Journal of Applied Science and Engineering, Vol. 19, No. 2, pp. 187-196 (2016)
DOI: 10.6180/jase.2016.19.2.09
Third-Order Sinusoidal Oscillator Using a Single
CMOS Operational Transresistance Amplifier
Hung-Chun Chien
Department of Electronic Engineering, Jinwen University of Science and Technology,
Xindian, Taiwan 231, R.O.C.
Abstract
This paper presents the design of a compact third-order sinusoidal oscillator based on an
operational transresistance amplifier (OTRA). The proposed circuit consists of a single OTRA
combined with three resistors and three capacitors. A review of relevant literature revealed that this is
the first study to design a third-order sinusoidal oscillator, constructed with a single OTRA and the
minimal number of passive components, with independent control of the oscillation condition and
frequency. This study involved a review of previous designs as well as related formulations, nonideal
analyses, and sensitivity discussions regarding the proposed circuit. Because the proposed circuit
features a low-impedance output, it can be applied in cascading and used without additional buffer
circuits. This study conducted simulations for the proposed circuit using HSPICE, and used
commercially integrated circuits (ICs) and discrete components for circuit implementation and testing
to verify its feasibility. Simulation and experimental results confirmed the validity of the proposed
oscillator circuit.
Key Words: Active-RC Circuit, Analog Circuit Design, Current-mode Circuit, Norton Amplifier,
Operational Transresistance Amplifier (OTRA), Sinusoidal Oscillator
1. Introduction
Early active-RC circuit designs have been widely
employed in voltage-dominated operational amplifiers
(OAs) as the active components in manufacturing various electronic circuits in applications [1]. Although OAs
have long been the predominant active components for
analog circuit designs, the conventional method they employ has been challenged by the requirements of high
performance circuit applications, such as higher signal
bandwidth, larger dynamic range and slew rate, lower
power consumption, functional versatility, and simple implementation. Previous research has revealed that the
current-mode circuit technique is more adequate for use
in analog circuit designs than in voltage-mode circuits.
Therefore, interest in designing analog circuits based on
the current-mode circuit technique has increased [2]. Sinusoidal oscillators are vital circuit cells, which are used
*Corresponding author. E-mail: hcchien@just.edu.tw
in numerous electronic and electrical applications, for
example, measurement meters, AC machine control systems, power converter control circuits, telecommunication modules, and medical equipment, etc. For decades,
OAs have been used as the active component in constructing sinusoidal oscillators [3]; however, their use
excessive amounts of passive components and have been
unable to provide an adequate performance in high-frequency operation (< 100 kHz). To overcome this problem
in designing sinusoidal oscillators, current active components based on the current-mode circuit technique
have become a feasible option, providing superior circuit performances compared with OA-based solutions.
Several typical studies have reported designs employing
various active components [4-9]. A three-terminal active component called the operational transresistance amplifier (OTRA), introduced in [10,11], was an improved
version of commercial Norton amplifiers. Because the
OTRA exhibited a higher operating speed, wider signal
bandwidth, and was free of the parasitic capacitance ef-
188
Hung-Chun Chien
fect [12], there was an increased interest in sinusoidal
oscillator designs based on this device. The first reported
single-phase sinusoidal oscillators employing OTRAs as
the active component appeared in [13]. However, the
circuits depicted in [13] (topologies in Figures 3(b) and
3(c) of [13]) required numerous passive components to
construct the second-order sinusoidal oscillators. Consequently, an improved topology designed with an OTRA,
three resistors, and two capacitors was proposed to reduce the number of passive component and enable a
resistance-controlled oscillation frequency [14]. However, that circuit still lacked an independent control for
adjusting the oscillation condition. Thus, a specific design process regarding the oscillation condition must be
first established for this circuit, to ensure the start of an
oscillation procedure. In 2010, researchers presented an
OTRA-based single-phase sinusoidal oscillator with an
independent control of the oscillation condition and frequency [15]. However, that topology exhibited the complex circuitry and was considered a noncanonical design, because it employed three capacitors to realize a
second-order sinusoidal oscillator. Recent studies have
reported feasible topologies to enrich the list of the
OTRA-based sinusoidal oscillator [16,17]. However, all
of the suggested oscillators presented in [16,17] still did
not enable an independent control of the oscillation condition and frequency. One study [18] presented another
feasible scheme featuring independent control of the
oscillation condition and frequency. However, the main
problem of this design is that it requires an additional
buffer circuit, thus indicating that the reported oscillator
cannot be treated as a single OTRA-based design. Other
types of sinusoidal oscillator using OTRAs as the active component include quadrature oscillators [13,19,20]
and multiphase oscillators [21]. An available method for
constructing quadrature oscillators is based on cascading
the active-RC filters and integrators in a closed loop connection [19,20], but requires more active and passive
components. For highly precise signal processing applications, third-order sinusoidal oscillators are preferable
to second-order sinusoidal oscillators. Third-order sinusoidal oscillators first appeared in 2002 [22]; subsequently, third-order sinusoidal oscillators have since attracted increased attention [23-26]. Although secondorder OTRA-based single-phase sinusoidal oscillators
have appeared in previous studies [13-18], third-order
single-phase sinusoidal oscillators employing a single
OTRA as the active component have not yet been discussed. Only two OTRA-based third-order quadrature
oscillators were explored in [27]. However, both of the
presented circuits employed three OTRAs, five resistors,
and three capacitors, and were therefore considerably
complex. This paper proposes the design of a third-order
OTRA-based single-phase sinusoidal oscillator. The proposed circuit is composed of a single OTRA, three resistors, and three capacitors, which enables independent
control of the oscillation condition and frequency. The
proposed oscillator, which features a compact circuitry,
contributes a novel approach to the study of third-order
sinusoidal oscillator designs. Table 1 displays a comparison of various solutions to illustrate the novelty and advancement of the proposed circuit. Among the existing
OTRA-based single-phase sinusoidal oscillators [1318], the proposed circuit exhibits the following advantages: 1) this is the first study to construct an OTRAbased third-order single-phase sinusoidal oscillator with
a single OTRA and low number of passive components;
2) independently controllable oscillation condition and
frequency (except for [15,18]); 3) low-impedance voltage output enabling cascading applications without supplementary buffer circuits; and 4) an operating frequency
higher than that exhibited by operational-amplifier-based
designs. The remainder of this paper is organized as follows. Section 2 introduces the OTRA followed by a circuit description of the proposed OTRA-based third-order sinusoidal oscillator and related governing equations.
Section 3 discusses the nonideal effects and sensitivity
analyses. Section 4 demonstrates the effectiveness of the
proposed circuit by presenting computer simulation and
experimental results. Finally, section 5 provides the conclusion to summarize this study.
2. Proposed OTRA-Based Third-Order
Sinusoidal Oscillator
An OTRA is a transimpedance-type active building
block with two current input terminals (I+ and I-) and a
voltage output terminal (Vo) [10,11]. The circuit symbol
of an OTRA is depicted in Figure 1. The terminal relations of an OTRA are characterized in (1), using a matrix
form, which indicates that the output voltage (Vo) is the
difference between two input currents (I+ and I-) multiplied by the transresistance gain (Rm). Ideally, the transresistance gain (Rm) approaches infinity, forcing the two
Third-Order Sinusoidal Oscillator Using a Single CMOS Operational Transresistance Amplifier
189
Table 1. Comparisons among various OTRA-based single-phase sinusoidal oscillators
Circuit
Component numbers
and connected type
Independent control
Circuit
of OC and OF/Type
order/Canonical
of OC and OF
design
control
Implement
technology/
Supply voltage
Measured highest
frequency/Total
harmonic distortion
Topology in
OTRA ´ 1
Figure 3(b) of
Resistor ´ 4
[13]
(Only one grounded)
Capacitor ´ 2
(Floating)
Second-order/
Yes
No (only of the
oscillation
condition)/
OC: by R
OF: NA
CMOS realization
(NA)/±2.5 V
Hundreds of kHz
(790 kHz)/NA
Topology in
Figure 3(c) of
[13]
OTRA ´ 1
Resistor ´ 3
(Floating)
Capacitor ´ 3
(Only one grounded)
Second-order/
No
No (only of the
oscillation
frequency)/
OC: NA
OF: by C
CMOS realization
(NA)/±2.5 V
Hundreds of kHz
(790 kHz)/NA
Topology in
Figure 1 of
[14]
OTRA ´ 1
Resistor ´ 3
(Only one grounded)
Capacitor ´ 2
(Floating)
Second-order/
Yes
No (only of the
oscillation
frequency)/
OC: NA
OF: by R
CMOS realization
(MIETEC 1.2-mm
CMOS process
technology)/±5 V
Several MHz
(1.59 MHz)/NA
Topology in
Figure 3 of
[15]
OTRA ´ 2
Resistor ´ 3
(Floating)
Capacitor ´ 3
(Floating)
Second-order/
No
Yes/
OC: by R
OF: by R
Commercial ICs
(AD844AN)/
NA
Hundreds of kHz
(159 kHz)/NA
Topology in
Figure 4(a) of
[16]
OTRA ´ 1
Resistor ´ 2
(Floating)
Capacitor ´ 2
(Floating)
Second-order/
Yes
NO/
OC: NA
OF: NA
CMOS realization
(TSMC 0.35-mm
CMOS process
technology)/
±2.5 V
Hundreds of kHz
(108 kHz)/2.6%
Topology in
OTRA ´ 1
Figure 4(b) of
Resistor ´ 3
[16]
(Only one grounded)
Capacitor ´ 2
(Floating)
Second-order/
Yes
No (only of the
oscillation
condition)/
OC: by R
OF: NA
CMOS realization
(TSMC 0.35-mm
CMOS process
technology)/±2.5 V
Hundreds of kHz
(150 kHz)/2.7%
OTRA ´ 1
Resistor ´ 2
(Floating)
Capacitor ´ 3
(Only one grounded)
Second-order/
No
No (only of the
oscillation
frequency)/
OC: NA
OF: by C
CMOS realization
(TSMC 0.35-mm
CMOS process
technology)/
±2.5 V
Hundreds of kHz
(260 kHz)/NA
Topology in
OTRA ´ 1
Figure 4(a) of
Resistor ´ 3
[17]
(Only one grounded)
Capacitor ´ 2
(Floating)
Second-order/
Yes
NO/
OC: NA
OF: NA
CMOS realization
(NA)/±2 V
Tens of kHz
(21.8 kHz)/NA
Topology in
Figure 4(c) of
[16]
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Hung-Chun Chien
Table 1. Continued
Component numbers
and connected type
Circuit
Topology in
Figure 6 of
[18]
Proposed
(Figure 3)
Independent control
Circuit
of OC and OF/Type
order/Canonical
of OC and OF
design
control
Implement
technology/
Supply voltage
Measured highest
frequency/Total
harmonic distortion
OTRA ´ 1
Voltage buffer ´ 1
Resistor ´ 3
(Only one grounded)
Capacitor ´ 2
(Floating)
Second-order/
Yes
Yes/
OC: by R
OF: by R
Commercial ICs
(AD844AN)/±5 V
Hundreds of kHz
(424 kHz)/NA
OTRA ´ 1
Resistor ´ 3
(Floating)
Capacitor ´ 3
(Only one floating)
Third-order/
Yes
Yes/
OC: by C
OF: by R
CMOS realization
(TSMC 0.35-mm
CMOS process
technology)/
±2.5 V
Several MHz
(1.08 MHz)/1.9%
OC: oscillation condition.
OF: oscillation frequency.
NA: not available or not tested.
Figure 1. Circuit symbol of an OTRA.
input currents (I+ and I-) to be equal. Several CMOS
OTRA realizations have been explored from the perspective of analog integrated-circuit implementation [10,11,
28].
using the commercially ICs (AD844ANs) [29] presented
in [16]. The AD844AN IC is a current-feedback amplifier, which differs from a conventional OA in that the
voltage at the noninverting input terminal is transferred
to the inverting input terminal. Thus, an inherent virtual
short exists between these two terminals. Also, the current flowing into the inverting input terminal is duplicated to the terminal Tz, and the output voltage is the
same as the voltage appearing at Tz. The implemented
circuit (Figure 3) was composed of two AD844ANs to
perform the functions of the virtual ground of the two input terminals (V+ and V-), and the output voltage (Vo)
proportional to the difference between the noninverting
(1)
Figure 2 depicts a feasible CMOS OTRA internal
circuit construction consisting of two main functional
blocks: a differential-current-controlled current source
(M1 - M12) followed by a voltage buffer circuit (M13 –
M20) [28]. In this circuit, the substrate terminals of all
PMOS and NMOS transistors are connected to positive
and negative supply voltages, respectively. Although previous studies have presented several implementations of
CMOS OTRA [10,11,28], a practical implementation of
OTRA is shown in Figure 3, which was constructed by
Figure 2. CMOS internal circuit construction of an OTRA
[28].
Third-Order Sinusoidal Oscillator Using a Single CMOS Operational Transresistance Amplifier
and inverting input currents (I+ and I-) multiplied by the
transresistance gain (Rm) of the OTRA [16]. If the Tz
node of the second AD844AN is open-circuited (Rm ®
¥), the circuit (Figure 3) is nearly an ideal OTRA. Therefore, the realization shown in Figure 3 can provide a viable method to practically implement an OTRA, which
is currently commercially unavailable.
Figure 4 depicts the circuit diagram of the proposed
third-order OTRA-based single-phase sinusoidal oscillator. The proposed circuit features compact circuitry that
employs a single OTRA, three resistors, and three capacitors to meet the minimum requirements of a third-order
oscillator. It is crucial to note that the output terminal of
the OTRA features a low-impedance voltage output. Thus,
the proposed oscillator does not require supplementary
voltage buffers for cascading applications.
Assuming an ideal OTRA, characterized in (1), a
routine circuit analysis using (1) yields the characteristic
equation of the circuit expressed in (2).
(2)
Based on (2), (3) and (4) express the formulas of the oscillation condition and frequency for the oscillator, respectively. Equations (3) and (4) indicate that the oscillation condition is independently controllable by using
the capacitor Cf without affecting the oscillation frequency, whereas the oscillation frequency can be adjusted by varying the resistor R. Thus, both the oscillation condition and oscillation frequency have non-interactive adjustment manners. Because the oscillation
frequency is independently controllable by using the resistor, a resistor-controlled sinusoidal oscillator is feasible.
C = 12Cf
(3)
Figure 3. OTRA constructed using commercially integrated
circuits (AD844ANs) [16].
191
(4)
Compared with the existing OTRA-based single-phase
sinusoidal oscillators reported in [13-18], this paper is
the first study to realize a noninteractive control oscillation condition and frequency third-order single-phase
sinusoidal oscillator employing a single OTRA with fewer passive components. No previously suggested single
OTRA-based single-phase sinusoidal oscillators have
enabled independent tuning of the oscillation condition
and frequency. Because OTRA-based signal-processing
and signal-generation circuits have been widely discussed in analog circuit designs, the new proposed circuit presents a practical design for OTRA application
and for teaching purposes in current analog circuit systems.
3. Non-Idealities Analysis
This section presents nonideal problems to determine
the influences of the nonideal effects on the proposed oscillator circuit (Figure 4). Non-idealities can originate
from finite transresistance gain and nonideal parasitic
elements of the OTRA. However, reports have shown
that because the input terminals of an OTRA are internally grounded, the influences of the parasitic elements
on the OTRA can be largely ignored [12]. Thus, the main
nonideality problem is caused by the finite transresistance gain of the OTRA [19]. As discussed in section 2,
ideally the transresistance gain of an OTRA approaches
infinity. In practice, the transresistance gain was a frequency-dependent finite value. This might have caused
the performance of the oscillator circuit (Figure 4) to deviate from the ideal scenario. A practical transresistance
gain Rm (s) can be expressed as in (5), which is determined from a single-pole model [19].
Figure 4. Circuit diagram of the proposed OTRA-based thirdorder sinusoidal oscillator.
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Hung-Chun Chien
(5)
where Rmo represents the DC transresistance gain, and
wo is the pole angular frequency of the OTRA. Rmo is
typically in the order of several hundreds of dBW and
wo is in the range of several hundreds of rad/sec. The
standard values of the parameters (Rmo and wo) for the
realized OTRA (Figure 2) are Rmo = 130 dBW (3.16
MW) and wo = 703.72 rad/sec (112 kHz) [19]. For highfrequency applications, the Rm (s) can be expressed as
in (6).
(6)
After the high-frequency single-pole model of the OTRA
was applied to the circuit (Figure 4) and a routine circuit analysis was repeated, tedious derivations yielded
the following modified characteristic equation:
(7)
Based on (7), the modified oscillation condition and
frequency were determined using (8) and (4), respectively. Equation (8) indicates that the finite transresistance gain of the OTRA caused the oscillation condition
to deviate from the expected values of the ideal scenario. This problem was overcome by slightly readjusting the value of the capacitor Cf, to ensure the oscillation starting procedure in the circuit (Figure 4).
C = 12(Cp + Cf)
(8)
The non-idealities analysis result shows that the oscillation frequency could still be independently controlled
by the resistor R without affecting the oscillation condition. By calculating (4), the sensitivities of the oscillator circuit were derived in (9).
(9)
Because the proposed third-order oscillator circuit used
a single OTRA and few passive components, the realization constituted a compact configuration. The following section presents computer simulation and experimental results to verify the feasibility of the proposed
circuit.
4. Simulation and Experimental Results
To verify the theoretical analyses, the proposed oscillator circuit (Figure 4) was simulated using HSPICE
circuit simulation program with a CMOS implementation
of the OTRA, as shown in Figure 2 with the supply voltages: VDD = -VSS = 2.5 V, the bias voltages Vg1 = -Vg2 =
1 V, and the bias currents IB1 = IB2 = 30 mA. The NMOS
and PMOS transistor models were adopted from the Taiwan Semiconductor Manufacturing Company (TSMC)
0.35 mm CMOS process parameters with the designed
aspect ratios of the MOS transistors: (W/L) M1-M18 = 2
mm /1.5 mm and (W/L) M19-M20 = 8 mm /0.8 mm. As an example, the oscillator circuit (Figure 4) was designed for
an oscillation frequency of fo = 100 kHz with the component values R = 227 W and Cf = 1 nF. In practice, the
value of a capacitor C = 12.1 nF was designed to be
slightly larger than the theoretical value (C = 12 nF) to
start the oscillations. This slight deviation was caused by
the finite-transresistance-gain effect, which was similar
to that anticipated based on (8). Figure 5 provides the
simulation results of the time waveform in a steady state,
using these component values, and its corresponding
frequency spectrum for the output. Simulation results
exhibited an oscillation frequency of fo = 99.1 kHz,
which was 0.9 % in disagreement with the designed value.
The percentage of the total harmonic distortion (THD)
and total power consumption were determined as 1.9 %
and 6.8 mW, respectively. However, the output voltage
swing (Figure 5) seems to be a low voltage level. To overcome this problem, an additional voltage amplifier is required to use the voltage across the output for circuit applications.
Figure 6 presents the simulation result of the transient response of the circuit (Figure 4) when building the
oscillations for the component values Cf = 1 nF, C = 12
nF, and R = 227 W. To investigate the lowest-and highest-applicable operating frequencies of the oscillator circuit (Figure 4), the specific simulation tests were conducted as follows. Figure 7(a) displays the simulation result for the output of the oscillator circuit in the steady
state with fo = 1.08 MHz, when adopting the component
values Cf = 0.5 nF, C = 6.06 nF, and R = 41.4 W . Under
this condition, the deviation of the oscillation frequency
between the theoretical value and the simulation result
was 1.82%. The highest operating frequency of the proposed oscillator was in the range of several MHz. To at-
Third-Order Sinusoidal Oscillator Using a Single CMOS Operational Transresistance Amplifier
193
Figure 5. Simulation results of the output Vo for the circuit (Figure 4) in the steady state, and its corresponding frequency spectrum.
Figure 6. Simulation result of the output Vo for the circuit (Figure 4) during the start-up oscillation.
tain the lowest-applicable operating frequency of the
oscillator circuit, a simulation example with component
values Cf = 0.1 mF, C = 1.21 mF, and R = 227 W was
conducted to explore this characteristic. Figure 7(b) presents the simulation result for the lowest frequency of
the oscillator with an oscillation frequency fo = 1.004
Hz which deviated from the theoretical value by 0.59%.
The simulation results demonstrated that the proposed
sinusoidal oscillator (Figure 4) operated in a wide frequency range from several hertz to several megahertz.
Although the output waveform seems immature in Figure
7(a), an advanced CMOS process technology or a highspeed OTRA component can be used to extend the frequency higher. For comparison, the limited frequency
range of the conventional OA-based oscillator is also ve-
rified by experiments. Experimental results demonstrate
that the conventional OA-based oscillator performs well
in a frequency range from several hertz to several tens of
kilohertz. Furthermore, the power consumption of the
OA-based oscillator is also determined as several tens of
mW. Thus, the proposed circuit (Figure 4) operates at a
higher operating frequency and a lower power consumption than that used in the OA-based design. To demonstrate
the property used to independently control the oscillation
frequency by using the resistor R, the capacitance values Cf = 1 nF, C = 12.1 nF were assigned to enable starting the oscillation procedure; R was varied from R = 0.1
kW to 1 kW in 0.1 kW steps to vary the oscillation frequency. Figure 8 presents the theoretical and simulation
results for the variation of the oscillation frequency. Sim-
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Hung-Chun Chien
ulation results confirmed that the proposed circuit (Figure 4) enabled independent control of tuning the oscillation frequency through resistor R without affecting the
oscillation condition. If concerned with the non-linearity problem of the proposed oscillator, the total harmonic
distortion analysis is required. The simulation result
was recorded in Figure 9. This study summarized that
the percentage of total harmonic distortion fluctuates
from 0.91% to 1.93% when the oscillation frequency
of the circuit (Figure 4) varies from 10 kHz to 100 kHz.
Due to very simple structure, the proposed oscillator enjoys an extremely low output harmonic distortion. Re-
garding the experimental tests for the proposed oscillator, Figure 3 presents a viable implementation of OTRA
for experimental testing. A prototype circuit of the proposed oscillator (Figure 4) was constructed using
AD844AN ICs and discrete passive components. The
AD844AN ICs were biased with a supply voltage of ±5
V. For example, the circuit (Figure 4) was designed for
the oscillation frequency of fo = 100 kHz with the component values of R = 215 W , Cf = 1 nF, and C = 12.8 nF.
In practice, the value of C was designed slightly larger
than the theoretical value (C = 12 nF) to start the oscillations. Figure 10 shows the experimental results of the
Figure 7. Simulation results of the circuit (Figure 4) for (a) the highest-applicable oscillations and (b) the lowest-applicable oscillations.
Figure 8. Variation of the oscillation frequency against R for
the circuit (Figure 4).
Figure 9. Simulation results of the total harmonic distortion
versus oscillation frequency for the circuit (Figure 4).
Third-Order Sinusoidal Oscillator Using a Single CMOS Operational Transresistance Amplifier
time waveform of the output voltage, and its output frequency spectrum. The experimental result in Figure
10(a) shows an oscillator frequency fo = 99.28 kHz,
which is close to the theoretical prediction (fo = 100 kHz).
The percentage error between theoretical and experimental result is 0.72%. The percentage total harmonic
distortion was measured as 2.55%. The experimental result shows the slight distortion on this circuit because
Figure 4 circuit suggests a compact circuit topology design. Simulation and experimental results supported the
theoretical analysis, confirming the feasibility of the proposed sinusoidal oscillator.
5. Conclusions
This study presented the design of a third-order
OTRA-based single-phase sinusoidal oscillator. The proposed circuit comprises a single OTRA, three resistors,
and three capacitors, and enables independent control of
the oscillation condition and frequency. This study pro-
Figure 10. Experimental results for the circuit (Figure 4): (a)
output waveform and (b) corresponding output frequency spectrum.
195
vided relevant governing equations of the oscillator and
investigated nonideal problems by using a single-pole
model of an OTRA. Simulation and experimental results
were provided to validate the effectiveness of the circuit.
The proposed topology provides a novel choice for realizing a compact third-order sinusoidal oscillator and will
have wide application in many electronic systems and industrial applications.
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Manuscript Received: Jul. 28, 2015
Accepted: Jan. 26, 2016