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Int. J. Electron. Commun. (AEÜ) 66 (2012) 933– 936

Contents lists available at SciVerse ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

jou rn al h omepage: www.elsev ier .de /aeue

ingle CDTA-based current-mode quadrature oscillator

ie Jin, Chunhua Wang ∗

ollege of Information Science and Engineering, Hunan University, Changsha 410082, PR China

r t i c l e i n f o

rticle history:eceived 21 February 2012

a b s t r a c t

This letter presents a single CDTA (Current Differencing Transconductance Amplifier)-based current-mode quadrature oscillator (QO). The proposed circuit structure is very simple, which only consists of one

ccepted 30 March 2012

eywords:urrent Differencing Transconductancemplifier

CDTA, one resistor and two capacitors, and it is easy for monolithic integration. The oscillation frequencyof the QO can be electronically controlled by the bias current of the CDTA. Moreover, the oscillatorcan provide two quadrature current outputs. PSPICE simulation results are provided to verify all thetheoretical analysis.

urrent-modeuadrature oscillator

. Introduction

Oscillator is a very important building block in modern com-unication systems. The realization of oscillator using a variety of

ctive elements has been reported in [1–3], the CDTA-based oscil-ators are also reported in [4–12]. However, the works in [4–9] useoo many active elements (more than two CDTAs); the works in11–13] only have one CDTA, but they use more than three passivelements; the works in [4,5,9,10] lack the electronic adjustability;he work in [10] which is similar to our work uses only one CDTA,ne resistor and one capacitor. But the resistor in [10] is floatingvirtually grounded), and the resistor used in our work is grounded.

In this letter, a single CDTA-based current-mode quadraturescillator (QO) is proposed. The proposed QO only consists of oneDTA, one resistor and two capacitors, and it is easy for monolithic

ntegration. The oscillation frequency of the QO can be electroni-ally controlled by the bias current Ib of the CDTA. The QO can alsorovide two quadrature current outputs.

. The circuit symbol of CDTA and its realization

Fig. 1 shows the symbol of CDTA, and the terminal relation ofhe CDTA can be characterized by the following set of equations:

vp = vn = 0iz = ip − inix+ = gmvz = gmZziz

(1)

here p and n are the input terminals, z and x are the outputerminals, gm is the transconductance gain, and Zz is an externalmpedance connected to the terminal z. From Eq. (1), the current iz

∗ Corresponding author. Tel.: +86 13973125061; fax: +86 731 8822417.E-mail address: wch1227164@sina.com (C. Wang).

434-8411/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.aeue.2012.03.018

© 2012 Elsevier GmbH. All rights reserved.

is the difference of the currents at p and n (ip − in), and it flows fromthe terminal z into the impedance Zz. The voltage at the terminal zis transferred to a current at the terminal x (ix) by a transconduc-tance gain (gm), which can be generally electronically controllableby an external bias current Ib.

The CDTA used in this work is shown in Fig. 2 [4]. Assumingtransistors M16 and M17 are operated in saturation region, thetransconductance gain (gm) can be expressed as:

gm =√

�Cox

(W

L

)|Ib| (2)

where u is carrier mobility, Cox is the gate oxide capacitance perunit area.

From Eq. (2), we can know that the transconductance gain canbe electronically controlled by adjusting the bias current Ib.

3. The proposed quadrature oscillator

The proposed new oscillator is shown in Fig. 3, and it employsonly one CDTA, one resistor and two capacitors. In order to haveindependent quadrature current output, an auxiliary Zc terminal isused for utilizing the current through the capacitor C2 [14].

A routine circuit analysis using Eq. (1), we can get the charac-teristic equation of the QO is:

s2RC1C2 + s(C2 − gmRC1) + gm = 0 (3)

where gm is the transconductance of the CDTA.From Eq. (3), the condition of oscillation (CO) and frequency of

oscillation (FO) can be expressed as:

C2 = gmRC1 (4)

ω0 =√

gm

RC1C2(5)

934 J. Jin, C. Wang / Int. J. Electron. Commun. (AEÜ) 66 (2012) 933– 936

CDTA

Vp

Vn

Ip

In

p

n

Ix-

Ix+ x+ z

Iz

x-

Ib

Fig. 1. Symbol for the CDTA.

n p M1 M2

M3

M4 M6

M12

M11

M8 M10

M5

IBa

IBb

M15

M16

M24

M18 M19 M20 M21 M22 M23

Ib

Z

I+ I-

M13

Vcc

M17

M25 M26 M27

Zc

IBc IBc IBc

ooc

c

i

p

4

ac

a

Ideal

CDTA

Z

n

P

Io1 x3+

x4+

C1

C2

R

x2+

Io2

x1+

Zc

Rp

Rn

Rz

Cz

0.5Rx

2Cx

Rx

Cx

M7 M9 M14 Vss

Fig. 2. CMOS-based CDTA in this work [4].

From Eqs. (4) and (5), we can know that by modifying the valuef R to maintain the oscillation condition, when Ib changing, thescillation frequency can be electronically controlled by the biasurrent Ib.

From Fig. 3, the current transfer function between Io1 and Io2 is:

Io1(s)Io2(s)

= gm

sC2= Io1(jω)

Io2(jω)= gm

ωC2e−j90◦

(6)

So, the phase difference between Io1 and Io2 is 90◦, and the twourrents are quadrature.

When the oscillator in the sinusoidal steady state, setting Eq. (5)nto (6), we can get:

Io1(s)Io2(s)

=√

gmRC1

C2e−j90◦

(7)

Taking into account oscillation condition (4), the oscillator canrovide equal magnitude quadrature signals.

. Non-ideal analysis

Taking the parasitics and transfer errors of the CDTA intoccount [10,15], the small-signal model of the proposed oscillator

an be presented in Fig. 4.

In Fig. 4, R, C1 and C2 are the working passive elements. Rp

nd Rn are the series input parasitic resistances at terminals p

CDTA

Z

n

P

Io1 x+

x+

C1

C2

R

x+

Io2

x+ Zc

Fig. 3. Proposed current-mode quadrature oscillator.

Fig. 4. Non-ideal model of the proposed oscillator.

and n, respectively. (Rz//Cz) and (Rx//Cx) are the output parasiticimpedances at terminals z and x.

The transfer errors of CDTA can be expressed as:

iz = ˛pip − ˛ninix+ = ˇgmVz

(8)

where ˛p = 1 − εp is the current tracking error from terminal p to z,˛n = 1 − εn is the current tracking error from terminal n to z, and ˇis transconductance inaccuracy factor from the z to x terminals ofthe CDTA, respectively. There are two x+ terminals connected to nterminal in the proposed QO. If the two ix is not exactly equal, wecan assume ix1 = � ix2 = �ˇgmVz.

The main parasitics of the proposed QO are the influences of cur-rent tracking errors, transconductance inaccuracy factor, parasiticresistance (Rp) of p terminal, parasitic resistances and capacitance(Rz and Cz) of z terminal. Using the similar parasitics analysismethod in [10], taking transfer errors, transconductance inaccuracyfactor and parasitic current gain between ix1 and ix2 into account,the oscillation frequency can be rewritten as:

ω′0 =

√1 + ˛nˇgmRz

C1C ′2(R + Rp)Rz

(9)

The condition of oscillation can be rewritten as:

C ′2 +

{R + Rp

Rz+ [˛n − (1 + �)˛p]ˇgmR + ˛nˇgmRp

}C1 = 0 (10)

where C ′2 = C2 + Cz; ˛p and ˛n are the current tracking errors; ̌ is

transconductance inaccuracy factor between the z to x terminals;� is parasitic current gain between ix1 and ix2; Rp is the series inputparasitic resistance at terminal p; Rz is the parasitic resistance atterminal z.

Because 1 � ˛pgmRz and (R + Rp) � Rz, for the ideal values˛p = ˛n = ̌ = 1, Eqs. (9) and (10) can be simplified as:

ω′0 =

√1 + gmRz

C1C ′2(R + Rp)Rz

≈√

gm

C1C ′2(R + Rp)

(11)

C ′2 = gm(�R − Rp)C1 (12)

From Eqs. (11) and (12), it is clearly that the FO will deviate

slightly because of the parasitic resistance Rp. Rp and � also haveinfluence on the CO, these parameters are the intrinsic parastics ofthe CDTA, and the CDTA should be designed carefully for minimiz-ing these errors.

J. Jin, C. Wang / Int. J. Electron. Commun. (AEÜ) 66 (2012) 933– 936 935

c

I∣∣∣∣tac

5

To

tp3(aGia

t

Fig. 5. The simulated Io1 during initial state.

From Fig. 4, the current transfer function between Io1 and Io2an be rewritten as:

Io1(jω′0)

Io2(jω′0)

=√

R + Rp(2˛p − ˛n

)R − ˛nRp

e−j90◦(13)

Choosing R = 10 k�, Rp = 500 �, the magnitude ratio of Io1 ando2 is:

Io1(jω′0)

Io2(jω′0)

∣∣∣∣ =√

R + Rp

R − Rp≈ 1.05 (14)

Because of the parasitic resistance Rp at terminal p, the magni-ude of Io2 will be a little smaller than Io1. The method to alleviatell the influence of Rp on the oscillator is making R � Rp, under theondition of maintaining the oscillation condition.

. Simulation results

The performance of proposed circuits is verified using PSpice.he CDTA is realized as showed in Fig. 2, and the parameter valuesf passive element in Fig. 3 is C1 = 8.5 pF, C2 = 9 pF, R = 10 k�.

Fig. 5 shows the simulated Io1 during initial state. Fig. 6 showshe quadrature output waveforms; Fig. 7 shows the simulated out-ut spectrums, where the THD (total harmonic distortion) is about%. The distortion is due to the fact that the nonlinear elementssuch as diode, thermal resistor etc.) are not used to stabilize themplitude of oscillation [16]. In fact, we can add the AGC(Automaticain-Control) for keeping the amplitude stabilization and minimiz-

ng the THD of the QO. The AGC can be realized by using CDTA with simple diode–resistor network [12].

The bias current is chosen as: Ib = 200 �A, which should result inhe oscillation frequency of fosc = ωo/2� = 1.87 MHz. From Eq. (11),

Fig. 6. The simulated quadrature outputs Io1 and Io2.

Fig. 7. The simulated output spectrum of Io1.

the theoretical value is fosc = w′o/2� = 1.83 MHz, while the simu-

lated oscillation frequency is 1.73 MHz. Obviously, there are someadditional influences which are not taken into account through-out the nonideal analysis, and they further decrease the oscillationfrequency.

The critical gm value from Eq. (12) for providing the steady-state oscillation is 1.12 × 10−4 A/V. When Ib = 200 �A, the gm wouldbe 2.23 × 10−4 A/V. Reconsidering Eq. (4) with C1 = 8.5 pF, C2 = 9 pF,R = 10 k� and gm = 2.23 × 10−4 A/V, we can see that C2 < gmRC1,which indicates that the loop-gain is a little greater than unity, andit is the self-starting condition of the oscillator.

6. Conclusion

A single CDTA-based current-mode quadrature oscillator (QO) isproposed in this letter. The proposed QO has following advantages:(a) It only consists of one CDTA, one resistor and two capacitors, andit is easy for monolithic integration; (b) the oscillation frequencyof the QO can be electronically controlled by the bias current of theCDTA; (c) The QO can provide two quadrature current outputs. Themain drawback of the proposed circuit are that there is a floatingcapacitor, and a CDTA with four x+ terminals is used. We hope therelevant scholars to further improve it.

Acknowledgements

This work is supported by the National Natural Science Founda-tion of China (No. 60776021), and the authors would like to thankthe editors and anonymous reviewers for providing valuable com-ments which helped in improving the manuscript.

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