Band-stop filter under diode off, even mode, with insertion loss >35dB
Figure 4.14 Simulated insertion loss and return loss of the band-stop filter in figure 4.13
Figure 4.15 Band-pass filter under diode on, even mode, with return loss >15dB
Figure 4.16 Simulated insertion loss and return loss of the band-pass filter in figure 4.15
Figure 4.17 Band-pass filter under diode off, odd mode, with return loss >15dB
Figure 4.18 Simulated insertion loss and return loss of the band-pass filter in figure 4.17
4.4 All-stop Filter
Below is the equivalent circuit of all-stop filter.
Figure 4.19 All-stop filter
Ω
= Ω Ω
= 0
//50 0
2 3 2
Zin Z (130)
Ω
= Ω
=0 // 0
2
Z2
Z
50 1 At the center frequency, all the incident power will be reflected. At the other frequency, the electrical length of each line segment will not be 90 degrees.
θ
If the circuit is symmetric, all of the incident power will be reflected by the short circuit.
In other word, it can be taken as an all-stop filter.
If we take capacitor and diode effect into consideration, the all-stop filter is as below, and the performance will be degraded as well.
Figure 4.20 All-stop filter under diode on, odd mode, with insertion loss >35dB
Figure 4.21 Simulated insertion loss and return loss of the all-pass filter in figure 4.20
Chapter 5 Fabrication and Measurements
5.1 Circuit Fabrication
Figure 5.1 illustrates the photograph of the dual mode switch filter. Figure 5.2 and 5.3 illustrate the insertion loss and return loss measurement result. There is frequency shift effect, which will result in bandwidth degradation. The bandwidth(pass-band insertion loss <3dB, stop-band insertion loss >32.685dB, pass-band return loss >13.843dB, stop-band return loss <3dB) is 2.445GHz~2.636GHz, which is about 7.64% fractional bandwidth.
Figure 5.1 Photograph of the dual mode switch filter
Figure 5.2 Measured insertion loss of dual mode switch filter
filters are frequency up-shifted. The return loss measurement of the all-stop filter and band-stop filter match the 3D EM simulation. The insertion loss measurement of the band-pass filters, band-stop filter, and all-stop filter also match the 3D EM simulation.
Figure 5.4 Measurement and 3D EM simulation correlation under diode on / odd mode
Figure 5.5 Measurement and 3D EM simulation correlation under diode on / even mode
Figure 5.6 Measurement and 3D EM simulation correlation under diode off / odd mode
will impact the performance of the dual-mode switch filter. From the simulation result, the length lc2 of the right half of the coupled line will impact the filter performance a lot. It is a critical to fine tune lc2 length in manufacturing.
The circuit miniaturization is a topic of future works. The DC blocking coupled line plays a role of DC blocking and load impedance transformation. Theoretically, we may combine the DC blocking coupled line and the filter section 1. We need 3 sections of coupled lines to implement the second order band-pass filter. However, we may still need to fine tune the layout and check if there is any side effect in 3D EM simulation or circuit fabrication.
References
[1] David M. Pozar, Microwave Engineering, 3’rd Edition, John Wiley & Sons, N.Y., 2005.
[2] Christophe Caloz, and Tatsuo Itoh, Electromagnetic Metamaterials, John Wiley & Sons, 2005.
[3] Hsin-Chia Lu and Tah-Hsiung Chu, “Multiport Scattering Matrix Using a Reduced-Port Network Analyzer“, IEEE Transactions on Microwave Theory and Techniques, Vol.51, No.5, May 2003
[4] Rivlin T.J., Chebyshev Polynomials, John Wiley & Sons, N.Y., 1999.
[5] George L. Matthaei, Leo Young, and E. M. T. Jones, Microwave Filters, Impedance- Matching Networks, and Coupling Structures, Artech House, M.A., 1980.