Mathematical Expressions Useful for Tunable Properties of Simple Square Wave Generators

Abstract

This paper compares two electronically controllable solutions of triangular and square wave generators benefiting from a single IC package including all necessary active elements (modular cells fabricated in I3T 0.35 µm ON Semiconductor process operating with ±1.65 V supply voltage). Internal cells are used for construction of building blocks of the generator (integrator and Schmitt trigger/comparator). Proposed solutions have adjustable parameters dependent on the values of DC control voltages and currents. Attention is given to the mathematical expressions for the advantageous tunability of these generators. Theoretical mathematical functions comparing the standard linear formula with special expression for the frequency adjustment are evaluated and compared with experimentally obtained results. Mathematical functions prove that the proposed topologies improve efficiency of tunability and reduce overall complexity of both generators. Features of proposed solutions were verified experimentally. Both single-parameter tunable designs target on the operation in bands from tens to hundreds of kHz (from 13 kHz up to 251 kHz for the driving voltage between 0.05 V and 1.0 V for the first solution; from 12 kHz up to 847 kHz for the driving current between 5 µA and 140 µA for the second solution). A comparison with similar solutions indicates beneficial performance of the proposed solutions in tunability ratio vs. driving parameter ratio and also because simplicity of circuitry is low. The qualitative evaluation and comparison of parameters of both circuits is given and confirms theoretical expectations.

1. Introduction

Compactness and simplicity of functional (triangular and square wave) generators is very important for minimization of circuitry and also for effectivity of the design [1]. Modern active devices [2,3] may increase overall simplicity and bring interesting adjustable features to these generators (by the driving voltage or current). However, the actual number of active and passive elements cannot serve for evaluation of overall circuit suitability.

Table 1. Comparison of the single-parameter electronically tunable generators providing triangular and square waveforms.

Reference	Number. of Passive Elements (in Lab. Experiments)	Number of Active Elements	Number of IC Packages	Compact IC Device Including All Active Elements	Driving Force	Driving Force Range	Frequency Range (If More Than One, the Widest Reported)	FOM Ratio (f0max: f0min)/(Vset_gmmax: Vset_gmmin)	Type of Active Element(s)	Classification of Active Element(s) Used in Verification	Power Consumption [mW]
[4]	3	3	3	No	I	5 µA→5 mA	1 kHz→900 kHz	0.90	OTA	standard	N/A
[5]	3	3	N/A	No	I	0.1 µA→10 mA	0.1 kHz→3 MHz	0.30	OTA	standard	N/A
[6]	1	2	N/A	No	I	1 µA→100 µA	2 kHz→150 kHz	0.75	MO-CTTA	hypothetical	N/A
[7]	1	2	N/A	No	I	1 µA→100 µA	3 kHz→200 kHz	0.67	MO-CCCDTA	hypothetical	N/A
[8]	1	2	N/A	No	I	1 µA→100 µA	3 kHz→200 kHz	0.67	MO-CCCCTA	hypothetical	1.6
[9]	8	2	1	Yes	V	0.05 V→0.7 V	200 kHz→1.6 MHz	0.57	UCC + CCII	fabricated	N/A
[10]	3	2	N/A	No	I	5 µA→100 µA	211 kHz→2.83 MHz	0.67	VDBA	hypothetical	N/A
[11]	3	1	5	No	I	93 µA→37 µA	1 MHz→4.39 MHz	1.75	ZC-CG-VDCC	hypothetical	6.3
[12]	1	1	8	No	V	0.04 V→0.63 V	11 kHz→110 kHz	0.63	ZC-CG-VDCC	hypothetical	N/A
[13]	4	3	3	No	V	0.1 V→3 V *	2.55 kHz→25 kHz	0.33	ECCII + VGA + OPAMP	standard	N/A
[14]	2	1	2	No	V	0.1 V→3.0 V	0.56 MHz→11.39 MHz	0.68	CG-CDVA	hypothetical	N/A
[15]	2	3	3	No	V	0.73 V→1.2 V	0.5 MHz→2.01 MHz	1.68	DT + VGA	standard	N/A
[16]	2	2	2	No	V	0.1 V→1.5 V	1.16 MHz→6.7 MHz	0.39	ECCII + VGA	standard	N/A
[17]	2 (4)	1	7	No	I	0.5 mA→2 mA	5 kHz→25 kHz	1.25	MO-DXCCTA	hypothetical	N/A
[18]	2 (4)	1	6	No	I	0.1 mA→0.2 mA	8 kHz→16.5 kHz	1.03	MO-DVCCTA	hypothetical	226
[19]	3	2	N/A	No	I	1 µA→1 mA	3 kHz→2 MHz	0.67	MO-VDTA	hypothetical	14.3
[20]	2 (4)	1	8	No	I	1 µA→55 µA	1.69 MHz→3.88 MHz	0.04	MO-DXCCTA	hypothetical	1
[21]	1(2)	1	5	No	I	10 µA→80 µA	13 MHz→75 MHz	0.72	MO-CIDITA	hypothetical	**
[22]	1 (5)	1	3	No	I	10 µA→80 µA	8 kHz→58 kHz	0.91	MO-CCCCTA	hypothetical	0.27
[23]	3	2	2	No	I	10 µA→100 µA	20 kHz→120 kHz	0.60	OTA + CFOA + inverter	standard	N/A
[24]	4	2	4	No	I	100 µA→500 µA	4 kHz→9 kHz	0.45	VDGA	hypothetical	1.39
[25]	4	6	N/A	No	V	0.58 V→1.2 V	27 MHz→48 MHz	0.86	CCCII	standard	11.32
[26]	2	1	3	No	I	50 µA→150 µA	1.27 kHz→3.65 kHz	0.97	MO-CFDITA	hypothetical	1.25
Proposed
Section 3.1	1	2 (3) #	1	Yes	V	0.05 V→1.0 V	13 kHz→251 kHz	0.96	OTA + DVCC (OTA + VDDB + CCCII)	fabricated	43
Section 3.2	1	2	1	Yes	I	5 µA→140 µA	12 kHz→847 kHz	2.52	CCCII + OTA	fabricated	41

* two electronically controllable parameters can be used for the tuning of frequency in increased range but it also causes changes of waveforms shape (triangular-exponential) [13]; ** result 0.5 mW available for simulations only (not for experiment); # two active elements in figures but comparator completed from 2 cells of one IC device (Figure 1), N/A—not available. CCCII—current controlled current conveyor, CCII—current conveyor of second generation, CFOA—current feedback operational amplifier, MO-CFDITA—multiple output current follower differential input transconductance amplifier, CG-CDVA—controlled gain current and differential voltage amplifier, DT—diamond transistor, ECCII—electronically controllable CCII, MO-CCCCTA—multiple-output current controlled current conveyor transconductance amplifier, MO-CCCDTA—multiple-output current controlled current differencing transconductance amplifier, MO-CTTA—multi-output output current through transconductance amplifier, MO-CIDITA—multiple-output current inverting differential input transconductance amplifier, MO-DVCCTA—multiple-output differential voltage current conveyor transconductance amplifier, MO-DXCCTA—dual-X current conveyor transconductance amplifier, OPAMP—operational amplifier, UCC—universal current conveyor, VF—voltage follower, VDBA—voltage differencing buffered amplifier, VDGA—voltage differencing gain amplifier, VGA—variable gain amplifier, ZC-CG-VDCC—z-copy controlled gain voltage differencing current conveyor.

brings overview of recent development in the field of electronically tunable generators [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The design of these analog subsystems with requirement of a simple electronic adjustability of the repeating (oscillation) frequency or the duty cycle (see for example [1,4]) started with the simplest active elements. See solutions [4,5] for example. They provide a single electronically adjustable parameter such as operational transconductance amplifiers (OTA) [2,3]. The development continued with improvement of the features of these active elements and by introducing the adjustable elements with the capability of setting several parameters for driving of parameters of their applications in generators [6,7,8,9,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26]. However, some of these active elements are hypothetical only. Their structures are usually constructed from individual transistor models and, then, final application is verified only by simulations with such structures [5,6,7,8,10,14,19,20,23,25]. Another possibility is to create an active device as so-called behavioral model consisting of several off-the-shelf commercially available standard devices [11,12,17,18,20,21,22,24,26]. High complexity of such structures is a key disadvantage in this case (5 IC packages or more are required [11,12,17,18,20]). In some cases, the number of these sub-elements is quite high [12,17,20] and it influences overall behavior of the circuit (overall power consumption increases; significant impact of real parasitic features of sub-elements such as bandwidth degradation, additional noise, offsets, etc.). It can be a typical issue for a discrete form of implementation using commercially available elements, but it is not valid in general. The bandwidth degradation is typical for these constructions but effects as noises are individual to the specific case of each discrete solution and cannot be directly generalized. In fact, integration of the system or its active parts and majority of components increases immunity of the system against these effects. In specific cases [9] the used active elements were fabricated in the form of an integrated circuit (IC) where all required operations are available within a single chip.

We compared solutions [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and came to the following conclusions: (a) Very low number of solutions use DC voltage as driving force [9,12,14,15,16,25] for tuning or adjustment of additional parameters; (b) Most circuits employ DC current as a driving force 4–8,10,11,17–24,26]; (c) Only one generator was designed with completely integrated active circuitry [9]; (d) The number of active elements (without consideration of “complex behavior models”) is acceptable in all cases (2–3) but number of passive elements can be very high [9]; and (e) The ratio of the frequency range vs. the driving parameter range (evaluated as figure of merit—FOM in Table 1) is quite low (<0.8) in many of the reported results [5,6,7,8,9,10,12,13,14,16,19,20,21,23,24]. There are also solutions with excellent features for a wideband tuning [11,15]. However, these results were usually obtained by implementation of nonlinear dependence (inversely proportional [11] or exponential [15]) of parameters on driving force as well as solutions that require an increased number of active elements or IC packages [15] and passive elements [11]. A special type of tunability range extension has been observed in the case of [13]. This concept combined simultaneous multiparametric adjustment (voltage and current gain). Note that power consumptions are reported for simulated results (despite existing real experiments using off-the-shelf active elements—behavioral model of hypothetical active device) in most cases.

Based on this overview, there is a space for improvements in the view of complexity and tuning efficiency. The so-called modular approach [27,28] in development of the integrated building blocks for various purposes may help in this case with the design of adjustable functional generators. We prepared two novel solutions of generators using mutually interchanged order of active elements representing integrators and comparators. In both cases, tunability of the repeating frequency is achieved by the DC driving voltage or the DC driving current. The intentions of this paper (contribution of this work) can be summarized to:

(a)

introduction of two new and very simple solutions of electronically tunable generators employing single IC,

(b)

improvement of the tunability range and the tunability efficiency (ratio of the frequency range vs. the driving parameter range),

(c)

study of mathematical dependences of the frequency on the driving parameters and further consequences,

(d)

tests of tunability and efficiency for the nonlinear driving of the adjustable parameter,

(e)

comparison of expected and real (experimentally obtained) features of both circuits.

Organization of this paper is as follows: Section 1 compares previous works in the field, establishes important conclusions about possible improvements of state-of-the-art and introduces the main intentions of this work. Section 2 deals with the used active elements of modular concept (available as integrated cells on single IC package) fabricated for these purposes. Section 3 deals with two solutions of the proposed generators and includes theoretical analysis supplemented by real experimental results. Section 4 compares obtained features of both solutions qualitatively, and Section 5 concludes this work.

2. Active Devices—Integrated Modular Cells

Proposed generators use fabricated active elements available as cells of IC. The cells are accessible independently at pins of the IC package. Figure 1a shows schematic symbols of these devices. These cells (taken from the left side) are voltage multipliers having a current output terminal (based on core with bipolar junction transistors or CMOS transistors, i.e., abbreviated as BJT MLT and CMOS MLT), a single current controlled current conveyor of second generation (CCCII), and a voltage differencing differential buffer (VDDB). Details about these cells (internal structures and features) can be found in [29]. The detailed CMOS topologies are shown in Appendix A. The overall chip die area takes 1526 × 1526 µm where cells used in this work occupy 761 × 204 µm (CMOS MLT), 340 × 204 µm (BJT MLT), 616 × 204 µm (CCCII) and 430 × 204 µm (VDDB).

The relation between output current of the multiplier (BJT MLT or CMOS MLT) and input pair of differential voltages, which is I_o = (V_X1 − V_X2)∙(V_Y1 − V_Y2)∙k, allows us to employ this cell also as an operational transconductance amplifier [2,3] with voltage adjustable transconductance when V_Y2 = 0 because then I_o = (V_X1 − V_X2)∙V_Y1∙k. The term V_Y1∙k = g_m has meaning of transconductance where k ≅ 1.3 mA/V² [29] is valid for the CMOS core and k ≅ 4.9 mA/V² [29] for BJT core of the internal structure. The output impedance (resistive part) of CMOS/BJT MLTs (OTAs), used in presented designs, reaches a higher value than 100 kΩ [29] that has an insignificant effect on the generator performances. Moreover, this value is independent of DC adjustment of the OTA by voltage of selected input pair [29] (intentional topological feature—see Appendix A, Figure A1). These output resistances are more important in linear applications (influence of high-impedance nodes). The current controlled current conveyor is defined by standard relations I_Y = 0, V_X = V_Y + R_XI_X (when current flowing from X terminal is defined as I_X) and four outputs of current I_Z of both polarities (indicated by arrows in Figure 1a): I_Z4 = I_Z3 = –I_Z2 = –I_Z1 = I_X. The R_X parameter can be adjusted by bias current I_B. The real measured dependence can be approximated (within the range of I_B from 5 µA up to 350 µA) as R_X ≅ 0.317∙I_B^−0.8 [Ω, A]) [29]. The last active device (VDDB) serves for voltage sum and subtraction in accordance with equation V_W = V_Y1 − V_Y2 + V_Y3 [29]. Our proposal in this paper described in Section 3 requires construction of so-called differential voltage current conveyor (DVCC) [2,3] from available cells. It can be easily solved by appropriate interconnection of CCCII and VDDB cells as shown in Figure 1b. Some of available terminals are not necessary for intended application, therefore, they can be grounded. DVCC provides following inter-terminal transfers: V_X = (V_Y1 − V_Y2) + R_XI_X, I_Z+ = I_X.

3. Proposed Designs

Each triangular and square wave generating topology (Figure 2) requires interconnection of two functional blocks into the single loop [1,4,7]. These blocks are: (a) lossless adjustable integrator ensuring linear charge of capacity (linear increase/decrease of voltage across the capacitor when charging current changes polarity immediately), (b) comparator with hysteresis (so-called Schmitt trigger [1,4,7]). Internal cells of the proposed modular device (Figure 1) offer two very simple ways for synthesis of the topology in Figure 2 where both of them bring simple electronic adjustment of time constant of the integrator by DC driving voltage or DC driving current. Both solutions are discussed in detail in following subsections.

3.1. Resistor-Less Generator Using Voltage Tunable OTA-Based Integrator and Differential Voltage Current Conveyor in Schmitt Trigger

Figure 3a introduces the first proposed solution based on the single-output OTA having electronically controllable transconductance g_m (MLT with CMOS core, Figure 1) and DVCC-based Schmitt trigger. The solution is completely resistor-less because R_X is a real part of real input impedance of terminal X (not external discrete or internal integrated passive element) that can be adjusted by bias current I_B (see Section 2). The threshold levels of the comparator are intentionally set to V_Zsat with full positive feedback (because V_Y2 = V_Zsat when V_Y1 = V_Z, I_Z → 0 and, then, theoretical gain of the loop→∞, while practical gain is in range of hundreds). Therefore, the amplitude of the square wave (V_SQ) reaches ±V_Zsat (slightly lower than supply voltage ±1.65 V, from ±1.5 to ±1.6 V in experiments). The change of voltage drop across the capacitor C (Figure 2b) charged by constant current I_Cmax can be expressed as:

$$\Delta v_{\mathrm{C}}=v_{\mathrm{C}}(t=T/2)-v_{\mathrm{C}}(t=0)=\frac{1}{C}{\displaystyle \underset{0}{\overset{T/2}{\int }}{i}_{\mathrm{C}}(t)dt}=\frac{I_{\mathrm{Cmax}}}{C}{\displaystyle \underset{0}{\overset{T/2}{\int }}dt}=\frac{I_{\mathrm{Cmax}}}{C}{\left[t\right]}_0^{T/2}=\frac{I_{\mathrm{Cmax}}}{C}\frac{T}{2}$$

(1)

where ∆v_C = 2V_TR (V_TR—amplitude of triangular wave). Then (1) can be modified into 2 V_TR = I_Cmax∙T/(C∙2). The resulting equation for repeating frequency f₀ supposes maximal charging current I_Cmax ≅ V_SQ∙V_{set_gm}⋅k, equality of V_TR and V_SQ and has the form:

$$f_0=\frac{I_{\mathrm{Cmax}}}{4C\cdot V_{\mathrm{TR}}}\cong \frac{V_{\mathrm{SQ}}\cdot V_{\mathrm{set}\text{\_}\mathrm{gm}}\cdot k}{4C\cdot V_{\mathrm{TR}}}\cong \frac{V_{\mathrm{set}\text{\_}\mathrm{gm}}\cdot k}{4C}$$

(2)

where V_{set_gm} is DC voltage for adjusting of g_m and transconductance constant k = 1.3∙10⁻³ A/V² represents the transconductance parameter of CMOS MLT defined in previous text (Section 2). The sufficient linearity of g_m (when CMOS MLT operates as OTA) is ensured for voltage input range of ±0.5 V. Increasing g_m value above these values causes the limitation of the linearity range of OTA. It significantly influences accuracy of all calculations with g_m values when V_{set_gm} is too high and dependence of g_m on V_{set_gm} starts to be nonlinear (this nonlinearity is visible in real dependence of frequency on V_{set_gm}). However, I_Cmax should not be expressed by g_m (g_m = V_{set_gm} · k) as a small-signal parameter but as operation of the multiplier (I_Cmax = V_SQ · V_{set_gm}⋅k). Then, the further modification of (2), supposing ideal equality of V_TR and V_SQ amplitudes, gives independence of the frequency on these voltages in principle because comparator thresholds are set to amplitude of V_SQ (V_SQ = V_TR) by full positive feedback of DVCC from Y₁ to z₊. This is thanks to having a different internal principle of operation in case of our “OTA” [29], when compared with standard bipolar OTAs [2,3] having highly nonlinear DC V-I transfer response (I_out ~ tanh (V_in) also for very small input voltages (tens of mV), being always oversaturated in this type of application. Then, g_m of standard OTA is a really small-signal parameter that cannot be used in expressions. Of course, also our expectation is less accurate with increasing V_{set_gm} because of k (then also g_m) dependent on the large-signal behavior. This behavior is observed in Figure 4. Taking into account this empirical (based on measured data) nonlinear function of g_m = f (V_{set_gm}), the following equation for frequency can be established:

$$f_{0\text{\_}\mathrm{estimated}}\cong \frac{V_{\mathrm{set}\text{\_}\mathrm{gm}}\cdot k_{\mathrm{nonlinear}}}{4C}\cong \frac{V_{\mathrm{set}\text{\_}\mathrm{gm}}\cdot \left(-3\cdot {10}^{-3}\cdot V_{\mathrm{set}\text{\_}\mathrm{gm}}{}^2+70\cdot {10}^{-6}\cdot V_{\mathrm{set}\text{\_}\mathrm{gm}}+1.3\cdot {10}^{-3}\right)}{4C}$$

(3)

Numerical results of this equation (called as “estimated”) are now plotted in Figure 5 together with experimental and ideal results and follows real behavior of the circuit.

The relative sensitivity (nondimensional) of the repeating frequency f₀ on parameters of (2) are: S_{r_f0}^Vset_gm = ∂f₀/∂V_{set_gm} · V_{set_gm}/f₀ = 1 and S_{r_f0}^C = ∂f₀/∂C · C/f₀ = − 1. These results mean that 1% change of parameter causes 1% change (sign ± indicates direction of change) of the targeted parameter (f₀ in our case). These values are typical for similar types of circuits and their respective expressions defining the repeating frequency.

The circuit in Figure 3 was designed with the following values of parameters: C = 1 nF, R_X = 450 Ω (I_B = 100 µA), while V_{set_gm} varied from 0.05 V up to 1 V. This design fits the band of tens and hundreds of kHz that represents intended operational ranges of fabricated device [29] (units of MHz expected for linear applications). Note that OTA employs CMOS MLT (Figure 1a). Theoretically expected tunability of f₀ defined by (2) yields 16 kHz → 325 kHz. The experimentally obtained dependence of f₀ on V_{set_gm} was gained to be between 13 kHz and 251 kHz as shown in Figure 5. The deviation of measured points from the theoretical trace is given by real effects (Equation (3)) described below equation (2). This nonlinearity causes a deviation of g_m value that is not directly proportional to the value of V_{set_gm} above 0.5 V, but the g_m value still rises. Slew-rate limitation and the limited speed of the comparator part contribute as well. The time domain responses of output waveforms are shown in Figure 6 for V_{set_gm} = 1 V (f₀ = 251 kHz). The overall power consumption 43 mW was obtained at supply voltage ±1.65 V. The experimental setup used in both measurements of the proposed generators is shown in Appendix B.

We tested the complete generator (example from Figure 3) for process, voltage and temperature variation (PVT) and reached the following results. The frequency varies in the corner simulation between 136 and 192 kHz (nominal 172 kHz for V_{set_gm} = 0.5 V), i.e., it is variation about 20%. The variation of the duty cycle was also within 20%. The temperature variation (between 10 and 60 °C) causes smaller deviation from nominal value (between 147 and 189 kHz, i.e., less than 15%). This temperature range was selected as the most typical for usage of the device and its applications operating close to room temperature 20–30 °C. The duty cycle is almost not affected. Note that these results are worst case (the most pessimistic). The supply voltage variation influences bias conditions and should not be lower than ±1.5 V (frequency variation 30%).

The CMOS process usually limits fabricated value of passive capacitor up to several tens of pF. However, we should consider also so-called capacitance multipliers [30,31]. Such devices (generally called also as impedance converter) are able to form very large values of capacities by utilization of active devices. For instance, these circuits use OTAs and small grounded capacitor as in [30] or feedback chain of OTAs and current conveyor together with small floating capacitor as shown in [31] and as shown in Figure 3b. Definitions are described directly in Figure 3b. This solution ensures complete on-chip integration of the system even when large values of capacitors are required. The equivalent capacity C_eq can be obtained as original capacity C multiplied by parameters g_m1⋅g_m2/(g_m3⋅g_m4) [30]. Note that values of g_m1 = g_m2 = 500 µS and g_m3 = g_m4 = 15 µS (these values are standardly available in many CMOS processes) leads to C_eq ≅ 1 nF from C = 1 pF [30]. The bandwidth of the capacitance multiplier (based on MLTs/OTAs) has sufficiently high value regarding the expected operational range of proposed generator. The acceptable (for integration) capacity value can be selected even larger (10, 100 pF). Then the multiplication factor can be decreased to 100, 10 and effects of additional parasitic capacitances of the chip substrate (in parallel with this capacity), causing possible inaccuracies, are suppressed significantly. Even the ratio of g_ms (Figure 3b) will be lower as well as it increases the bandwidth of functionality of the capacitance multiplier. The impact of the decreasing quality factor of capacitor may influence the accuracy of frequency about 5% as the maximum (the duty cycle insignificantly).

3.2. Resistor-Less Generator Using Current Controlled Current Conveyor of Second Generation-Based Integrator and OTA-Based Schmitt Trigger

The second solution is shown in Figure 7. When compared to Figure 3 active blocks are interchanged. The comparator is now solved by OTA and the integrator by CCCII i.e., DVCC. The integrator was constructed from CCCII cell that allows R_X adjusting by I_B current and Schmitt trigger by simple OTA cell (as widely used in standard solutions [4]) based now on BJT MLT (see Section 2). The behavior is significantly different than the previous type of Schmitt trigger as shown in real results. The maximal current of the OTA is limited by R_X (I_B), then we can write: I_SQ = V_SQ/R_X. The threshold voltages (i.e., amplitude of V_TR) of the comparator can be found as:

$$V_{\mathrm{TR}}=V_{\mathrm{SQ}}\left[\frac{g_{\mathrm{m}}{R}_{\mathrm{X}}-1}{g_{\mathrm{m}}{R}_{\mathrm{X}}}\right]$$

(4)

The change of the voltage drop across the C can be expressed in general form also by (1). However, I_Cmax current charging C is defined differently than in previous case. This current I_Cmax is given by the bias current I_B and the current gain of the internal CMOS mirror in the biasing of the differential pair topology of CCCII [29]. Our CCCII provides I_Cmax = 10∙I_B. Then, we can write the final form (with help of (4)) for repeating frequency as:

$$f_0=\frac{I_{\mathrm{Cmax}}}{4C\cdot V_{\mathrm{TR}}}\cong \frac{10\cdot I_{\mathrm{B}}}{4C\cdot V_{\mathrm{SQ}}}\left(\frac{g_{\mathrm{m}}{R}_{\mathrm{X}}}{g_{\mathrm{m}}{R}_{\mathrm{X}}-1}\right)$$

(5)

that can be modified (in accordance to empirical equation for R_X = f(I_B) ≅ 0.317 · I_B^−0.8 [Ω, A]) as:

$$f_0\cong \frac{10\cdot I_{\mathrm{B}}}{4C\cdot V_{\mathrm{SQ}}}\left(\frac{g_{\mathrm{m}}\cdot 0.317\cdot I_{\mathrm{B}}{}^{-0.8}}{g_{\mathrm{m}}\cdot 0.317\cdot I_{\mathrm{B}}{}^{-0.8}-1}\right)=0.793\cdot \frac{I_{\mathrm{B}}{}^{0.2}}{C\cdot V_{\mathrm{SQ}}}\left(\frac{g_{\mathrm{m}}}{g_{\mathrm{m}}\cdot 0.317\cdot I_{\mathrm{B}}{}^{-0.8}-1}\right)$$

(6)

We can see that I_B also modifies the relation between V_TR and V_SQ. Therefore, increasing I_B causes a decrease in the V_TR output level in direction to higher frequencies (f₀).

The maximal available g_m of BJT MLT for V_{set_gm} = 1 V reaches approximately 3.3 mS, but dependence of g_m on V_{set_gm} is highly nonlinear and asymmetrical (linearity ensured up to 0.5 V). The numerical parameters of this design are C = 1 nF and V_{set_gm} = 1 V. Note that BJT MLT (Figure 1a) now serves as an OTA-based comparator. Dependence of f₀ on driving current I_B is shown in Figure 8. The range of tunability for I_B adjusted from 5 µA up to 140 µA allows a theoretical change of f₀ from 8 kHz up to 1010 kHz. Experimentally obtained results indicated tunability between 12.2 kHz and 847 kHz. The grounded terminal Y of the CCCII may serve for duty cycle adjustment if DC voltage is connected at this terminal (similarly as in [16]). The example of output waveforms in time domain (Figure 9) confirms expected behavior (I_B = 50 µA for f₀ = 137 kHz). Dependence of the output levels of generated signals on f₀ in Figure 9b confirms the discussed theoretical influence of I_B on the triangular waveform (decreasing trend). The maximal power consumption reaches 41 mW. The BJT MLT has almost constant power consumption, approximately 10 mW, but the CCCII needs a majority of the total power consumption. The power consumption of the generator is variable (based especially on the current I_B value) from approximately 30 mW up to 41 mW.

The relative sensitivities of the repeating frequency f₀ on parameters from (5) are S_{r_f0}^C = ∂f₀/∂C · C/f₀ = − 1, S_{r_f0}^VSQ = ∂f₀/∂V_SQ · V_SQ/f₀ = − 1. The sensitivity on g_m reaches S_{r_f0}^gm = ∂f₀/∂g_m · g_m/f₀ = − 1/(R_X⋅g_m − 1) ≅ − 1/(0.317⋅I_B^−0.8⋅g_m − 1). The sensitivity of f₀ on R_X has the following form S_{r_f0}^gm = S_{r_f0}^RX. Considering maximal available voltage amplitude of V_SQ = 1.65 V (saturation of the output equal to ±1.65 V) and design parameters noted above, the numerical values are between −0.06 and −3.75 when I_B varied between 5 µA and 140 µA (as indicated above for tunability purposes). It means that f₀ is more sensitive on fluctuations of g_m when I_B is higher. Fortunately, when g_m · R_X >> 1 (see right part of (5)), the whole term in parenthesis goes close to 1 (and generated waveforms have identical amplitude as well). This fact is determined by the performance of used active elements. Increasing the value of g_m causes a lower sensitivity range (g_m = 10 mS and the same range of I_B yields sensitivity from −0.02 to −0.35). The sensitivity on I_B theoretically reaches S_{r_f0}^IB = ∂f₀/∂I_B · I_B/f₀ = 1. However, the R_X value is also influenced by I_B, therefore S_{r_f0}^IB = ∂f₀/∂I_B · I_B/f₀ ≅ (317 · g_m − 200 · I_B^0.8)/(317 ⋅ g_m − 1000 · I_B^0.8). The numerical value of sensitivity varies between 1.05 and 4.00 (for 5 µA and 140 µA). This impact on the value of the repeating frequency is significant and very useful, because a higher value of I_B increases sensitivity of f₀ on its value. We can observe this behavior in Figure 8. It supports the initial idea about possible tunability extension and it represents an interesting conclusion for further methodologies of tunability control. Therefore, this paper has a theoretical (mathematic) impact observable in specific circuitry and it has important practical consequences. The first circuit (Figure 3) has a novel character of topology too, but standard tunable properties (linear) useful for comparison purposes (see

Table 2. Evaluation of experimental performances of both studied solutions.

	Figure 3	Figure 7
Parameter for f0 tuning	gm (Vset_gm)	RX (IB)
DC control range (full tested)	0.05 V→1.0 V	5 µA→140 µA
Range of f0 tuning (full tested)	13 kHz→251 kHz	12 kHz→847 kHz
Ratio (f0max: f0min)/(Vset/IB max: Vset/IB min)	0.96	2.52
Number of passive elements	1	1
Number of cells of IC used	3	2
Suitable for high-speed applications	No	Yes
Power consumption	43 mW	41 mW
Note	high-impedance output node of comparator	product gm∙RX of the comparator influences triangular level when tuned

). Moreover, the second circuit (Figure 7) also shows interesting consequences of special (untypical) expression for the repeating frequency on tunability.

The nonlinear dependence of the frequency on the DC control current (Figure 8) is an intentional feature of our circuit because of the requirement of tunability range extension in low-voltage designs [32,33]. These features have not been presented in case of the square waveforms before. It is known from sine wave oscillators [32,33]. The nonlinear dependence as presented in our work (Figure 7, Equation (6)) allows a wider range of the frequency adjustability than in case of standard methods.

We set the nominal frequency 136 kHz (I_B = 50 mA) and tested corner analysis. The frequency varies between 128 kHz and 158 kHz (15%). The duty has a variation of 10%. The temperature (between 10 and 60 °C) effects do not create significant deviations (less than 1%) in frequency and duty cycle. There is an impact on the triangular wave level (but expected as discussed in theory). The supply voltage variation has a less significant effect then in the previous case. The frequency variation is up to 30% maximally for supply no lower than ±1.4 V. The duty cycle is not influenced.

4. Comparison of Both Solutions

Table 2 summarizes features of both proposed solutions under very similar conditions. The voltage controlled f₀ in solution presented in Figure 3 brings significant simplification for driving system (microprocessor, digital to analog converter—DAC) because the generators can be tuned directly from driving system without any additional complex active circuitry as in case of operational amplifier-based solutions (tuned by the active replacement of resistor(s)). However, nonlinearity of dependence of g_m on V_{set_gm} (especially for V_{set_gm} > 0.5 V) causes a decrease in the tunable range of f₀ in comparison to the theory (deviation from linear dependence) as evident from Figure 5. The second solution (Figure 7) seems to be significantly better in range of f₀ readjustment by DC bias current I_B despite nonlinear dependence of R_X on I_B. However, the voltage to current conversion must be implemented (transforming resistor) for direct driving from common systems. Number of passive elements is similar in both solutions. In fact, both circuits require only one grounded capacitor. Nevertheless, significant difference of both concepts can be found in number of integrated parts (cells) of IC. The first solution (Figure 3) requires tree cells and second solution (Figure 7) only two. Level of output waveform of V_SQ output (Figure 3) reaches almost supply voltage (due to type of used Schmitt comparator—no sink for output current of the DVCC) but the second case of the Schmitt trigger (Figure 7) supplies integrator block by current flowing from OTA output to the X terminal of CCCII. Therefore, principal dependence of V_TR level (given by product of g_m∙R_X) is evident when R_X is used for f₀ tuning. The second circuit (Figure 7) seems to be more suitable for high-speed operations than the first circuit (Figure 3) because the comparator output has quite low-impedance (direct connection of R_X) in comparison to the high impedance node of DVCC-based solution of Schmitt trigger in Figure 3. Then, the parasitic capacities in nodes have a not so significant impact as in the case of high-impedance nodes.

As the conclusion of this discussion, we can observe that the change of number of active or passive elements cannot guarantee obtainment of high-quality solutions. Parameters other than complexity should be more important than differences in the amount of elements in circuit topologies. Nevertheless, both solutions are suitable for appropriate application based on particular requirements.

5. Conclusions

We proposed and designed two electronically tunable solutions of the triangular and square wave generator of similar complexity employing active subparts (cells) available in single IC package fabricated in I3T 0.35 µm ON Semiconductor technology. Both solutions are designed for operation over one decade of repeating frequency from 13 kHz (12 kHz, respectively) up to more than 250 kHz. Detailed analysis of obtained features and their comparison predetermines the first type (Figure 3) for simple linear voltage control from driving system. It is worth to use this generator (Figure 3) in applications below 1 MHz otherwise parasitic features start to be important sources of inaccuracies. The approximate value of 1 MHz is taken as the maximal frequency, where circuit in Figure 3 works. This maximal available frequency was tested with C = 100 pF, but dependences cannot be considered as accurate and operation as reliable because of waveform shapes degradation and invalidity of determined equations due to complex effects (limited slew-rate, nonlinearities, parasitic effects, etc.) acting together that are not analytically describable. This also includes high deviation of theoretically expected and real dependence of the frequency on the driving voltage. Similarly, the control voltage V_{set_gm} should be used up to 0.5 V (limitation of designed and fabricated cell [29]) when strictly linear dependence of repeating frequency on the driving voltage is required. Similar consideration is valid for the second solution of the generator (I_b approximately up to 120 µA in designed specification). The second solution (Figure 7) indicates better features than the circuit in Figure 3 also for higher frequency bands (no significant effects of parasitic features influencing integrator—value of C, or comparator) and nonlinear tuning of repeating frequency in wide range. Significantly wider range of f₀ readjustment than in the first solution results from nonlinear dependence of f₀ on I_B). However, nonlinear bias current (I_B) adjustment of R_X is not common method that can be immediately used by peripheral devices (microcontroller, DAC, etc.) because additional voltage to current converter should be used in advance. Based on comparison with known solutions in literature (Table 1), both proposed generators offer beneficial features. The dependence of triangular-wave amplitude on tuning process (Figure 7) does not mean significant issue because square-wave outputs are used much more frequently than triangular waveforms in standard mixed-mode systems especially, i.e., extension of frequency tunability range of square-wave can be found as important advantage. Special concepts of active elements (carbon nanotube FET based devices) have been reported quite recently also in the design of waveform generators [34]. However, the example presented in [34] is not utilizing the full potential of electronic adjustability so far (tuning by value of resistor only) and the resulting circuitry is complex. The complexity consists in three active and nine passive elements used for construction of generator of the triangular and square waveforms.

The general application field of square wave generators can be found in the circuit testing (wideband signal of various frequency), digital systems (clock), signal processing and sensing systems (phase locked loops, modulators and demodulators) [5,12,32,33], etc. The presented structures can be directly used, for example, in the measuring and sensing systems for the signal transmission reported in [35,36]. The applications of presented tunability methods are useful for wideband readjustment of the frequency over a large band because decreasing the supply voltage and the voltage used for adjustment is limited [32,33]. These issues are solved by the nonlinear exponential dependence of the frequency on the driving parameter [32] or power of the driving parameter [33]. As observed for the second solution (Figure 7), the increased sensitivity of the key parameter for the frequency tuning results in extended tunability range that was not reported before.