3. Microwave foundation of designing UWB bandpass filters

#### 3.1 Performance specifications

The performance is a critical factor for UWB filters even in any engineering device, and the major parameters for UWB filter performance evaluating are as follows:

1. Insertion loss (IL): insertion loss is the attenuation caused by the introduction of the device between the in port and out port, usually expressed in dB. The insertion loss can be calculated as follows:

$$IL = 10\log\frac{P\_{in}}{P\_{out}}\ (\text{dB})\tag{4}$$

where Pin is the input transmitted power and the Pout is the output received power. In addition to the mismatching loss, the actual bandpass filters have a series of other losses. Firstly, dielectric loss, can expressed as

$$a\_d' = \text{27.3} \frac{\tan \delta}{\lambda\_{\text{g}}} \text{ dB/cm} \tag{5}$$

adjust the resonant modes to fulfill the design specifications of UWB bandpass filter. For the purpose of establishing the expression of the resonant frequencies and each electrical lengths or impedances, the Yin of UWB filter needs to be derived, and the resonance condition can be calculated by using the following

As demonstrated in Figure 4, the detailed steps for solving resonant frequencies

Step 2, Recalculating the electrical lengths. When the frequency fi is considered,

Step 3, if Eq. (1) is satisfied, that is, f<sup>1</sup> = fi is the first resonant frequency that we are searching for, then the resonant frequency f<sup>1</sup> should be saved and turn to the next step. If Eq. (1) is not satisfied, the program turns to the next step directly.

Step 5, is the new value of fi++ beyond the frequency sweep range? If the answer

is yes, then quit and end the program. If the answer is no, then go to step 2.

Step 1, Initialization. The electrical lengths θ<sup>i</sup> (i = 1, 2, …, n), f<sup>0</sup> (reference frequency for electrical length calculation), and frequency sweep range should be

Im ð Þ¼ Yin 0 (11)

<sup>0</sup> ¼ θnfi

=f <sup>0</sup>. Then, we

expression:

Review on UWB Bandpass Filters

given.

Figure 4.

83

Flow chart of solving the resonate frequency.

with numerical calculation are as follows:

DOI: http://dx.doi.org/10.5772/intechopen.87204

all of the electrical lengths should be recalculated as θ<sup>n</sup>

Step 4, considering the next frequency fi++.

substitute the updated electrical lengths into Eqs. (2)–(8).

where the λ<sup>g</sup> is the guide wavelength of 50 Ω microstrip line at frequency f. Secondly, the conductor loss, can be derived from

$$a\_a = \frac{\sqrt{\pi \oint \mu\_0 \sigma}}{4(w+t)\sigma \cdot Z\_0} \tag{6}$$

where μ<sup>0</sup> is the permeability of vacuum, σ is the conductivity, w is the width of conductor, and t is the conductor thickness.

Thirdly, the dielectric loss, can be written as

$$a\_d = \tan \delta \cdot \pi / \lambda\_{\text{g}} \tag{7}$$

2. Return loss (RL): return loss is the ratio of the reflected power to the incident wave power, expressed in dB:

$$RL = 10\log\frac{P\_{\text{ref}}}{P\_{\text{in}}}\,\text{(dB)}\tag{8}$$


$$ROR = \frac{|\delta\_{-20dB} - \delta\_{-3dB}|}{|f\_{-20dB} - f\_{-3dB}|} \tag{9}$$

where δ�20 dB and δ�3 dB are attenuation point at �20 and �3 dB, respectively. f�20 dB and f�3 dB are, respectively, �20 and �3 dB stopband frequency.

5. Group delay: the ratio of phase variation to frequency variation is utilized to describe the overall delay of signal though the device. The group delay can be derive as

$$
\pi = -\frac{\partial \rho\_{21}(\boldsymbol{\alpha})}{\partial \boldsymbol{\alpha}} \tag{10}
$$


#### 3.2 Foundation of conventional transmission line filter analysis

The critical step in the design of a conventional transmission line UWB filter is to select the appropriate electrical lengths/impedances of transmission lines to

α0

α<sup>a</sup> ¼

Secondly, the conductor loss, can be derived from

conductor, and t is the conductor thickness. Thirdly, the dielectric loss, can be written as

UWB Technology - Circuits and Systems

wave power, expressed in dB:

3. FBW and center frequency

derive as

82

filters with sharp shirt.

selectivity and can be defined as follows:

<sup>d</sup> <sup>¼</sup> <sup>27</sup>:<sup>3</sup> tan <sup>δ</sup> λg

where the λ<sup>g</sup> is the guide wavelength of 50 Ω microstrip line at frequency f.

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup><sup>f</sup> <sup>μ</sup>0<sup>σ</sup> <sup>p</sup> 4ð Þ w þ t σ � Z<sup>0</sup>

where μ<sup>0</sup> is the permeability of vacuum, σ is the conductivity, w is the width of

2. Return loss (RL): return loss is the ratio of the reflected power to the incident

Pin

4.Roll-off rate (ROR): the ROR is a critical specification for evaluating passband

ROR <sup>¼</sup> <sup>∣</sup>δ�20dB � <sup>δ</sup>�3dB<sup>∣</sup>

where δ�20 dB and δ�3 dB are attenuation point at �20 and �3 dB, respectively.

5. Group delay: the ratio of phase variation to frequency variation is utilized to describe the overall delay of signal though the device. The group delay can be

<sup>τ</sup> ¼ � <sup>∂</sup>φ21ð Þ <sup>ω</sup>

6.Out-of-band suppression level: the stopband suppression level is applied to evaluate the out-of-band performance of the UWB bandpass filter.

7. Upper stopband bandwidth: it is worth noting that there is no spike in the stop band.

8.Transmission poles: multi-transmission poles prone to achieve UWB bandpass

9.Transmission zeros: the UWB bandpass filter with multi-transmission zeros tends to process excellent out-of-band rejection and high selectivity of passband.

The critical step in the design of a conventional transmission line UWB filter is to select the appropriate electrical lengths/impedances of transmission lines to

3.2 Foundation of conventional transmission line filter analysis

f�20 dB and f�3 dB are, respectively, �20 and �3 dB stopband frequency.

RL <sup>¼</sup> 10 log Pre

dB=cm (5)

α<sup>d</sup> ¼ tan δ � π=λ<sup>g</sup> (7)

ð Þ dB (8)

<sup>∣</sup> <sup>f</sup> �20dB � <sup>f</sup> �3dB<sup>∣</sup> (9)

<sup>∂</sup><sup>ω</sup> (10)

(6)

adjust the resonant modes to fulfill the design specifications of UWB bandpass filter. For the purpose of establishing the expression of the resonant frequencies and each electrical lengths or impedances, the Yin of UWB filter needs to be derived, and the resonance condition can be calculated by using the following expression:

$$\text{Im } (Y\_{\text{in}}) = \mathbf{0} \tag{11}$$

As demonstrated in Figure 4, the detailed steps for solving resonant frequencies with numerical calculation are as follows:

Step 1, Initialization. The electrical lengths θ<sup>i</sup> (i = 1, 2, …, n), f<sup>0</sup> (reference frequency for electrical length calculation), and frequency sweep range should be given.

Step 2, Recalculating the electrical lengths. When the frequency fi is considered, all of the electrical lengths should be recalculated as θ<sup>n</sup> <sup>0</sup> ¼ θnfi =f <sup>0</sup>. Then, we substitute the updated electrical lengths into Eqs. (2)–(8).

Step 3, if Eq. (1) is satisfied, that is, f<sup>1</sup> = fi is the first resonant frequency that we are searching for, then the resonant frequency f<sup>1</sup> should be saved and turn to the next step. If Eq. (1) is not satisfied, the program turns to the next step directly.

Step 4, considering the next frequency fi++.

Step 5, is the new value of fi++ beyond the frequency sweep range? If the answer is yes, then quit and end the program. If the answer is no, then go to step 2.

Figure 4. Flow chart of solving the resonate frequency.

## 3.2.1 Classical even-odd-mode analysis method

Since the odd-mode resonant frequencies of the symmetrical structure are orthogonal to the even-mode resonant frequencies, the whole transmission line model can be divided into odd-mode and even-mode circuits. Therefore, the resonant frequencies are then derived separately, which dramatically reduces the computation of resonant modes. It is worth noting that with odd-mode excitation, the symmetrical planes are considered to be grounded and with even-mode excitation, the symmetrical planes are considered to be open.

Even-/odd-mode input admittance can be obtained from the even�/odd-mode equivalent circuit, and Eq. (11) can be replaced by the following equations:

$$\text{Im}\left(Y\_{\text{ine}}\right) = \mathbf{0} \tag{12}$$

$$\text{Im}\left(Y\_{\text{ino}}\right) = \mathbf{0} \tag{13}$$

U<sup>1</sup> U<sup>2</sup> U<sup>3</sup> U<sup>4</sup>

Z<sup>11</sup> ¼ Z<sup>22</sup> ¼ Z<sup>33</sup> ¼ Z<sup>44</sup> ¼ �j

Z<sup>12</sup> ¼ Z<sup>21</sup> ¼ Z<sup>34</sup> ¼ Z<sup>43</sup> ¼ �j

Z<sup>13</sup> ¼ Z<sup>31</sup> ¼ Z<sup>24</sup> ¼ Z<sup>42</sup> ¼ �j

Z<sup>14</sup> ¼ Z<sup>41</sup> ¼ Z<sup>23</sup> ¼ Z<sup>32</sup> ¼ �j

and ultra-wide stopband of the UWB bandpass filter (Figure 9).

Z<sup>11</sup> Z<sup>12</sup> Z<sup>13</sup> Z<sup>14</sup> Z<sup>21</sup> Z<sup>22</sup> Z<sup>23</sup> Z<sup>24</sup> Z<sup>31</sup> Z<sup>32</sup> Z<sup>33</sup> Z<sup>34</sup> Z<sup>41</sup> Z<sup>42</sup> Z<sup>43</sup> Z<sup>44</sup>

A quintuple-mode resonator is proposed to design UWB bandpass filter, and the physical layout of the presented UWB filter is sketched in Figure 7 [19]. Since the whole structure is symmetrical along the T–T' line, classical odd-even-mode method is adopted to analyze the quintuple-mode resonator. As demonstrated in Figure 8, five resonant modes can be generated by quintuple-mode resonator; besides, owing to the loaded stub, two transmission zeros are realized both at lower and upper cutoff frequencies; thus, high selectivity is approached. As shown in Figure 9, the measurement results are in good agreement which shows sharp skirt

As illustrated in Figure 10, dual short stub-loaded resonator is presented to construct UWB transmission characteristics [31]. Owing to symmetrical structure

Z0<sup>e</sup> þ Z0<sup>o</sup>

Z0<sup>e</sup> � Z0<sup>o</sup>

Z0<sup>e</sup> � Z0<sup>o</sup>

Z0<sup>e</sup> þ Z0<sup>o</sup>

I1 I2 I3 I4

<sup>2</sup> cot <sup>θ</sup> (15)

<sup>2</sup> cot <sup>θ</sup> (16)

<sup>2</sup> csc<sup>θ</sup> (17)

<sup>2</sup> csc<sup>θ</sup> (18)

(14)

Electrical diagram of parallel-coupled line.

Review on UWB Bandpass Filters

DOI: http://dx.doi.org/10.5772/intechopen.87204

4. Common UWB bandpass filters

4.1 UWB bandpass filters using MMR

4.2 UWB bandpass filters using SLMMR

where

85

Figure 6.
