Design and Experiment of Electronically Tunable Voltage-Mode Biquad and Output Current Amplitude Oscillator

Abstract

This study presents an electronically tunable configuration for the design of a voltage-mode (VM) biquad with four input terminals and three output terminals. The proposed circuit employs four operational transconductance amplifiers (OTAs) and two grounded capacitors. Depending on the selections of the four input voltage signals, all the standard filtering functions can be realized. The proposed configuration simultaneously provides VM inverting band-pass, non-inverting low-pass, and non-inverting band-reject filtering functions without any component-matching choices. It offers the features of a resistorless structure, high-input impedance, electronic control of the pole frequency and quality factor, and low active and passive sensitivities. The measured power dissipation of the biquad is 0.96 W under 32 mA constant output current. The measured 1 dB power gain compression point of the output inverting band-pass filter is −7 dBm. The measured value of the third-order intercept point is 5.136 dBm, and the measured value of the third-order intermodulation distortion is −50.83 dBc. Moreover, the measured value of the spurious-free dynamic range is 53.49 dB, and the figure-of-merit of the biquad is 268.75 × 10³. In addition, an electronically controllable quadrature oscillator (QO) with amplitude of output current can be realized using the proposed biquad. The proposed electronically controllable QO can provide an amplitude modulation signal or an amplitude shift keying signal, and is widely applied in signal processing systems and electronic communication systems. PSpice simulations and experimental results are accomplished.

1. Introduction

With increasing emphasis on voltage-mode (VM) biquads and oscillators, there is still a need to develop active components for VM biquads and oscillators. Oscillators [1,2,3,4,5,6] and biquads [7,8,9,10,11,12,13,14] are the most crucial circuit blocks in numerous applications, such as electronic communications, signal processing systems, and measurement systems. Especially in communication modulation systems, the sinusoidal oscillator is utilized to generate the carrier signals for amplitude modulation (AM) and amplitude shift keying (ASK) modulation. Most VM continuous-time biquad designs require the following features: (a) Achievement of five generic filtering signals, that is, low-pass filter (LPF), band-pass filter (BPF), high-pass filter (HPF), band-reject filter (BRF), and all-pass filter (APF) responses. (b) Utilization of grounded capacitors, which is attractive for absorbing shunt parasitic capacitance of the active component and suitable for integrated circuit (IC) implementation. (c) High-input impedance, which is suitable for easy cascading in VM operation without needing any extra voltage followers. (d) Orthogonal tunability of resonance angular frequency (ω_o) and quality factor (Q). (e) Low ω_o and Q sensitivities with respect to the active and passive components. On the other hand, most quadrature oscillator (QO) designs require the following features: realization of two quadrature sinusoidal voltage outputs with 90° phase difference, independent control of frequency of oscillation (FO) and condition of oscillation (CO), electronic controllability, employment of grounded capacitors, and so on. However, the electronically controllable QO with amplitude of output current should also be considered. The amplitude of output current can be electronically adjusted by fine-tuning the direct-current (DC) bias current (I_B), which is suitable for signal processing systems and AM/ASK systems.

The VM electronically tunable, high-input impedance configurations used in the design of operational transconductance amplifier and capacitor (OTA-C) biquads are worthy of study, because they are suitable for the cascade in VM operation and the fine-tuning of the deviation of conventional electronic components. OTA provides highly linear electronic controllability and has a wide tunable range of transconductance gain (g_m), which is an attractive feature for circuit designers [15,16,17,18,19,20,21,22,23]. OTAs are commercially available ICs, and LT1228 from Linear Technology Corporation [24] has been widely used to implement OTA-based circuits. The transconductance of LT1228 is electronically controlled by an external DC bias current I_B. The OTA-C biquad does not require a resistor, so it is very suitable for IC compared to other conventional active elements. Recently, some types of electronically tunable VM OTA-based biquads have been developed [25,26,27,28,29,30]. The circuits in [25,26] employ six OTAs, but the proposed circuits cannot simultaneously realize LPF, inverting BPF (IBPF) and BRF responses from the same structure. In 2019, three interesting VM electronically tunable OTA-C biquads were proposed, which employ five OTAs along with two grounded capacitors [27,28,29]. The circuit proposed in [27] cannot use the same structure to realize LPF, IBPF, and BRF responses. The circuits proposed in [28,29] can realize LPF, IBPF, and BRF responses with the same configuration, and can be simply converted to QO. However, none of the circuits proposed in [25,26,27,28,29] can be transformed into an electronically controllable QO with high-output impedance and amplitude-controllable sinusoidal current output signal. In 2020, another VM biquad using one dual-output OTA (DO-OTA), three OTAs, and two grounded capacitors was proposed [30]. The proposed circuit can realize LPF, IBPF, and BRF responses with the same configuration, and can be simply transferred to QO. However, the proposed circuit cannot electronically control the amplitude of the high-output impedance sinusoidal current signal. The amplitude of electronically controlled high-output impedance sinusoidal current signals can be applied in AM/ASK systems.

In this study, a new electronically tunable VM biquad with four inputs and three outputs is presented. The biquad has the following advantages: (i) using one DO-OTA, three OTAs, two grounded capacitors, and resistorless realization, which is suitable for integration, (ii) simultaneously realizing LPF, IBPF, and BRF responses without changing the circuit structure, (iii) realizing all the standard biquad functions from the same circuit structure, (iv) high-input impedance, (v) orthogonal electronic-tuning the characteristic parameters of ω_o and Q, (vi) easily transformed into a QO with high-output impedance and amplitude-controllable sinusoidal current output signal, and (vii) easily realizing AM/ASK of the oscillator.

Table 1. Comparison of the proposed biquad and the biquads previously reported in the literature [25,26,27,28,29,30].

Features	[25]	[26]	[27]	[28]	[29]	[30]	Proposed
Active and passive elements	6 OTAs, and 2 C	6 OTAs, 2 Rmos, and 2 C	5 OTAs, and 2 C	5 OTAs, and 2 C	5 OTAs, and 2 C	1 DO-OTA, 3 OTAs, and 2 C	1 DO-OTA, 3 OTAs, and 2 C
Resistorless structure	yes	no	yes	yes	yes	yes	yes
Component-matching condition (R = 1/gm)	no	yes	no	no	no	no	no
Using grounded passive elements	yes	yes	yes	yes	yes	yes	yes
No inverting-type input signal	yes	yes	yes	yes	yes	yes	yes
High-input impedances	yes	yes	yes	yes	yes	yes	yes
Orthogonal control of ωo and Q	yes	yes	yes	yes	yes	yes	yes
Versatile input/output functions	no	no	no	yes	yes	yes	yes
Simultaneously realizing three filtering functions	no	no	no	yes	yes	yes	yes
Offering five standard filters	yes	yes	yes	yes	yes	yes	yes
Easily transformed into a QO with output current amplitude	no	no	no	no	no	no	yes
Easily realizing oscillator with AM/ASK signals	no	no	no	no	no	no	yes

Note: QO: quadrature oscillator; AM: amplitude modulation; ASK: amplitude shift keying.

summarizes the performance comparison between the proposed biquad and the previously reported biquads [25,26,27,28,29,30]. Regarding the OTA-based circuits proposed in [25,26,27,28,29,30], the proposed circuit has electronically controllable amplitude of sinusoidal current output with high-output impedance and can be used in an AM/ASK systems.

Table 2. Characteristic comparisons of the proposed biquad with the biquads previously reported in the literature [25,26,27,28,29,30].

Features	[25]	[26]	[27]	[28]	[29]	[30]	Proposed
Power supply	±15 V	±5 V	±15 V	±15 V	±15 V	±15 V	±15 V
Verification of simulation/measurement	yes/yes	yes/yes	yes/yes	yes/yes	yes/yes	yes/yes	yes/yes
Simulated the power dissipation (W)	none	none	none	none	0.86	0.92	0.92
Measured the power dissipation (W)	none	none	none	0.86	1.2	1.23	0.96
Central frequency of the filter (kHz)	10	32.9	238.7	159.15	217	144.7	144.7
Maximum output noise (nV/$\sqrt{\mathrm{Hz}}$)	none	none	none	none	none	none	7.937
Measured the output P1dB point (dBm)	none	none	none	−9.2	none	−5.9	−7
Measured the TOI point (dBm)	none	none	none	−5.66	none	−2.597	5.136
Measured the IMD3 (dBc)	none	none	−30.03 (@ 80 mVpp)	−42.35 (@ 60 mVpp)	−31.16 (@ 90 mVpp)	−48.67 (@ 63 mVpp)	−50.83 (@ 126 mVpp)
FoM (×103)	none	none	none	none	none	none	268.75

Note: P1dB: 1 dB power gain compression; TOI: third-order intercept; IMD3: third-order intermodulation distortion; FoM: figure-of-merit.

compares the additional performance of the proposed biquad and the previously reported biquads [25,26,27,28,29,30].

2. Proposed Electronically Tunable VM OTA-C Biquad

OTA offers a high linear electronic controllability and its transconductance gain has a wide tunable range [19]. The electrical symbol of the OTA is shown in Figure 1. According to Figure 1, the single-ended output OTA consists of an input differential pair, an output current mirror, and a g_m control unit. The output current of a single-ended OTA is characterized by I_O = g_m(V+ − V−), where V+ and V− are the input signals applied to the non-inverting and inverting terminals, respectively [19]. The g_m value is electrically controllable by an external DC bias current I_B, which leads electronic controllability to the design biquad or oscillator parameters. The DO-OTA has two identical current outputs, which can be described by I_O = I_Q = g_m(V+ − V−). Figure 2 shows the proposed biquad with four inputs and three outputs. It is configured by only one DO-OTA, three OTAs, and two grounded capacitors. Routine analysis of the proposed electronically controllable OTA-C biquad in Figure 2 yielded the following three-output voltage signals.

$${\mathrm{V}}_{\mathrm{o}1}=\frac{1}{\mathrm{D}\left(\mathrm{s}\right)}{[\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}{\mathrm{V}}_{\mathrm{i}1}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}3}{\mathrm{g}}_{\mathrm{m}4}{(\mathrm{V}}_{\mathrm{i}2}-{\mathrm{V}}_{\mathrm{i}4})+{\mathrm{sC}}_2{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}3}{\mathrm{V}}_{\mathrm{i}3}]$$

(1)

$$\begin{array}{ll}{\mathrm{V}}_{\mathrm{o}2}=\frac{1}{\mathrm{D}\left(\mathrm{s}\right)}[-{\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{V}}_{\mathrm{i}1}-& {\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}4}{(\mathrm{V}}_{\mathrm{i}2}-{\mathrm{V}}_{\mathrm{i}4})\\ & +{(\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}{)\mathrm{V}}_{\mathrm{i}3}]\hfill \end{array}$$

(2)

$$\begin{array}{c}{\mathrm{V}}_{\mathrm{o}3}=\frac{1}{\mathrm{D}\left(\mathrm{s}\right)}{\left[\right(\mathrm{s}}^2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}{)\mathrm{V}}_{\mathrm{i}1}-{\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}3}{\mathrm{g}}_{\mathrm{m}4}{(\mathrm{V}}_{\mathrm{i}2}-{\mathrm{V}}_{\mathrm{i}4})\\ -{\mathrm{s}}^2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}3}{\mathrm{V}}_{\mathrm{i}3}]\end{array}$$

(3)

where the denominator D(s) is:

$$\mathrm{D}\left(\mathrm{s}\right)={\mathrm{s}}^2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}$$

(4)

From Equations (1)–(4), the VM transfer functions can be obtained according to input and output conditions as follows:

Part I: If only V_i1 is applied to the input voltage signal, V_in, the output voltage functions V_o1, V_o2, and V_o3 can be simultaneously obtained, as shown in Equations (5)–(7):

$$\frac{{\mathrm{V}}_{\mathrm{o}1}}{{\mathrm{V}}_{\mathrm{in}}}=\frac{{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{D}\left(\mathrm{s}\right)}$$

(5)

$$\frac{{\mathrm{V}}_{\mathrm{o}2}}{{\mathrm{V}}_{\mathrm{in}}}=\frac{-{\mathrm{sC}}_1{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}}{\mathrm{D}\left(\mathrm{s}\right)}$$

(6)

$$\frac{{\mathrm{V}}_{\mathrm{o}3}}{{\mathrm{V}}_{\mathrm{in}}}=\frac{{\mathrm{s}}^2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{D}\left(\mathrm{s}\right)}$$

(7)

Equations (5)–(7) show that the LPF, IBPF, and BRF signals can be simultaneously implemented.

Part II: According to Equation (3), the proposed biquad can realize six different generic filtering signals. Assuming that g_m1 = g_m3 and g_m2 = g_m4 = kg_m3, where k is the scale factor, Equation (3) can be rewritten as:

$${\mathrm{V}}_{\mathrm{o}3}=\frac{{(\mathrm{s}}^2+\frac{1}{{\mathsf{\tau }}_1{\mathsf{\tau }}_2}{)\mathrm{V}}_{\mathrm{i}1}-\mathrm{s}\frac{\mathrm{k}}{{\mathsf{\tau }}_2}{(\mathrm{V}}_{\mathrm{i}2}-{\mathrm{V}}_{\mathrm{i}4})-{\mathrm{s}}^2{\mathrm{V}}_{\mathrm{i}3}}{{\mathrm{s}}^2+\mathrm{s}\frac{\mathrm{k}}{{\mathsf{\tau }}_2}+\frac{1}{{\mathsf{\tau }}_1{\mathsf{\tau }}_2}}$$

(8)

where in Equation (8), τ₁ = (C₁/g_m2) and τ₂ = (C₂/g_m3) are the realized time-constants. Examining Equation (8) shows that the LPF, BPF, IBPF, inverting HPF (IHPF), BRF, and APF signals can be implemented by appropriately applying the input signals without the need for an inverting-type input signal. According to Equation (8), the following six different generic filtering signals can be obtained as:

(1)

LPF: V_i2 = V_i4 = 0 and V_i1 = V_i3 = V_in.

(2)

BPF: V_i1 = V_i2 = V_i3 = 0 and V_i4 = V_in.

(3)

IBPF: V_i1 = V_i3 = V_i4 = 0 and V_i2 = V_in.

(4)

IHPF: V_i1 = V_i2 = V_i4 = 0 and V_i3 = V_in.

(5)

BRF: V_i2 = V_i3 = V_i4 = 0 and V_i1 = V_in.

(6)

APF: V_i3 = V_i4 = 0 and V_i1 = V_i2 = V_in.

From Parts I and II, the biquad structure can be applied as a single-input and three-output VM multifunction biquad, and the LPF, IBPF, and BRF functions from the three output terminals can be realized without any component-matching conditions. It can also be applied as a four-input and single-output VM universal biquad, and can realize six different generic filtering signals in the same structure. The characteristic parameters ω_o and Q of the biquad can be calculated as:

$${\mathsf{\omega }}_{\mathrm{o}}=\sqrt{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3} }{{\mathrm{C}}_1{\mathrm{C}}_2}}, \mathrm{Q}={\mathrm{g}}_{\mathrm{m}1}\sqrt{\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}}$$

(9)

3. Proposed Electronically Tunable VM QO with Amplitude of Output Current

The biquad in Figure 2 can be used as the proposed QO with output current amplitude, when V_i1 and V_i2 are connected to the ground while V_i3 and V_i4 are connected to V_o1 and V_o2, respectively. The current output terminal of g_m4 in Figure 2 can be the AM/ASK modulated current output terminal, as shown in Figure 3. The characteristic equation of the electronically controllable OTA-C QO with amplitude of output current can be obtained as in Equation (10).

$${\mathrm{s}}^2{\mathrm{C}}_1{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{s}(\mathrm{C}}_1{\mathrm{g}}_{\mathrm{m}2}-{\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}{)\mathrm{g}}_{\mathrm{m}3}+{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}=0$$

(10)

Thus, CO and FO of the electronically controllable OTA-C QO are as follows:

$$\mathrm{CO}: {\mathrm{C}}_1{\mathrm{g}}_{\mathrm{m}2}\le {\mathrm{C}}_2{\mathrm{g}}_{\mathrm{m}1}$$

(11)

$$\mathrm{FO}: {\mathrm{f}}_{\mathrm{o}}=\frac{1}{2\mathsf{\pi }}\sqrt{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3} }{{\mathrm{C}}_1{\mathrm{C}}_2}}$$

(12)

From Equations (11) and (12), CO can be controlled independently from FO by varying g_m1, and FO can be controlled by g_m3. Hence, the proposed QO in Figure 3 has independent electronic control of CO and FO. The two output voltages, V_o1 and V_o3, of the proposed QO can be expressed as:

$$\frac{{\mathrm{V}}_{\mathrm{o}1}}{{\mathrm{V}}_{\mathrm{o}3}}=-\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{sC}}_1}$$

(13)

For sinusoidal steady state, Equation (13) becomes:

$$\frac{{\mathrm{V}}_{\mathrm{o}1}}{{\mathrm{V}}_{\mathrm{o}3}}=\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathsf{\omega }}_{\mathrm{o}}{\mathrm{C}}_1}{\mathrm{e}}^{\mathrm{j}90°}$$

(14)

Based on Equation (14), the proposed QO has two sinusoidal signal voltages with 90° phase difference. Furthermore, the output current I_AM/ASK can be obtained as:

$${\mathrm{I}}_{\mathrm{AM}/\mathrm{ASK}}={\mathrm{g}}_{\mathrm{m}4}{\mathrm{V}}_{\mathrm{o}2}$$

(15)

The amplitude of the sinusoidal signal I_AM/ASK can be independently controlled by the OTA’s g_m4 transconductance gain. The value of g_m4 can be varied through its I_B4. This means that the value of g_m4 can be tuned by I_B4 without affecting CO and FO. Therefore, if a modulation signal is applied to I_B4, the AM/ASK signals can be realized from I_AM/ASK. In addition, the output current I_AM/ASK port has high impedance. The load can be easily driven without using a buffering device.

4. Effect of Non-Idealities Analysis

In this section, the non-ideal circuit model of OTA will be considered. The non-ideal terminal effects of OTA are considered, including non-ideal current transfer error parameters and non-ideal finite input and output parasitic elements. Figure 4 illustrates a non-ideal OTA model with various parasitic elements [19], where R_Pj, R_nj, R_oj, R_qj, C_Pj, C_nj, C_oj, and C_qj represent the finite parasitic resistances and capacitances of the input and output terminals of the jth OTA, respectively. In Figure 4, the non-ideal dual-output currents of OTA are given as I_O = γg_m(V+ − V−) and I_Q = ηg_m(V+ − V−), where γ = 1 − ε_γi and η = 1 − ε_ηi. Here, ε_γi (|ε_γi|«1) and ε_ηi (|ε_ηi|«1) indicate the non-ideal dual-output currents of OTA current tracking errors. Taking the non-ideal OTA model into account, the biquad shown in Figure 2 can be modified to become the biquad presented in Figure 5. The effect of the non-ideal current transfer error parameters and the parasitic impedance elements on the characteristic parameters of ω_o and Q has been investigated to indicate the performance of the proposed biquad. Accounting for the parasitic elements in Figure 5, the non-ideal characteristic equation, D₁(s), of the equivalent circuit model in Figure 5 becomes:

$$\begin{array}{c}{\mathrm{D}}_1\left(\mathrm{s}\right)={\mathsf{\beta }}_1{\mathrm{s}}^2{\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }{\mathrm{g}}_{\mathrm{m}1}(1+\frac{\mathrm{P}}{{\mathsf{\beta }}_1{\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }{\mathrm{g}}_{\mathrm{m}1}})+{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{sC}}_1^{\prime }{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}(1+\frac{\mathrm{Q}}{{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{C}}_1^{\prime }{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}) \\ +{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}(1+\frac{\mathrm{R}}{{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}})\end{array}$$

(16)

where $\mathrm{P}={\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }{\mathrm{G}}_{3\mathrm{p}}+{\mathrm{C}}_1^{\prime }{\mathrm{C}}_{3\mathrm{p}}{\mathrm{G}}_{2\mathrm{p}}+{\mathrm{C}}_2^{\prime }{\mathrm{C}}_{3\mathrm{p}}{\mathrm{G}}_{1\mathrm{p}}$, $\mathrm{Q}={\mathrm{C}}_1^{\prime }{\mathrm{G}}_{2\mathrm{p}}{\mathrm{G}}_{3\mathrm{p}}+{\mathsf{\beta }}_1{\mathrm{C}}_1^{\prime }{\mathrm{G}}_{2\mathrm{p}}{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{C}}_2^{\prime }{\mathrm{G}}_{1\mathrm{p}}{\mathrm{G}}_{3\mathrm{p}}+{\mathsf{\beta }}_1{\mathrm{C}}_2^{\prime }{\mathrm{G}}_{1\mathrm{p}}{\mathrm{g}}_{\mathrm{m}1}+{\mathrm{C}}_{3\mathrm{p}}{\mathrm{G}}_{1\mathrm{p}}{\mathrm{G}}_{2\mathrm{p}}+{\mathrm{s}}^2{\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }{\mathrm{C}}_{3\mathrm{p}}$, $\mathrm{R}={\mathrm{G}}_{1\mathrm{p}}{\mathrm{G}}_{2\mathrm{p}}{\mathrm{G}}_{3\mathrm{p}}+{\mathsf{\beta }}_1{\mathrm{G}}_{1\mathrm{p}}{\mathrm{G}}_{2\mathrm{p}}{\mathrm{g}}_{\mathrm{m}1}+{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}{\mathrm{G}}_{1\mathrm{p}}$, ${\mathrm{C}}_1^{\prime }={\mathrm{C}}_1+{\mathrm{C}}_{1\mathrm{p}}$, ${\mathrm{C}}_2^{\prime }={\mathrm{C}}_2+{\mathrm{C}}_{2\mathrm{p}}$, ${\mathrm{C}}_{1\mathrm{p}}={\mathrm{C}}_{\mathrm{o}1}+{\mathrm{C}}_{\mathrm{p}2}$, ${\mathrm{C}}_{2\mathrm{p}}={\mathrm{C}}_{\mathrm{o}2}+{\mathrm{C}}_{\mathrm{o}4}+{\mathrm{C}}_{\mathrm{p}3}$, ${\mathrm{C}}_{3\mathrm{p}}={\mathrm{C}}_{\mathrm{q}1}+{\mathrm{C}}_{\mathrm{o}3}+{\mathrm{C}}_{\mathrm{n}1}+{\mathrm{C}}_{\mathrm{n}2}$, ${\mathrm{G}}_{1\mathrm{p}}=\frac{1}{{\mathrm{R}}_{1\mathrm{p}}}$, ${\mathrm{G}}_{2\mathrm{p}}=\frac{1}{{\mathrm{R}}_{2\mathrm{p}}}$, ${\mathrm{G}}_{3\mathrm{p}}=\frac{1}{{\mathrm{R}}_{3\mathrm{p}}}$, ${\mathrm{R}}_{1\mathrm{p}}={\mathrm{R}}_{\mathrm{o}1}{//\mathrm{R}}_{\mathrm{p}2}$, ${\mathrm{R}}_2{}_{\mathrm{p}}={\mathrm{R}}_{\mathrm{o}2}{//\mathrm{R}}_{\mathrm{o}4}{//\mathrm{R}}_{\mathrm{p}3}$, and ${\mathrm{R}}_3{}_{\mathrm{p}}={\mathrm{R}}_{\mathrm{q}1}{//\mathrm{R}}_{\mathrm{o}3}{//\mathrm{R}}_{\mathrm{n}1}{//\mathrm{R}}_{\mathrm{n}2}$.

According to Equation (16), non-ideal parasitic elements and non-ideal current transfer error parameters influence ω_o and Q. Neglecting the third-order effects, the biquad can operate as an ideal case by satisfying the following conditions:

$$\frac{1}{{\mathrm{sC}}_1^{\prime }}<<{\mathrm{R}}_{1\mathrm{p}}$$

(17)

$$\frac{1}{{\mathrm{sC}}_2^{\prime }}<<{\mathrm{R}}_{2\mathrm{p}}$$

(18)

$${\mathrm{sC}}_{3\mathrm{p}}+{\mathrm{G}}_{3\mathrm{p}}<<{\mathrm{g}}_{\mathrm{m}1}$$

(19)

Hence, D₁(s) of Figure 5 can be approximated as:

$${\mathrm{D}}_2\left(\mathrm{s}\right)={\mathsf{\beta }}_1{\mathrm{s}}^2{\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }{\mathrm{g}}_{\mathrm{m}1}+{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{sC}}_1^{\prime }{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}+{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{g}}_{\mathrm{m}1}{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}$$

(20)

In this case, ω_o and Q were modified as:

$${\mathsf{\omega }}_{\mathrm{o}}=\sqrt{\frac{{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}{{\mathsf{\beta }}_1{\mathrm{C}}_1^{\prime }{\mathrm{C}}_2^{\prime }}}$$

(21)

$$\mathrm{Q}={\mathrm{g}}_{\mathrm{m}1}\sqrt{\frac{{\mathsf{\alpha }}_1{\mathrm{C}}_2^{\prime }}{{\mathsf{\beta }}_1{\mathsf{\alpha }}_2{\mathsf{\alpha }}_3{\mathrm{C}}_1^{\prime }{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}3}}}$$

(22)

The sensitivities of the parameters ω_o and Q in Equations (21) and (22) were not larger than unity in magnitude. Therefore, the proposed biquad can be classified as an insensitive circuit.

5. Simulation and Experimental Results

To demonstrate the feasibility of the proposed OTA-C configurations, the proposed biquad and QO were simulated in PSpice using the macro-model of LT1228 IC and were measured in the laboratory using commercially available IC from Linear Technology Inc., LT1228. The power supply was ±15 V. The transconductance gain is given by g_m = 10 I_B [24]. This feature makes it possible to electronically control the transconductance gain.

5.1. The Electronically Tunable VM Biquad

As shown in Figure 2, the capacitors C₁ and C₂ were chosen as 2.2 nF and the bias currents were set to I_B1 = I_B2 = I_B3 = I_B4 = 200 µA (that is, g_m = 2 mS). This setting has been designed for LPF, BPF, IBPF, IHPF, BRF, and APF responses, with f_o = 144.7 kHz and Q = 1. Hence, the simulation and measurement functions of the LPF (V_o1), IBPF (V_o2), and BRF (V_o3) in Part I are depicted in Figure 6, Figure 7 and Figure 8, respectively. Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 represent the simulated and measured functions of the LPF, BPF, IBPF, IHPF, and APF in Part II, respectively. The power consumption for simulation and measurement is 0.92 and 0.96 W under ±15 V of supply voltage, respectively.

To illustrate the tunable property of natural frequency, f_o, without affecting the Q, the bias currents I_B1 = I_B2 = I_B3 were given as 100, 200, and 300 µA respectively, while keeping C₁ = C₂ = 2.2 nF and I_B4 = 200 µA. The simulation and measurement responses of the BPF on the V_o1 output terminal corresponding to different DC bias currents are shown in Figure 14, where V_i3 = V_in and V_i1 = V_i2 = V_i4 = 0. As depicted in Equation (9), Q can be used to adjust the transconductance gain, g_m1, without affecting the f_o. The feature of varying the Q without affecting the f_o is shown in Figure 15. Q varies (Q = 1, 1.5, and 2) when f_o is maintained at 144.7 kHz. In Figure 15, the I_B1 was given as 200, 300, and 400 µA respectively, while keeping C₁ = C₂ = 2.2 nF and I_B2 = I_B3 = I_B4 = 200 µA.

To verify the dynamic range, the proposed biquad was tested by repeating simulation of the sinusoidal input signal at f_o = 144.7 kHz. Figure 16 shows the input dynamic range of the BPF on the V_o3 output terminal when only V_i4 is applied to the input signal, which indicates an amplitude of 60 mV_pp (peak-to-peak) without signification distortion. The total harmonic distortion (THD) result for the BPF response is shown in Figure 17. From Figure 17, it can be obtained that the THD is about 3.92% when the input signal increases to 80 mV_pp. Figure 18 shows the measurement of the BPF response, which can be extended to an amplitude of 60 mV_pp without signification distortion. Figure 19 shows the simulation output noise of the BPF versus frequency. As shown in Figure 19, the maximum output noise reaches 7.937 nV/$\sqrt{\mathrm{Hz}}$. Hence, the output noise was small and did not affect the proposed filter.

To show the linearity of the biquad proposed in Figure 2, a 1 dB power gain compression point (P1dB) of the IBPF response has been measured. Figure 20 shows the measured output P1dB of the IBPF response on the V_o2 output terminal when only V_i1 is applied to the input signal, which indicates that the measured output P1dB point is about −7 dBm. The spectrum of the IBPF response is illustrated in Figure 21. The measured angular frequency is about 144.64 kHz, which is very close to the theoretical value of 144.69 kHz. To represent the nonlinearity of the biquad proposed in Figure 2, a two-tone testing of intermodulation distortion (IMD) has been used to characterize the nonlinearity of the IBPF response on the V_o2 output terminal. Figure 22 shows the spectrum of IBPF responses obtained by applying two-tone signals, f₁ and f₂, near the center frequency of f_o = 144.69 kHz for intermodulation characterization. In Figure 22, a low-frequency tone with f₁ = 143.69 kHz and a high-frequency tone with f₂ = 145.69 kHz are used, and the input amplitude is equal to 126 mV_pp. The measured value of the third-order IMD (IMD3) is about −50.83 dBc, and the measured value of the third-order intercept (TOI) point is about 5.136 dBm. The figure-of-merit (FoM) of the biquad was defined as [14]:

$$\mathrm{FoM} = \frac{\mathrm{Dynamic} \mathrm{range}\times {\mathrm{f}}_{\mathrm{o}}}{\mathrm{Power} \mathrm{dissipation}\times \mathrm{Supply} \mathrm{voltage}}$$

(23)

where f_o is the central frequency. According the measured IBPF frequency spectrum in Figure 21, the dynamic range between the fundamental tone and the largest spur of the spurious-free dynamic range is approximately 53.49 dB. Hence, the FoM of the proposed IBPF is approximately calculated as 268.75 × 10³.

5.2. The QO with Amplitude of Output Current

To demonstrate the theoretical study, the behavior of the proposed electronically controllable OTA-C QO with amplitude of output current has been verified by simulations and experimental measurements. The QO, as shown in Figure 3, was designed for the f_o = 72.3 kHz, with g_m1 = 1.18 mS, g_m2 = g_m3 = g_m4 = 1 mS, and C₁ = C₂ = 2.2 nF. As mentioned earlier, g_m1 was designed to be greater than the theoretical value to guarantee start-up of oscillation. Figure 23 shows the simulation quadrature output voltage waveforms. It can be seen from the simulation results that the FO is approximately equal to 71 kHz, close to the theoretical value, and the error rate is −1.8%. Figure 24 represents the experimental quadrature outputs V_o1 and V_o3. According to the experimental results, the measured FO is equal to 73.78 kHz, which is close to the theoretical value, and the error rate is 2%. Figure 25 shows the simulated and experimental results of the FO by varying the values of g_m3. Figure 26 shows how to calculate the phase noise using the Agilent phase noise measurement solution. The phase noise in the proposed electronically controllable OTA-C QO is lower than −74.93 dBc/Hz (at 1 kHz offset). To show the usability of the electronically controllable OTA-C QO in Figure 3, the applications of AM and ASK generators are confirmed in Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32, where I_B4 is sine wave, triangle wave, and pulse signals respectively, with a frequency of 7.4 kHz. These simulation and experimental measurements are close to the theoretical prediction, confirming the feasibility of the proposed electronically controllable OTA-C QO.

6. Conclusions

This paper presented an electronically tunable biquad and an output current amplitude, electronically tunable QO. The proposed circuits employed three OTAs, one DO-OTA, and two grounded capacitors. The single-input three-output VM multifunction biquad or the four-input single-output VM universal biquad can work in the same configuration. The proposed biquad can simultaneously realize LPF, IBPF, and BRF responses. By choosing the appropriate input terminals, the proposed biquad can realize all standard filtering functions without inverting-type voltage input signals. The characteristic parameters ω_o and Q are orthogonally controllable. The measured value of the power dissipation was 0.96 W, under 32 mA constant output current. The measured P1dB value of the output IBPF was −7 dBm, and the measured IMD3 and TOI values were −50.83 dBc and 5.136 dBm, respectively. The measured value of the spurious-free dynamic range was 53.49 dB, and the FoM of the filter was 268.75 × 10³. The proposed biquad configuration can be used as a type of electronically tunable QO with high-output impedance and amplitude-controllable sinusoidal current output signal. The proposed electronically tunable QO generated two sinusoidal output voltages with 90° phase difference, and CO and FO can be adjusted independently by adjusting the I_B of the OTA. Moreover, the proposed QO with electronically controllable amplitude of the sinusoidal current output signal can provide an AM signal or ASK signal, and is widely utilized in modulation systems. The proposed circuits have been verified by experiments and verified by OrCAD PSpice simulations to confirm the theoretical and expected assumptions.