**1. Introduction**

Integration plays an important role in many scientific and engineering applications. An integrator is an electronic circuit which produces the output that is the integral of the input applied. Electronic analogue integrators are the basis of analog computers and charge amplifiers, which are performed in the continuous time domain. The integrator is widely used in analog computers, analog-to-digital converters and wave-shaping circuits. Initially, an RC integrator is a circuit that approximates the mathematical process of integration. A simple R-C integrator circuit is shown in **Figure 1**, in which a capacitor (C) in series with a resistor (R) and the source (*Vin*). The output (*Vout*) of the circuit is taken across the capacitor (C).

Let i is the resulting current. Applying Kirchhoff's voltage law to the circuit,

$$V\_{in} = iR + \frac{1}{C} \int\_0^t i\,dt\tag{1}$$

$$V\_{out} = \frac{1}{C} \int\_0^t i\,dt\tag{2}$$

**Figure 1.** *R-C Integrator circuit.*

Multiplying throughout by C, we get

$$\text{CV}\_{in} = \text{iRC} + \int\_0^t i.d.t \tag{3}$$

ultra-wideband (frequency range 3.1–10.6 GHz) applications [3–5]. In the zdomain, Hsue et al. have introduced a first-order trapezoidal-rule microwave integrator using a chain-scattering transmission matrix with an operating frequency range from 1 to 10 GHz [6]. By cascading equal-length transmission line sections in the z-domain, three microwave integrators and differentiators have been designed and implemented with different time constants [7, 8]. The second-order microwave integrator (SOMI) was designed by Tsai et al. [9] in the z-domain. A further firstorder microwave integrator was designed by Gautam et al. using the ABCD transmission matrix with a bandwidth of 4 to 10 GHz [10]. SOMI was designed by Gupta et al. in the z-domain over the 1.5–5.5 GHz frequency range [11]. Another SOMI was designed by Gautam et al. in the z-domain over the frequency range 3–15 GHz [12]. Nowadays, analyzers are gravitating towards the use of population-based metaheuristic algorithms to optimize system coefficients in order to design complex or multi - modal systems [13–17]. This chapter introduces modern and compact SOMI designs that lead to wide bandwidth. These designed SOMIs are accomplished by cascading three transmission line sections and two single section stubs of equal length. By population-based meta heuristic algorithms, the optimum value of characteristic impedances of these line elements are obtained. A global cost-function solution is achieved by minimizing the error gap between the ideal second-order integrator (SOI) and the designed SOMIs. The design-1 SOMI approximates the ideal SOI over the 2.5 to 16 GHz frequency range, and the design-2 SOMI approximates the ideal SOI over the 3 to 15 GHz frequency range. These designed wideband SOMIs would operate with a wider frequency band to be used in mobile communication on a mobile network such as 4 G and 5 G (above 3 GHz) [18]. All the

*Analysis of Wideband Second-Order Microwave Integrators*

*DOI: http://dx.doi.org/10.5772/intechopen.94843*

simulated results are obtained by MATLAB and ADS. These simulated outcomes are

designed SOMIs exists in terms of wide bandwidth and miniaturization of hardware.

To realize second-order microwave integrator in microwave range, consider a

<sup>¼</sup> *<sup>S</sup>*<sup>11</sup> *<sup>S</sup>*<sup>12</sup> *<sup>S</sup>*<sup>21</sup> *<sup>S</sup>*<sup>22</sup> *<sup>a</sup>*<sup>1</sup>

<sup>¼</sup> *<sup>T</sup>*<sup>11</sup> *<sup>T</sup>*<sup>12</sup> *<sup>T</sup>*<sup>21</sup> T22 *<sup>b</sup>*<sup>2</sup>

where *a*<sup>1</sup> and *a*<sup>2</sup> are incident waves at port 1 and port 2, respectively, and *b*<sup>1</sup> and *b*<sup>2</sup> are reflected waves at port 1 and port 2, respectively [4]. The chain scattering matrix of a two-port network can be established from the scattering matrix (S-matrix). The chain-scattering matrix of two-port network is defined as [5].

*a*2

*a*2

(8)

(9)

formulated to be in close agreement with the ideal one. The novelty of these

**2. Problem formulation of SOMIs**

**Figure 2.** *Two-port network.*

**131**

two-port network, which is illustrated in **Figure 2**. Its scattering matrix is defined as [4].

> *b*1 *b*2

*a*1 *b*1 

as *RC* ≫ *t*, the term Ð*<sup>t</sup>* <sup>0</sup>*i:dt* may be neglected

$$\text{CV}\_{\text{in}} = \text{iRC} \tag{4}$$

Integrating with respect to *t* on both sides of Eq. (3)

$$\int\_{0}^{t} CV\_{in} = RC \int\_{0}^{t} i.d t \tag{5}$$

$$\frac{1}{C} \int\_{0}^{t} i.dt = \frac{1}{RC} \int\_{0}^{t} V\_{in} \, dt \tag{6}$$

From Eq. (2),

$$V\_{out} = \frac{1}{RC} \int\_0^t V\_{in} \, dt \tag{7}$$

Eq. (7) shows that the output of an integrator circuit is the integral of the input signal. These analog integrators are limited for low frequency application. Thus, the researcher moved to design digital integrators. Digital integrator is a system that performs mathematical operations on a sampled discrete time signal to reduce or enhance certain aspects of that signal. It is commonly used for applications such as waveform shaping, coherent detection, edge detection, and accumulator analysis in biomedical engineering and signal processing. It is widely utilized in biomedical engineering and signal processing applications, for example, as waveform shaping, coherent detection, edge detection, and accumulator analysis. It is also used in radar applications such as the allocation of mobile satellites, enterprise networks, commercial television services and digital services [1]. In order to design the wideband digital integrators, various methods were intended. Using the Newton-cotes integration rule and various digital integration techniques, the Recursive wideband digital integrators have been designed [2]. For low-speed applications up to barely a few hundred MHz, the integrators are primarily designed and implemented. Therefore, to cover wideband applications such as radar and wireless communication, the design and implementation of integrators for high-frequency applications is necessary. The microwave integrator is essentially used to measure the time integral of the input signal at microwave frequencies (0.3–300 GHz). Using wideband integrators, the high-frequency active filters can be introduced, and these wideband integrators can also be used for industrial and real-time applications for

## *Analysis of Wideband Second-Order Microwave Integrators DOI: http://dx.doi.org/10.5772/intechopen.94843*

ultra-wideband (frequency range 3.1–10.6 GHz) applications [3–5]. In the zdomain, Hsue et al. have introduced a first-order trapezoidal-rule microwave integrator using a chain-scattering transmission matrix with an operating frequency range from 1 to 10 GHz [6]. By cascading equal-length transmission line sections in the z-domain, three microwave integrators and differentiators have been designed and implemented with different time constants [7, 8]. The second-order microwave integrator (SOMI) was designed by Tsai et al. [9] in the z-domain. A further firstorder microwave integrator was designed by Gautam et al. using the ABCD transmission matrix with a bandwidth of 4 to 10 GHz [10]. SOMI was designed by Gupta et al. in the z-domain over the 1.5–5.5 GHz frequency range [11]. Another SOMI was designed by Gautam et al. in the z-domain over the frequency range 3–15 GHz [12]. Nowadays, analyzers are gravitating towards the use of population-based metaheuristic algorithms to optimize system coefficients in order to design complex or multi - modal systems [13–17]. This chapter introduces modern and compact SOMI designs that lead to wide bandwidth. These designed SOMIs are accomplished by cascading three transmission line sections and two single section stubs of equal length. By population-based meta heuristic algorithms, the optimum value of characteristic impedances of these line elements are obtained. A global cost-function solution is achieved by minimizing the error gap between the ideal second-order integrator (SOI) and the designed SOMIs. The design-1 SOMI approximates the ideal SOI over the 2.5 to 16 GHz frequency range, and the design-2 SOMI approximates the ideal SOI over the 3 to 15 GHz frequency range. These designed wideband SOMIs would operate with a wider frequency band to be used in mobile communication on a mobile network such as 4 G and 5 G (above 3 GHz) [18]. All the simulated results are obtained by MATLAB and ADS. These simulated outcomes are formulated to be in close agreement with the ideal one. The novelty of these designed SOMIs exists in terms of wide bandwidth and miniaturization of hardware.
