# DDCC-Based Quadrature Oscillator with  Grounded Capacitors and Resistors

Authors: Montree Kumngern; Kobchai Dejhan
Journal: Active and Passive Electronic Components
 (2009)

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## Abstract

A new voltage-mode quadrature oscillator using two differential difference current conveyors (DDCCs), two grounded capacitors, and three grounded resistors is presented. The proposed oscillator provides the following advantages: the oscillation condition and oscillation frequency are orthogonally controlled; the oscillation
frequency is controlled through a single grounded resistor; the use of only grounded capacitors and resistors makes the proposed circuit ideal for IC implementation; low passive and active sensitivities. Simulation results verifying the theoretical analysis are also included.


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## Body

## 1. Introduction

Second-generation  current conveyors (CCIIs) have been found
very useful in many applications. This is attributed to their  higher signal
bandwidths, greater linearity, and larger dynamic range  than those of the operational-amplifiers (op-amps) based ones. Recently,  Chiu et al. [1] proposed a
new current conveyor circuit called the differential difference current
conveyor (DDCC). The DDCC has the advantages of both the CCII and the
differential difference amplifier (DDA) (such as high input impedance and arithmetic
operation capability) [1].A quadrature oscillator typically provides two sinusoids with90° phase difference for a variety of applications, such as in
telecommunications for quadrature mixers, in single-sideband generators, in
direct-conversion receivers, or for measurement purposes in vector generators
or selective voltmeters [2, 3].
As a result, a number of circuits have been presented in technical literature [4–8].In this paper, a new voltage-mode quadrature
oscillator based on DDCCs is presented. The proposed circuit employs two DDCCs,
two grounded capacitors,  and three grounded resistors. The proposed circuit enjoys independent oscillation control
through a single grounded resistor and independent frequency control through
another single grounded resistor.
The use of grounded capacitors and resistors makes the proposed circuit
suitable for integrated circuit implementation [9]. The theoretical results are
verified by PSpice simulation.


## 2. Proposed Circuit

The electrical symbol of DDCC is
shown in Figure1. It was proposed in 1996 by Chiu et al. [1], and it enjoys  the advantages of
CCII and DDA such as larger signal bandwidth, greater linearity, wider dynamic
range, simple circuitry, low power consumption, and high input impedance [1].
The DDCC has three voltage input terminals:
Y1, Y2, and  Y3,
which have high input impedance. Terminal X is a low-impedance current input
terminal. There is a high-impedance current output terminal Z. The input-output
characteristics of the DDCC is described as [1] (1)(VXIY1IY2IY3IZ)=(1−1100000000000000001)(VY1VY2VY3IX). The CMOS realization of the DDCC
used in this paper for the quadrature oscillator circuit is shown in Figure 2.Figure 1
Electrical symbol of DDCC.Figure 2
The CMOS implementation of DDCC.The
proposed quadrature oscillator is shown in Figure3. It is composed of two DDCCs,
two grounded capacitors,  and three grounded resistors. The characteristic
equation of the circuit can be expressed as (2)s2C1C2R1R2R3+sC2R3(R1−R2)+R22R1=0. The oscillation condition and oscillation frequency can be
obtained as (3)R1≤R2,(4)ωo=R2C1C2R3R12. It can be seen from (3)
and (4)  that the oscillation condition can be adjusted by grounded resistor R1 or/and R2 and the oscillation frequency can be controlled by varying the grounded
resistor R3 without disturbing  the oscillation condition. This means that the oscillation
frequency and oscillation condition are orthogonaly controlled. By using a JFET
to replace R3, a voltage-controlled oscillator can be obtained [10]. From (4),
the passive sensitivities of proposed quadrature oscillator are low. From Figure 3,  DDCC2 along with  C2  and  R3
form the lossless integrator.
Hence, the phase difference ϕ between Vo1 and Vo2 is given by (5)ϕ=π−tan−1(ωR3C2). At ω=ω0, (5) can be obtained as ϕ=π/2, ensuring that the currents Vo1 and   Vo2 are in quadrature.Figure 3
Proposed voltage-mode quadrature oscillator.


## 3. Nonideal Effects

To  consider the nonideal effect of
a DDCC, taking the nonidealities of the DDCCs into account, the relationship
of the terminal voltages and currents  can be rewritten as [11](6)(VXIY1IY2IY3IZ)=(αk1−αk2αk30000000000000000βk)(VY1VY2VY3IX), where
αk1=1−εk1v and  εk1v(|εk1v|≪1) denotes the voltage tracking error from VY1  terminal to VX  terminal of the kth DDCC, αk2=1−εk2v and εk2v(|εk2v|≪1) denotes the voltage tracking error from VY2 terminal to VX terminal of the kth DDCC, αk3=1−εk3v  and εk3v(|εk3v|≪1)  denotes the voltage tracking error from VY3  terminal to VX terminal of the kth DDCC, and βk=1−εi and  εi(εi≪1) denotes the output current tracking error of the kth DDCC. The
characteristic equation of Figure 3 becomes (7)s2C1C2R1R2R3+sC2R3(R1−R2α13β1)+R22α12α21β1β2R1=0. The modified oscillation
condition and oscillation frequency are (8)R1≤R2α13β1,(9)ωo=R2α12α21β1β2C1C2R3R12. From (7) and (8), the tracking errors slightly change the oscillation  condition  and oscillation  frequency. However, the oscillation condition and oscillation
frequency still can be orthogonally controllable.


## 4. Simulation Results

To verify the theoretical  prediction of the proposed circuit, Figure3 has been simulated using PSpice simulation program. The DDCC in Figure 2 was
simulated using the 0.5 μm MIETEC as tabulated in Table 1 [12]. The
transistor aspect ratios of DDCC are listed in Table 2 [12], and the supply
voltages were VDD=−VSS = 2.5 V. The biasing voltage  VB  was taken as −1.7 V. The quadrature oscillator was designed with C1 = C2 =  50 pF,
R1 =  R3  = 5 kΩ, and R2 = 5.2 kΩ for the
oscillation frequency of f0 =  649 kHz that where
R2 was varied with R1 by (3) to ensure the oscillator will start. The
simulation results are shown in Figure 4. In this figure, the oscillation frequency of
640 kHz
is obtained. The oscillation frequency is 640 kHz instead of 649 kHz owing the
effect described in Section 3. According to (9), this drop-off would be caused
by voltage and current tracking errors. Figure 5 shows the simulated frequency
spectrums of  Vo1 and Vo2  in Figure 4. The result of the  Vo2 total harmonic distortion analysis is 1.02%. Figure 6 shows the simulation results of the proposed quadrature
oscillator in Figure 3 by varying the values of the resistor R3 with C1 = C2 = 50 pF,
R1 = 5 kΩ, and R2 = 5.2 kΩ. The nonidealities may be due to the ignored
tracking errors of the DDCC.Table 1
CMOS
model used in simulation.
.MODEL NMOS LEVEL = 3 UO  =  460.5 TOX = 1.0E-8 TPG = 1 VTO = +0.62JS = 1.08E-6  XJ = 0.15U  RS = 417 RSH = 2.73 LD = 0.04U VMAX = 130E3
NSUB = 1.71E17PB = 0.761 ETA = 0.00 THETA = 0.129 PHI = 0.905 GAMMA = 0.69 KAPPA = 0.10
CJ = 76.4E-5MJ = 0.357 CJSW = 5.68E-10 MJSW = 0.302 CGSO  = 1.38E-10 CGDO = 1.38E-10CGBO = 3.45E-10 KF = 3.07E-28 AF = 1 WD = +0.11U DELTA = +0.42 NFS = 1.2E11 DELL = 0ULIS = 2 ISTMP = 10 TT = 0.1E-9.MODEL PMOS LEVEL = 3 UO = 100 TOX = 1.0E-8 TPG = 1 VTO =  −0.58JS = 0.38E-6 XJ = 0.10U RS = 886 RSH = 1.81 LD = 0.03U VMAX = 113E3
NSUB = 2.08E17PB = 0.911 ETA = 00 THETA =  0.120 PHI =   0.905 GAMMA  = 0.76 KAPPA =  2 CJ = 85E-5MJ = 0.429 CJSW = 4.67E-10 MJSW = 0.631 CGSO = 1.38E-10 CGDO = 1.38E-10CGBO = 3.45E-10 KF = 1.08E-29 AF = 1 WD = +0.14U DELTA = 0.81 NFS = 0.52E11
DELL = 0ULIS = 2 ISTMP = 10 TT = 0.1E-9Table 2
Transistor
aspect ratios of the used DDCC.
MOS transistorW/LM1–M41.6/1M5-M68/1M7-M820/1M9-M1029/1M11-M1290/1Figure 4
The simulated quadrature output waveforms.Figure 5
The simulated frequency spectrum ofVo1 and  Vo2.Figure 6
Simulated results of the oscillation frequency.


## 5. Conclusions

In this paper, a new DDCC-based voltage-mode quadrature oscillator
circuit has been presented. The proposed circuit employs two DDCCs, two
grounded capacitors, and three grounded resistors. The proposed circuit enjoys
independent oscillation control through a single grounded resistor and
independent frequency control through another single grounded resistor. The use
of grounded capacitors makes the circuit attractive for integration, and the use
of grounded resistor for independent control of the oscillation frequency makes
the circuit attractive for the realization of voltage-controlled oscillators.
The active and passive sensitivities are no more than unity. Simulation
results, which confirm the theoretical analysis, are obtained.



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*Source: 987304-2009-04-12.xml*