Keywords

1 Introduction

Current-mode technique has been received commendable attention, as they offer following advantages: (i) high slew-rate, (ii) low-power consumption, (iii) operating frequency range is superior, (iv) enhanced accuracy and linearity. Interest in design of current-mode (CM) [1, 2] filters has developed, but for operation at higher frequencies (in MHz range) [3, 4], the main challenges faced during the design of analog filters are (i) consistent high-frequency performance, (ii) automatic on-chip availability of tuning and (iii) varying operating conditions.

Thus, for high-frequency applications, the output must be current, and it should be given as

$$I_{\text{out}} = g_{\text{m}} *V_{\text{in}}$$
(1)

where gm is the transconductance of the active analog building block. For application as continuous-time filters, the transconductance (gm) of the circuit should meet the following properties: linear, simpler and have wide frequency response, should have high output and input impedance to simplify circuit design, operate at low-voltage preserve power and to make it compatible with the digital technology. The gm depends on DC bias voltage (VB) or current (IB) to make circuits electronically tunable against the environmental variation.

Depending on the choice of the technology, the operating frequency range of the gm circuits [3] can be extended to be greater than 50 MHz (for CMOS technology), greater than 500 MHz (for bipolar technology) or even greater than 1 GHz (for GaAs technology) for designing high-frequency continuous-time telecommunication circuits.

As capacitors and the transconductors are the only components required for realizing filters, gm-C [5, 6] filters can readily be implemented in fully integrated form [7], with desired technology. For active simulations, we consider identical transconductors and grounded capacitors for simple layout and processing. In addition to this, the large bandwidth of transconductance and reduced parasitic effects of device and circuit on filter performance results in higher working frequency range of the circuits.

2 Biquadratic Filter

An electric filter is a two-port network which shapes the frequency band of the input in order to obtain an output with the preferred frequency. Thus, it has pass band and stop band, respectively, in which the frequencies are transmitted and rejected at the output.

2.1 Current-Mode KHN-Equivalent BiQUAD

It can be implemented by cascading two lossless integrators [8,9,10] to obtain five filter functions namely: low-pass, high-pass, band-pass, band-elimination and all-pass responses.

Figure 1 shows the block schematic representation of current-mode KHN-equivalent biquad , and the routine analysis has been carried out to find the current transfer functions. Figure 2 shows the CMOS realization of the proposed CM KHN-equivalent biquad using complementary current-mirror pair [11].

Fig. 1
figure 1

Block diagram of current-mode KHN-equivalent biquad

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Fig. 2
figure 2

Circuit diagram of CM KHN-equivalent biquad using MOS complementary current-mirror pairs

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2.2 Mathematical Analysis

From Fig. 1, we obtain the following three equations:

$$I_{{{\text{o}}3}} = I_{\text{in}} - I_{{{\text{o}}2}} - I_{{{\text{o}}1}}$$
(2)
$$I_{{{\text{o}}1}} = \frac{{I_{{{\text{o}}2}} }}{{sC_{2} R_{2} }}$$
(3)
$$I_{{{\text{o}}2}} = \frac{{I_{{{\text{o}}3}} }}{{sC_{1} R_{1} }}$$
(4)

From Eqs. (3) and (4), we have

$$I_{{{\text{o}}1}} = \frac{{I_{{{\text{o}}3}} }}{{s^{2} C_{1} C_{2} R_{1} R_{2} }}$$
(5)

Now, substituting the value of Io1 and Io2 from Eqs. (4) and (5) in Eq. (1), we obtain an expression for the current Io3 in terms of input current Iin as

$$\frac{{I_{{{\text{o}}3}} }}{{I_{\text{in}} }} = \frac{{s^{2} }}{{s^{2} + s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}$$
(6)

Thus, Eq. (6) represents a high-pass response.

Similarly, when the value of Io3 is substituted from Eq. (6) in Eqs. (4) and (5), respectively, we obtain the expression for the currents Io1 and Io2 in terms of input current Iin as follows:

$$\frac{{I_{{{\text{o}}2}} }}{{I_{\text{in}} }} = \frac{{s\left( {\frac{1}{{C_{1} R_{1} }}} \right)}}{{s^{2} + s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}$$
(7)
$$\frac{{I_{{{\text{o}}1}} }}{{I_{\text{in}} }} = \frac{{\left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}{{s^{2} + s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}$$
(8)

Thus, Eqs. (7) and (8), respectively, represent a band-pass response and a low-pass response.

Similarly, the currents Io4 can be written as

$$I_{{{\text{o}}4}} = I_{{{\text{o}}3}} + I_{{{\text{o}}1}}$$
(9)

Now, substituting the value of the currents Io3 and Io1 from Eqs. (6) and (8), respectively, in Eq. (9), we obtain an expression for the current transfer function Io4/Iin.

$$\frac{{I_{{{\text{o}}4}} }}{{I_{\text{in}} }} = \frac{{s^{2} + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}{{s^{2} + s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}$$
(10)

Thus, Eq. (10) represents a notch response.

Similarly, for the current Io5, we have the following expression:

$$I_{{{\text{o}}5}} = I_{{{\text{o}}4}} - I_{{{\text{o}}2}}$$
(11)

Now, substituting the value of the currents Io4 and Io2 from Eqs. (10) and (7), respectively, in Eq. (11), we obtain the expression for Io5/Iin.

$$\frac{{I_{{{\text{o}}5}} }}{{I_{\text{in}} }} = \frac{{s^{2} - s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}{{s^{2} + s\left( {\frac{1}{{C_{1} R_{1} }}} \right) + \left( {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \right)}}$$
(12)

Thus, Eq. (12) represents an all-pass response.

The expressions for the filter parameters namely the cut-off frequency (ωo) and the quality factor (Q) can be obtained from the above current transfer function, and they are given by:

$$\omega_{\text{o}} = \frac{1}{{\sqrt {C_{1} C_{2} R_{1} R_{2} } }}$$
(13)
$$Q = \sqrt {\frac{{C_{1} R_{1} }}{{C_{2} R_{2} }}}$$
(14)

where

$$R_{1} = \frac{1}{{g_{{{\text{m}}1}} }}\;{\text{and}}\;R_{2} = \frac{1}{{g_{{{\text{m}}2}} }}$$
(15)

where gm1 and gm2 are the transconductance of the diode-connected transistors Q12 and Q17, respectively. Also, IB1 and IB2 are the DC bias currents shown as IB2 and IB4 in Fig. 2.

But the transconductances gm1 and gm2 are directly related to the square root of the bias currents IB2 and IB4 and are given as

$$g_{{{\text{m}}1}} = \sqrt {2\mu_{\text{n}} C_{\text{ox}} \left( {\frac{W}{L}} \right)_{12} I_{{{\text{B}}2}} } \;{\text{and}}\;g_{{{\text{m}}2}} = \sqrt {2\mu_{\text{n}} C_{\text{ox}} \left( {\frac{W}{L}} \right)_{17} I_{{{\text{B}}4}} }$$
(16)

Therefore, the expression for filter parameters ωo and Q becomes

$$\omega_{\text{o}} = \sqrt {\frac{{I_{{{\text{B}}2}} I_{{{\text{B}}4}} }}{{C_{1} C_{2} }}}$$
(17)
$$Q = \sqrt {\frac{{C_{1} I_{{{\text{B}}4}} }}{{C_{2} I_{{{\text{B}}2}} }}}$$
(18)

If C1 = C2 = C, then Eqs. (17) and (18) become

$$\omega_{\text{o}} = \frac{1}{C}\sqrt {I_{{{\text{B}}2}} I_{{{\text{B}}4}} }$$
(19)
$$Q = \sqrt {\frac{{I_{{{\text{B}}4}} }}{{I_{{{\text{B}}2}} }}}$$
(20)

Also, if IB2 = IB4 = IB, then Eqs. (19) and (20) become

$$\omega_{\text{o}} = \frac{{I_{\text{B}} }}{C}$$
(21)
$$Q = 1$$
(22)

3 Simulation Results

The proposed circuit of Fig. 2 was verified using SPICE with 0.5 µm CMOS process parameters provided by MOSIS (AGILENT). These parameters are listed in Table 1 [12,13,14,15,16].

Table 1 CMOS process parameters
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For the circuit shown in Fig. 2, the analysis was carried out with DC bias current IB1 = IB3 = IB5 = 24 µA, IB2 = IB4 = 15 µA, C1 = 0.01 pF, C2 = 0.1 pF, (W/L)P ratio = 1 µm/1 µm, (W/L)N ratio = 1 µm/1 µm and supply voltage VDD = 1.5 V.

The simulated value of the cut-off frequencies for low-pass, high-pass and band-pass response is found to be: (f0)LPF = 51.67 MHz, (f0)HPF = 95.404 MHz and (f0)BPF = 88.444 MHz, respectively. The band-pass filter has a bandwidth of 186.894 MHz.

The simulated results are very well in agreement with the theoretical results that were found to be: (f0)LPF = 52 MHz, (f0)HPF = 95 MHz, (f0)BPF = 88 MHz and BW = 185 MHz, respectively. The SPICE simulation for the proposed circuit has been shown in Fig. 3.

Fig. 3
figure 3

Simulated response of current-mode KHN-equivalent biquad

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Figures 4 and 5, respectively, represent the variation in the cut-off frequency and gain of the band-pass and the low-pass responses of the current-mode KHN-equivalent biquad circuit shown in Fig. 2.

Fig. 4
figure 4

Variations in cut-off frequency of band-pass response with capacitance value for current-mode KHN-equivalent biquad

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Fig. 5
figure 5

Variations in gain of low-pass response with bias current for current-mode KHN-equivalent biquad

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4 Conclusions

This paper presents high-frequency CM KHN-equivalent biquad filter suitable for operation at high frequencies they can operate at a low voltage of 1.0 V [17,18,19,20,21,22,23,24]. It can also be concluded that by varying the capacitor or the bias current, improves the gain and the operating frequency of the filter. Therefore, frequency of the filter can be controlled through a single DC bias current, thus providing good electronic tunability. The circuit was verified using SPICE, and simulation results confirm theoretical results.