1 Introduction

Current-mode universal filters employing current conveyors have been received considerable attentions. Due to their convenience and versatility in terms of signal processing for practical applications [1–43]. Also, a number of design of multiple-input single-output and single-input multiple-output current-mode and voltage-mode filters have been presented [18–27], universal filters are able to achieve more than one basic filter functions simultaneously with the same topology. A number of simultaneously versatile realized functions have been developed by a slight modification of the circuit. This is mainly attributable to the facility with which current outputs and current feedback can be developed when multiple current outputs are used.

Many CC-based voltage-mode universal biquadratic filters of three inputs and one output were proposed [4, 5, 20]. However, the circuit configurations in [5, 6] require too many current conveyors, for example, circuit proposed by H.Y. Wang [20] has three current conveyors. Compared with voltage-mode counterpart, current-mode universal filters are more attractive for their higher signal bandwidth, greater linearity and larger dynamic range. In the year 1991, C.M. Chang and P.C. Chen proposed a circuit configuration with one input and three outputs using seven current conveyors, grounded capacitors and resistors [4], In the year 2001, circuit proposed by H.Y. Wang and C.T. Lee [20] employs three current conveyors, two grounded capacitors and two resistors. These circuits can realize all the standard filter functions, namely, low-pass, band-pass, high-pass. Simultaneously, other functions of notch and all-pass can also be realized by properly connecting outputs without using additional active elements. However, many applications require filter tuning, in above cases, it is desirable to vary the filter coefficients electronically.

The major intention of this paper is to present new current-mode configurations, based on minimum number of grounded passive elements, current conveyors, Operational Transconductance Amplifiers. These circuits can simultaneously realize the five filter functions directly or indirectly. Firstly, a new current-mode three inputs and one output universal filter is proposed, which employs of two current conveyors and minimum number of passive elements of only two grounded capacitors and two resistors. The proposed circuit can realize low-pass, band-pass and high-pass filters all at high impedance outputs, and is very convenient for cascadability. Notch and all-pass functions can also be simply realized by connecting the appropriate output nodes. Secondly, another new current-mode low sensitivities one input and three outputs universal filter is also proposed, which uses of two current conveyors, one multi-inputs, multi-outputs OTA and minimum number of passive elements of two grounded capacitors. Through tuning the bias current value of OTA in order to change the gm value of OTA, a tunable cutoff frequency universal filter can be developed, which also possesses a low input impedance and high output impedance. The realization of a notch function does not require additional current conveyors, as such a realization can simply be achieved by connecting the appropriate node. Simulation results are given to illustrate the performances of the proposed circuits.

The paper is organized as follows. Section 2 presents two new universal filters based on multiple outputs second generation current conveyor(MOCCII) and multiple outputs OTA, the value of OTA can be tuned by changing its bias current, which makes these filters’ cutoff frequency tunable. In Sect. 3, an automatic frequency tuning implementation using a switched-capacitor resistor and Operational Transconductance Amplifier is discussed. Section 4 contains simulations of universal current-mode filters, while Sect. 5 concludes the paper.

2 Circuit description

2.1 Basic building block of CCII ± and OTA

The practical terminal relations of MOCCII can be described as:

$$ \left[ \begin{gathered} i_{y} \hfill \\ V_{x} \hfill \\ i_{z1} \hfill \\ \cdot \hfill \\ i_{zn} \hfill \\ i_{{\overline{z} 1}} \hfill \\ \cdot \hfill \\ i_{{\overline{z} n}} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} 0 \hfill \\ \beta \hfill \\ 0 \hfill \\ \cdot \hfill \\ 0 \hfill \\ 0 \hfill \\ \cdot \hfill \\ 0 \hfill \\ \end{gathered} \right.\left. \begin{gathered} 000 \hfill \\ 000 \hfill \\ \alpha 00 \hfill \\ \cdot \cdot \cdot \hfill \\ \alpha 00 \hfill \\ ( - \alpha )00 \hfill \\ \cdot \cdot \cdot \hfill \\ ( - \alpha )00 \hfill \\ \end{gathered} \right] \times \left[ \begin{gathered} V_{y} \hfill \\ i_{x} \hfill \\ V_{z} \hfill \\ Z_{{\overline{z} }} \hfill \\ \end{gathered} \right] $$
(1)
$$ V_{x} = \beta V_{y} ;i_{z} = \alpha i_{x} ;i_{{\overline{z} }} = - \alpha i_{x} $$
(2)

From Fig. 1, Multiple outputs of z terminals of current conveyors can be developed employing current mirror technique.

Fig. 1
figure 1

Multiple outputs CCII (MOCCII)

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From Fig. 2, The ideal terminal relations of multiple inputs and multiple outputs operational amplifier (MOOTA) can be described as:

$$ i_{0} = - g_{m} V_{i0} \;\;\;\;i_{1} = - g_{m1} V_{i1} $$
(3)
$$ i_{2} = - g_{m2} V_{i2} \;\;\;\;i_{3} = - g_{m3} V_{i3} $$
(4)
Fig. 2
figure 2

Multiple outputs OTA (MOOTA)

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Taking the non-idealities of MOCCIIs into account, namely, \( V_{x} = \beta V_{y} \), \( \beta = 1 - \varepsilon_{v} \) (\( \varepsilon_{v} < < 1 \)) denotes the voltage tracking error of port y. \( i_{z} = \alpha i_{x} \), \( \alpha = 1 - \varepsilon_{i} \)(\( \varepsilon_{i} < < 1 \)) denotes the current tracking error of the port z. The first proposed three-input one-output filter is constructed with two current conveyors, and four grounded passive elements, as shown in Fig. 3.

Fig. 3
figure 3

The proposed current-mode three-input, one-output universal filter

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If the two MOCCIIs are integrated in the same chip, the tracking error of each MOCCII is the same, that is:

$$ \varepsilon_{i}^{(1)} = \varepsilon_{i}^{(2)} = \varepsilon_{i} ,\;\varepsilon_{v}^{(1)} = \varepsilon_{v}^{(2)} = \varepsilon_{v} $$
(5)

For \( \varepsilon < < 1 \), ignore high order items of equation, the output functions can be derived as:

$$ \begin{gathered} I_{0} = \frac{{I_{in1} s^{2} - I_{in2} s\frac{1}{{\alpha_{2} \beta_{2} C_{1} R_{1} }} + I_{in3} \frac{1}{{\alpha_{2} \beta_{1} \beta_{2} C_{1} C{}_{2}R_{1} R_{2} }}}}{{s^{2} + s\frac{1}{{\alpha_{2} \beta_{2} C_{1} R_{1} }} + \frac{1}{{\alpha_{1} \alpha_{2} \beta_{1} \beta_{2} C_{1} C_{2} R_{1} R_{2} }}}} \hfill \\ \approx \frac{{s^{2} I_{in1} - \frac{1}{{(1 - \varepsilon_{i} - \varepsilon_{v} )C_{1} R_{1} }}sI_{in2} + \frac{1}{{(1 - 2\varepsilon_{i} - 2\varepsilon_{v} )C_{1} C_{2} R_{1} R_{2} }}I_{in3} }}{{s^{2} + \frac{1}{{(1 - \varepsilon_{i} - \varepsilon_{v} )C_{1} R_{1} }}s + \frac{1}{{(1 - 2\varepsilon_{i} - 2\varepsilon_{v} )C_{1} C_{2} R_{1} R_{2} }}}} \hfill \\ \end{gathered} $$
(6)

Compared with following equation:

$$ I_{0} \approx \frac{{s^{2} I_{in1} - \frac{{\omega_{0}^{'} }}{{Q^{'} }}sI_{in2} + \omega_{0}^{'2} I_{in3} }}{{s^{2} + \frac{{\omega_{0}^{'} }}{{Q^{'} }}s + \omega_{0}^{'2} }} $$
(7)
$$ \omega_{0}^{'} = \frac{1}{{\sqrt {\alpha_{1} \alpha_{2} \beta_{1} \beta_{2} } }}\omega_{0} \approx \frac{1}{{1 - \varepsilon_{i} - \varepsilon_{v} }}\omega_{0} $$
(8)
$$ Q^{'} = \sqrt {\frac{{\alpha_{2} \beta_{2} }}{{\alpha_{1} \beta_{1} }}} Q = Q $$
(9)

From (7), (8), (9), We can see that ɛ has little influence on parameter ω ′0 and no influence on Q ′.

The ω ′0 and Q ′ sensitivities of the filter are found as:

$$ S_{{R_{1} }}^{{\omega_{0}^{'} }} = S_{{R_{2} }}^{{\omega_{0}^{'} }} = S_{{C_{1} }}^{{\omega_{0}^{'} }} = S_{{C_{2} }}^{{\omega_{0}^{'} }} = - \frac{1}{2} $$
(10)
$$ S_{{C_{1} }}^{{Q^{'} }} = S_{{R_{1} }}^{{Q^{'} }} = \frac{1}{2},\;S_{{C_{2} }}^{{Q^{'} }} = S_{{R_{2} }}^{{Q^{'} }} = - \frac{1}{2} $$
(11)

Which are no more than one in magnitude.

Moreover, the configuration in Fig. 4 can function as a single-input, three-output universal filter if and by taking as the single input terminal. From (7), the current transfer functions can be derived as low-pass, band-pass and high-pass responses, respectively, when

$$ \frac{{I_{HP} }}{{I_{in} }} = - \frac{{s^{2} }}{A(s)} $$
(12)
$$ \frac{{I_{BP} }}{{I_{in} }} = \frac{{\frac{{g_{m2} }}{{C_{1} }}s}}{A(s)} $$
(13)
$$ \frac{{I_{LP} }}{{I_{in} }} = - \frac{{\frac{{g_{m} g_{m2} }}{{C_{1} C_{2} }}}}{A(s)} $$
(14)
$$ \frac{{I_{NH} }}{{I_{in} }} = - \frac{{s^{2} + \frac{{g_{m1} g_{m2} }}{{C_{1} C_{2} }}}}{A(s)} $$
(15)
$$ \frac{{I_{AP} }}{{I_{in} }} = \frac{{s^{2} - s\frac{{g_{m3} }}{{C_{1} }} + \frac{{g_{m1} g_{m2} }}{{C_{1} C_{2} }}}}{A(s)} $$
(16)
$$ A(s) = s^{2} + s\frac{{g_{m3} }}{{C_{1} }} + \frac{{g_{m1} g_{m2} }}{{C_{1} C_{2} }} $$
(17)
$$ \omega_{0} = \sqrt {\frac{{g_{m1} g_{m2} }}{{C_{1} C_{2} }}} \;Q = \frac{1}{{g_{m3} }}\sqrt {\frac{{C_{1} }}{{C_{2} }}g_{m1} g_{m2} } $$
(18)
Fig. 4
figure 4

The proposed current-mode single-input, three-output universal filter

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The ω ′0 and Q ′ sensitivities of the filter are found as:

$$ \begin{gathered} S_{{g_{m} }}^{{\omega_{0} }} = 0,\,S_{{g_{m3} }}^{{\omega_{0} }} = 0,\;S_{{g_{m1} }}^{{\omega_{0} }} = S_{{g_{m2} }}^{{\omega_{0} }} = \frac{1}{2},\; \hfill \\ S_{{C_{1} }}^{{\omega_{0} }} = S_{{C_{2} }}^{{\omega_{0} }} = - \frac{1}{2} \hfill \\ \end{gathered} $$
(19)
$$ S_{{g_{m} }}^{Q} = 0,\;S_{{g_{m3} }}^{Q} = - 1,\,S_{{g_{m1} }}^{Q} = S_{{g_{m2} }}^{Q} = \frac{1}{2} $$
(20)
$$ S_{{C_{1} }}^{Q} = \frac{1}{2},\;S_{{C_{2} }}^{Q} = - \frac{1}{2} $$
(21)

Which are also no more than one in magnitude.

We can now discuss practical problems in design of filters employing OTA and MOCCII. In particular we will deal with the effects of OTA nonidealities on filter performance. The methods for the evaluation and reduction of effects will be proposed as follows.

An OTA macro-model with finite input and output impedances and transconductance frequency dependence is shown in Fig. 5. We use g i and C i to represent the differential input conductance and capacitance and drop subscript d (for differential) for simplicity. g 0 and C 0 are those at the output. The common-mode input conductance g ic and capacitance C ic are ignored because they are usually very small in practice compared with differential counterparts and can be absorbed as many filter structures have a grounded capacitor or a grounded OTA resistor from OTA input terminals to ground.

$$ \alpha_{1} = \alpha_{2} = \alpha ;\;\beta_{1} = \beta_{2} = \beta $$
(22)
$$ Y_{i} = g_{i} + sC_{i} \;Y_{0} = g_{0} + sC_{0} $$
(23)
$$ C_{i0} = C_{i1} = C_{i2} = C_{i3} = C_{i} $$
$$ g_{i0} = g_{i1} = g_{i2} = g_{i3} = g_{i} $$
(24)
$$ C_{00} = C_{01} = C_{02} = C_{03} = C_{0} $$
$$ g_{00} = g_{01} = g_{02} = g_{03} = g_{0} $$
(25)
$$ \frac{{I_{HP}^{'} }}{{I_{in} }} = \frac{{N_{HP} (s)}}{{D_{n} (s)}} $$
(26)
$$ \frac{{I_{BP}^{'} }}{{I_{in} }} = \frac{{N_{BP} (s)}}{{D_{n} (s)}} $$
(27)
$$ \frac{{I_{LP}^{'} }}{{I_{in} }} = \frac{{N_{LP} (s)}}{{D_{n} (s)}} $$
(28)
$$ \begin{gathered} D_{n} (s) = s^{2} (C_{2} + C_{i} )[(C_{1} + \alpha \beta (2C_{0} + C_{i} )] + \{ \alpha \beta [(C_{0} + C_{i} )(g_{i} + \alpha \beta g_{m1} ) \hfill \\ + (C_{2} + C_{i} )(g_{m3} + 2g_{0} + g_{i} )] + g_{i} (C_{1} + \alpha \beta C_{0} )\} s \hfill \\ + \alpha \beta [g_{i} (g_{m3} + 2g_{0} + g_{i} ) + \alpha \beta g_{m1} (g_{m2} + g_{0} + g_{i} )] \hfill \\ \end{gathered} $$
(29)
$$ N_{HP} (s) = - s\alpha C_{1} [g_{i} + s(C_{2} + C_{i} )] $$
(30)
$$ N_{BP} (s) = \alpha^{2} \beta^{2} [g_{m2} + g_{0} + g_{i} + s(C_{0} + C{}_{i})][g_{i} + s(C_{2} + C_{i} )] $$
(31)
$$ N_{LP} (s) = - \alpha^{2} \beta^{2} g_{m} [g_{m2} + g_{0} + g_{i} + s(C_{0} + C_{i} )] $$
(32)
$$ \omega_{0}^{'} = \left\{ \frac{{\alpha \beta [g_{i} (g_{m3} + 2g_{0} + g_{i} ) + \alpha \beta g_{m1} (g_{m2} + g_{0} + g_{i} )]}}{{(C_{2} + C_{i} )[C_{1} + \alpha \beta (2C_{0} + C_{i} )]}}\right\}^{1/2} $$
(33)
$$ Q^{'}=\frac{\left\{\frac{{\alpha \beta [g_{i} (g_{m3} + 2g_{0}+ g_{i} ) + \alpha \beta g_{m1} (g_{m2} + g_{0} + g_{i} )]}}{{(C_{2}+ C_{i} )[C_{1} + \alpha \beta (2C_{0} + C_{i})]}}^{1/2}\right\}}{\alpha \beta [(C_{0} + C_{i} )(g_{i} + \alpha\beta g_{m1} ) + (C_{2} + C_{i} )(g_{m3} + 2g_{0} + g_{i} )] + g_{i}(C_{1} + \alpha \beta C_{0} )} $$
(34)
Fig. 5
figure 5

Practical OTA macro-model

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It can be seen that the low-pass frequency responses of filter are little influenced by nonideal factors of OTA, the higher frequency, the higher influence.

The CMOS realization of MOCCII is shown in Fig. 6. However, it is possible to represent an MOCCII using Operational Amplifier (OA) and a small-signal current mirror as shown in Fig. 7.

Fig. 6
figure 6

CMOS implementation of MOCCII

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Fig. 7
figure 7

Implementation of an ideal MOCCII

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The proposed circuit implementation of multiple inputs and multiple outputs OTAs are shown in Fig. 2 if input and output conductances of non-idealities of OTAs are taken into account, the input and output immittances can be written as (23). The CMOS implementation of single g m cell of MOOTA is shown in Fig. 8.

Fig. 8
figure 8

CMOS implementation of single g m cell of Multiple Outputs OTA (MOOTA) in Fig. 2

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3 Automatic frequency tuning by switched capacitor and Operational Transconductance Amplifier (OTA)

A fully differential tuning circuit is used to generate the control voltages V C+ and V C− [28–40]. The tuning circuit is designed to allow tuning of the filter’s cutoff frequency and minimize the influence of the process parameters and operating conditions. Since the active filter’s frequency characteristic is determined by g m value of MOOTA, C 1 and C 2, any process variation and temperature dependencies can be compensated by tuning the g m value of the MOOTA. As explained above, the value of g m can be tuned by bias current which is determined by controlling the gate voltages of current mirrors of transistors M12 and M13 in Fig. 8. The required control voltages V C+ and V C− are generated in the tuning circuit shown in Fig. 9, which is based on the balancing principle that a switched-capacitor network is accurately to implement a reference time constant, which depends solely on capacitor matching. Shown in the left portion of the schematic Fig. 9 is the time-constant matching integrator of the continuous-time equivalent resistor path and the switched-capacitor equivalent resistor path. The time constant of the continuous-time equivalent resistor path is \( C_{\text{int}} /g_{m} \), and that of the switched-capacitor equivalent resistor patch is \( C_{\text{int}} /(f_{clk} C_{1} ) \). The mismatch of the two time constants, which simplifies as the mismatch of equivalent resistors 1/g m and 1/(f clk C 1), is reflected at the output of the integrator. That voltage is then translated to the control voltage of the MOOTA. Equilibrium is reached when g m  = f clk C 1. The right portion of the schematic is the g m value controlling voltage generating circuit of MOOTA.

Fig. 9
figure 9

Automatic frequency tuning circuit

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At each of the integrator inputs, two currents, \( I_{{g_{m} }} \)and I SC are summed. The currents flow from the reference voltages, V ref+ and V ref− , to the integrator inputs, which are at virtually the same potential as analog ground. The positive and negative reference voltages are symmetrical around analog ground. I SC is the current through the switched capacitor resistor realization R SC , while \( I_{{g_{m} }} \) is the current through transconductor structure. The transconductor is identical to the ones used in the filter and controlled by the same voltages V C+ and V C− . These control voltages are fed back from the outputs of the tuning circuit to the transconductor structure and control its equivalent transconductor g m in such a way that \( I_{SC} = I_{{g_{m} }} \) in steady state condition. Equilibrium is reached when g m  = f clk C 1. The output voltage of the integrator is symmetrical around a constant common voltage, V common = V DD /2, and therefore V C+ and V C− are symmetrical around a constant \( \bar{V}_{C} \).

In order to tune the cutoff frequency of the filter, the resistance implemented by the switched-capacitor circuit, \( R_{SC} = 1/(f_{clk} C_{1} ) \), can be changed by using a variable clock frequency. The integrator will then adjust the control voltages V C+ and V C− until the equilibrium of the tuning circuit is reached again.

The unity-gain frequency of an integrator can be written as:

$$ \omega_{\text{unity}} = \frac{{g_{m} }}{{C_{\text{int}} }} $$
(35)

In the tuning circuit, the equivalent resistance of transconductor and the one implemented by the switched-capacitor are equal in steady–state:

$$ R_{eq} = \frac{1}{{f_{clk} C_{1} }} $$
(36)

Using above Eqs. 35 and 36, the equation for the unity-gain frequency can be rewritten as:

$$ \omega_{\text{unity}} = f_{clk} \frac{{C_{1} }}{{C_{\text{int}} }} $$
(37)

Shown as above Eq. 37, mismatch in capacitors C 1 and \( C_{\text{int}} \)directly introduce frequency deviation of ω unity of integrator which is proportional to ratio of \( C_{1} /C_{\text{int}} \), assuming ideal matching between the components of the filter and the tuning circuit, the unity-frequency ω unity is accurately set to \( (f_{clk} C_{1} /C_{\text{int}} ) \), and hence precise frequency tuning can be achieved without any external components. Among the fabricated filter in Fig. 4, the ω unity standard deviation is 5%, measured under a fixed capacitor, C 1, and the same reference frequency, f clk . If a greater accuracy of the corner frequency is desired, a fine adjustment can be provided by means of either digitally trimming the capacitor C 1 or varying the clock frequency f clk . The fine tuning of capacitor of C 1 for the corner frequency of the proposed filter in Fig. 4 can be accomplished within a ±1% absolute accuracy. Note that an additional low-pass filter, R LP C LP , has been added here to remove the high-frequency ripple voltage due to using a switched-capacitor circuit. It should also be noted that, since this method requires a clock signal, there is a strong possibility that some of the clock signal will leak into the continuous-time filter, either mainly through the Operational Transconductance Amplifier controlling signal, differential operational amplifier of Fig. 9 or through the IC substrate, in the mean time, since the stability of the tuning voltage can be solved through placing a dominant pole in the control loop by choosing the product R LP C LP very large (alternately a large \( C_{\text{int}} \) can be used), a stable dc control voltage without ac ripple voltage is established. Any mismatch between the equivalent g m cell resistor and switched-capacitor equivalent resistor causes current to flow into the integrator, which changes the control voltage V C+ and V C−. The negative feedback loop controls the frequency of clock and adjusts the g m cell equivalent resistor forcing it to be equal to the average resistance of the SC branch. Once the feedback loop reaches equilibrium, the two equivalent resistances will be equal. The tuning action applies equally to the tuning of the continuous-time filter employing MOCCII and OTA.

4 Simulation results

Universal filters are simulated using schematics of MOCCII and Operational Transconductor Amplifier implementation as depicted in Figs. 3 and 4. All MOS transistors are operated in saturation region and all of the bulks are connected to the sources. The simulations are performed using SPICE based on 0.35 μm Chartered CMOS process model parameters (PMOS_3p3, V THP  = −0.837 V, μ P  = 277 cm2/V s, T OX  = 8.69 nm), (NMOS_3p3, V THN  = 0.6053 V, μ N  = 413.7172 cm2/V s, T OX  = 7.69 nm). The MOCCII is constructed using the schematic implementation in Fig. 6 with dc supply voltages equal to 3.3 V and 0 V and bias current of transistor M17 is 10 μA, similarly the supply voltage of single OTA is the same as that of MOCCII, and bias current of transistor M12 in Fig. 8 is 20 uA. The dimensions of the MOS transistors used in MOCCII (shown in Fig. 6) and Operational Transconductance Amplifier (shown in Fig. 8)’s implementations are given in Table 1, 2.

Table 1 Dimensions of CMOS transistors of MOCCII
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Table 2 Dimensions of CMOS transistors of Operational Transconductance Amplifier of Fig. 8
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To investigate what is the frequency for the designed MOCCII, AC simulations have been performed. The frequency responses of i z /i x , and V x /V y for MOCCII are depicted in Figs. 10 and 11. The frequency behavior reported in Fig. 12 suggests that for high frequency applications (frequencies of more than 90 MHz), a compensation is needed. The DC transfer characteristic of MOCCII is shown in Fig. 11.

Fig. 10
figure 10

The simulated frequency response of i z /i x for the MOCCII (VDD = 3.3 V, VSS = 0 V, Ibias = 10 uA)

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Fig. 11
figure 11

V x  − V y DC transfer characteristic of MOCCII(VDD = 3.3 V, VSS = 0 V, Ibias = 10 uA)

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Fig. 12
figure 12

The simulated frequency response of V x /V y for the MOCCII(VDD = 3.3 V, VSS = 0 V, Ibias = 10 uA)

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The characteristic of input voltage Vin and output current Iout of OTA is shown in Fig. 13, when the supply voltages used for OTAs of Fig. 8 are 3.3 V and 0 V, the bias currents of transistor M12 are selected as I bias = 20 uA, It can be seen that gm value of OTA is 0.05174 mS.

Fig. 13
figure 13

Characteristic of input voltage Vin and output current Iout of Operational Transconductance Amplifier (OTA) (VDD = 3.3 V, VSS = 0 V, Ibias = 20 uA, g m  = 0.05174 mS)

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To verify the theoretical analyses, active and passive bandpass responses of three-input, single-output filter are shown in Fig. 14. We selected the following setting to obtain the lowpass, bandpass and highpass responses in Fig. 3, with cutoff frequency of f 0 = 15.92 kHz, and a pole-quality factor of Q = 1: R 1 = R 2 = 100 K, C 1 = C 2 = 100 pf (circuit A); Q = 2: R 1 = R 2 = 200 K, C 1 = 100 pf, C 2 = 25 pf (circuit B); Q = 3: R 1 = R 2 = 300 K, C 1 = 100 pf, C 2 = 11.11 pf (circuit C); In the circuit of Fig. 3, the simulated values of f 0 and Q are 16 kHz and 1 for circuit A, 16 kHz and 2 for circuit B, and 16 kHz and 3 for circuit C, respectively. Figure 16 shows the responses for the lowpass, bandpass and highpass functions of the three-input single-output filter. All the results are in good agreement with the theoretical predictions.

Fig. 14
figure 14

Active and passive band-pass responses of three-input, single-output filter of Fig. 3

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Also the second-order current-mode filter of Fig. 4 is simulated. The active elements and biasing currents of MOCCII and MOOTA have been selected as g m  = g m1 = g m2 = g m3 = 0.05174 mS, C 1 = C 2 = 100 pf, I bias (MOCCII) = 20 uA, I bias (MOOTA) = 20 uA to obtain unity gain low-pass, band-pass, and high-pass responses with a natural pole frequency of f 0 = 82.38 kHz. The frequency responses of single-input, three-output universal filter with tunable frequency are shown in Fig. 17. When it functions as low-pass, band-pass and high-pass modes. The simulated results for MOCCII and OTA based single-input three-output universal filter of Fig. 4 are shown in Fig. 15. Employing above automatic frequency tuning technique by switched capacitor and Operational Transconductance Amplifier, and through adjusting the clock frequency of switched-capacitor, the cutoff frequency of single-input three-output universal filter can be tuned from 10 kHz to 300 kHz conveniently (Fig. 17). As shown in Fig. 17, the results also confirm the theoretical ones.

Fig. 15
figure 15

Active band-pass responses of three-input, single-output filter of Fig. 3 in different quality factors

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Fig. 16
figure 16

Frequency responses of the three-input, single-output universal filter of Fig. 3 employs MOCCII and CCII-

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Fig. 17
figure 17

Responses of the single-input, three-output filter of Fig. 4 employs MOCCII and OTA with tunable frequency

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The transient responses of the three-input, one-output and single-input, three-output universal filters are shown in Figs. 18 and 19, respectively. Also, the large signal behavior of these two circuits in Figs. 3 and 4 are tested with investigation of the dependence of the output harmonic distortion for the low-pass response on the amplitude of the sinusoidal input signals at 1 kHz. The obtained THD simulation results for above two configurations in Figs. 3 and 4 are given in Tables 3 and 4, respectively. From Table 3, it can be seen that the harmonic distortion rapidly increases for the input current signal amplitude is beyond 3 uA, because much more larger input signal will result MOS transistors of MOCCII operating in triode region, when the bias currents of MOCCII and CCII- are 20 uA, respectively. whereas, Table 4 shows that larger input current signal amplitude results in higher value of THD due to the open-loop nature of MOOTA in single-input, three-output configuration. generally, the input voltage signal range of MOOTA should be smaller than 100 mV or so, which can also be seen from Fig. 13.

Fig. 18
figure 18

Simulated transient responses of the three-input, one-output universal filter (the bias currents I bias of MOCCII and CCII- are 20 uA, respectively, I in is 3 uA, THD value of I rl1 is 6.2657%)

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Fig. 19
figure 19

Simulated transient responses of the single-input, three-output universal filter(the bias currents I bias of MOCCII and MO are 20 uA, respectively, I in is 5 uA, THD value of I rl1 is 1.1161%)

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Table 3 Dependence of output harmonic distortion of three-input, one-output filter of Fig. 3 on input current signal amplitude
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Table 4 Dependence of output harmonic distortion of single-input, three-output filter of Fig. 4 on input current signal amplitude
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Since output currents of MOCCII are composed of multiple current mirrors, process mismatches between the current mirror transistors of MOCCII could be analyzed to determine the effects of THD on proposed universal filters. First, the current distortion caused by only threshold voltage mismatches of current mirrors is considered with ΔV t  = V t1 − V t2. Assume transistors of current mirror are identical except for the threshold voltage mismatch. The output current i out is ideally identical to the signal current i, shown in (38). The expression for the output current with threshold voltage mismatch is:

$$ i_{out} = i + \frac{{2\Updelta V_{t} I}}{{(V_{GS1} - V_{t1} )}}\sqrt {\left( {1 + \frac{i}{I}} \right)} + \frac{\beta }{2}\Updelta V_{t}^{2} $$
(38)
$$ i_{out,dc} = \frac{\beta }{2}\Updelta V_{t}^{2} + \frac{{2\Updelta V_{t} I}}{{(V_{GS1} - V_{t1} )}} $$
(39)

Where \( \beta = k^{'} (W/L) \)

$$ i_{out,ac} = i\left[ {1 + \frac{{\Updelta V_{t} }}{{(V_{GS1} - V_{t1} )}}} \right] + \frac{{2\Updelta V_{t} I}}{{(V_{GS1} - V_{t1} )}} $$
$$ \cdot \left[ { - \frac{1}{8}\left( {\frac{i}{I}} \right)^{2} + \frac{1}{16}\left( {\frac{i}{I}} \right)^{3} - \frac{5}{128}\left( {\frac{i}{I}} \right)^{4} + \cdot \cdot \cdot } \right] $$
(40)

Where i is the peak signal current. It is seen from (39) (40) that threshold voltage mismatch in the current-mirror transistors distorts the output current. The current is separated into a dc term in (39) and an ac polynomial term in (40). The dc offset shifts the bias point as indicated by i out,dc term in (39). The magnitude of ac gain error and harmonic distortion are determined by the first and second i out,ac terms, respectively. The harmonic distortion is a strong function of the peak signal-to-bias-current ratio i/I, and therefore, harmonic distortion is minimized by reducing the i/I ratio. Which is also the explanation that the simulated results of proposed filters (Figs. 3 and 4) have THD characteristics of monotonicity according to the amplitude of current input signals (0.1–5 u) when the bias currents of MOCCII, CCII- and MOOTA are 20 uA, respectively.

5 Conclusions

Versatile multi-input multi-output current-mode biquad configurations are introduced in this paper. The proposed universal filters has been exhibited by application on the implementation of three-input, single-output and single-input, three-output filters, which realize low-pass, high-pass, and band-pass responses. In addition, by changing the bias current of OTA through technique of switched capacitor, a tunable frequency single-input three-output universal filter construction has been obtained. Only two current conveyors and several grounded passive elements are necessary in both proposed filters. The proposed circuits enjoy the following features: (1) using minimum number of passive elements; (2) employing only grounded capacitors and resistors, which is convenient for monolithic integration; (3) frequency tunability; (4) high impedance outputs enable easy cascading in current-mode operation; (5) low sensitivity and simple in structure. The simulation results, which confirm the theoretical analysis, are given.