Two integrator loop quadrature oscillators: A review.

A review of the two integrator loop oscillator circuits providing two quadrature sinusoidal output voltages is given. All the circuits considered employ the minimum number of capacitors namely two except one circuit which uses three capacitors. The circuits considered are classified to four different classes. The first class includes floating capacitors and floating resistors and the active building blocks realizing these circuits are the Op Amp or the OTRA. The second class employs grounded capacitors and includes floating resistors and the active building blocks realizing these circuits are the DCVC or the unity gain cells or the CFOA. The third class employs grounded capacitors and grounded resistors and the active building blocks realizing these circuits are the CCII. The fourth class employs grounded capacitors and no resistors and the active building blocks realizing these circuits are the TA. Transformation methods showing the generation of different classes from each other is given in details and this is one of the main objectives of this paper.

Introduction

Sinusoidal oscillators are basic building blocks in active circuits and have several applications in electronics and communication circuits [1-3]. The oscillator circuits are available in different forms which includes Colpitts oscillator, Hartely oscillator, two integrator loop oscillators, Wien bridge oscillator and phase shift oscillators [1-3]. This paper is limited to the two integrator loop oscillators realized using different active building blocks.

The first active building block used in this paper is the Operational Amplifier (Op Amp) and four circuits are given [1-5]. Next the Operational Transresistance Amplifier (OTRA) [6-8] is used as the active element and two circuits are reviewed both employ two floating capacitors and four floating resistors [9]. Both of the Op Amp and the OTRA oscillators represent the first class of oscillators considered in this paper. The second class of oscillator circuit considered employ grounded capacitors [10] and includes floating resistors. The active element employed is the Differential Current Voltage Conveyor (DCVC) [11] also known as the Current Differencing Buffered Amplifier (CDBA) [12], the unity gain cells and the Current Feedback Operational Amplifier (CFOA) [13]. The third class of oscillator circuit considered employ grounded capacitors and grounded resistors. The active element employed is the single output current conveyor (CCII) [14] or the two output CCII. The fourth class considered employs grounded capacitors and no resistor, the active element employed is the transconductance amplifier (TA) [15]. Transformation methods using Nodal Admittance Matrix (NAM) expansion to show how the CFOA oscillator circuits leads to the generation of CCII oscillators are given.

Op Amp based oscillators

The analysis in this section is based on assuming ideal Op Amp having infinite gain which forces the two input voltages to be equal. The two input currents are zero due to the very high input impedance at both input nodes.

Four alternative circuits are given in Fig. 1a–d and will be summarized next.

Three Op Amp oscillator circuits

The first circuit given in this paper is the three Op Amp two integrator loop circuit shown in Fig. 1a. It employs three single input Op Amps, two floating capacitors and four floating resistors [1], its NAM equation is given by:

Fig. 1a

Three Op Amp oscillator circuit [1–4].

The radian frequency of oscillation is given by:

This circuit has the disadvantage of having no independent control on the oscillation condition. A modification to the circuit of Fig. 1a to provide control on the oscillation condition was given in Refs. [4,5].

Fig. 1b represents the recently reported three Op Amp oscillator circuit with independent control on both the condition of oscillation and on the frequency of oscillation, its NAM equation is given by:

Fig. 1b

Modified three Op Amp oscillator circuit [5].

It should be noted that the signs of G1 and G2 are opposite to their signs in Eq. (1).The characteristic equation in this case is given by:

The condition of oscillation is given by:

The condition of oscillation is controlled by G3 or G4 without affecting the frequency of oscillation which is given by Eq. (2) and is independently controlled by G1 or G2without affecting the condition of oscillation.

The third circuit is shown in Fig. 1c which was recently reported by Horng [16]. This circuit is non-canonic as it employs three capacitors. The condition of oscillation and the radian frequency of oscillation are summarized in Table 1. The condition of oscillation is controlled by G3 or G4 or G5 without affecting the frequency of oscillation. There is no independent control on the frequency of oscillation. Another non-canonic quadrature oscillator circuit which uses five capacitors was also introduced by Horng [16].

Fig. 1c

Three Op Amp oscillator circuit [16].

Table 1

Summary of the class I oscillator circuits.

Circuit figure number	Active element	Number of C	Number of G	Condition of oscillation	ωO	Ref.
1(a)	3 Op Amp	2 Floating	4 Floating	None	G_1G_2C_1C_2	[1–4]
1(b)	3 Op Amp	2 Floating	6 Floating	G4 = G3	G_1G_2C_1C_2	[5]
1(c)	3 Op Amp	3 Floating	5 Floating	G_3G_4G_5G_1G_2C_3=G_1C_1+G_2C_2	G_1G_2C_1C_2	[16]
1(d)	2 Op Amp	1 Grounded 1 Floating	1 Grounded 4 Floating	G3 = G1	G_2G_3C_1C_2	[1]
2(a),(b)	2 OTRA	2 Floating	4 Floating	G4 = G3	G_1G_2C_1C_2	[9]

Two Op Amp oscillator circuit

The fourth circuit shown in Fig. 1d is the two Op Amp two integrator loop circuit based on using Deboo integrator [1]. It uses one grounded capacitor, one floating capacitor, one grounded resistor and four floating resistors [1]; its NAM equation is given by:

Fig. 1d

Two Op Amp oscillator circuit [1].

The characteristic equation in this case is given by:

The condition of oscillation is given by:

The condition of oscillation is controlled by G1 without affecting the frequency of oscillation and the frequency of oscillation is controlled by G2 without affecting the condition of oscillation.

OTRA based oscillators

The second active building block considered in this paper is the OTRA which in the ideal case has the two inputs virtually grounded leading to circuits that are insensitive to stray capacitances. Also ideally the transresistance gain, Rm approaches infinity which forces the input currents to be equal.

The first circuit shown in Fig. 2a is the two OTRA two integrator loop originally introduced in [9]. It uses two floating capacitors and four floating resistors; its NAM equation is given by:

Fig. 2a

Two OTRA oscillator circuit I [9].

The characteristic equation in this case is given by:

The condition of oscillation is given by:

The condition of oscillation is controlled by G3 or G4 without affecting the frequency of oscillation and the radian frequency of oscillation is given by Eq. (2) and is controlled by G1 or G2 without affecting the condition of oscillation.

The second circuit which employs the same number of elements is shown in Fig. 2b [9]; its NAM equation is given by:

Fig. 2b

Two OTRA oscillator circuit II [9].

The condition of oscillation and the frequency of oscillation are the same as in the previous circuit and are summarized in Table 1.

The oscillators reported in the next sections of the paper employ grounded capacitors which provide advantages in integrated circuits [10].

DCVC based oscillators

The DCVC was introduced in [11] as a four terminal building block and defined by:

The currents IX1 and IX2 are pointing inwards at nodes X1 and X2 whereas the current IZ is out words from node Z as shown in Fig. 3a.

Fig. 3a

Two DCVC oscillator circuit I [17,19].

The DCVC was independently introduced and defined as a CDBA by Acar and Özoguz [12].

The first oscillator circuit considered in this section is shown in Fig. 3a which uses two DCVC, two grounded capacitors and four virtually grounded resistors and was introduced by Horng [17]; its NAM equation is given by:

The condition of oscillation is controlled by varying G1 or G3 and the frequency of oscillation is controlled by varying G2 or G4 without affecting the condition of oscillation.

The second oscillator circuit considered in this section is shown in Fig. 3b which uses two DCVC, two grounded capacitors and three virtually grounded resistors was introduced [18,19]; its NAM equation is given by:

Fig. 3b

Two DCVC oscillator circuit II [18,19].

The condition of oscillation is controlled by varying G1 without affecting the frequency of oscillation and the frequency of oscillation is controlled by varying G3 without affecting the condition of oscillation.

Unity gain cells based oscillators

The four types of the unity gain cells are defined as follows [20,21].

The Voltage Follower (VF) is defined by:

The Voltage Inverter (VI) is defined by:

The Current Follower (CF) is defined by:

The Current Inverter (CI) is defined by:

Oscillators using unity gain cells were introduced in the literature [22].

The first oscillator circuit considered in this section is shown in Fig. 4a. This is equivalent to the circuit shown in Fig. 3a and Eq. (15) applies to this circuit.

Fig. 4a

Unity gain cells oscillator circuit equivalent to Fig. 3a [5,19].

The second oscillator circuit considered in this section is shown in Fig. 4b.

Fig. 4b

Unity gain cells oscillator circuit equivalent to Fig. 3b [19].

This is equivalent to the circuit shown in Fig. 3b and Eq. (16) applies to this circuit.

CFOA based oscillators

The CFOA is a four-terminal active building block and is described by the following matrix equation [13]:

The first oscillator circuit considered in this section is shown in Fig. 5a [23-25]; its NAM equation is given by:

Fig. 5a

Two CFOA grounded capacitor oscillator circuit I [23–25].

The condition of oscillation is given by Eq. (8) and the radian frequency of oscillation is given by Eq. (9).

The second oscillator circuit considered in this section is shown in Fig. 5b [25]. This is equivalent to the circuit shown in Fig. 1d and Eq. (6) applies to this circuit. The results of this class of oscillators are summarized in Table 2.

Fig. 5b

Two CFOA grounded capacitor oscillator circuit II [25].

Table 2

Summary of the class II oscillator circuits.

Circuit Figure Number	Active Element	Number of C	Number of G	Condition of oscillation	ωO	Ref.
3(a)	2 DCVC	2 Grounded	4 Floating	G3 = G1	G_2G_4C_1C_2	[17,19]
3(b)	2 DCVC	2 Grounded	3 Floating	G2 = G1	G_2G_3C_1C_2	[18,19]
4(a)	2VF,CF,CI	2 Grounded	1 Grounded 3 Floating	G3 = G1	G_2G_4C_1C_2	[5,19]
4(b)	VF, 2 CF, VI	2 Grounded	3 Floating	G2 = G1	G_2G_3C_1C_2	[19]
5(a)	2 CFOA	2 Grounded	2 Grounded 1 Floating	G3 = G1	G_2G_3C_1C_2	[23–25]
5(b)	2 CFOA	2 Grounded	1 Grounded 2 Floating	G3 = G1	G_2G_3C_1C_2	[25]

Single output CCII based oscillators

The single output CCII is defined by:The positive sign in the third row applies to CCII+ whereas the negative sign is for CCII−.

In this section, the conventional systematic synthesis framework using NAM expansion [26-30] to synthesize oscillator circuits is used to transform the CFOA oscillators to single output CCII oscillators using grounded capacitors and grounded resistors.

Starting from Eq. (22) and add a third blank row and column and then connect a nullator between columns 1 and 3 and a current mirror (CM) between rows 1 and 3 in order to move −G3 from the 1, 1 position to the diagonal position 3, 3 to become G3 as follows:

Adding a fourth blank row and column to the above equation and then connect a nullator between columns 1 and 4 and a CM between rows 2 and 4 in order to move −G2 from the 2, 1 position to the diagonal position 4, 4 to become G2, the following NAM is obtained:

Adding a fifth blank row and column to the above equation and then connect a nullator between columns 2 and 5 and a norator between rows 1 and 5 in order to move G3 from the 1, 2 position to the diagonal position 5, 5 the following NAM is obtained:

The above equation is realized as a five node circuit using three nullators, one norator and two CM. Noting that the nullator and norator with a common terminal realize a CCII−, the nullator and CM with a common terminal realize a CCII+, the circuit shown in Fig. 6a using two CCII+ and one CCII− is obtained. It should be noted that the number of resistors is increased to four instead of three in the circuit of Fig. 5a. Note also that the resistors at nodes 3 and 5 are of the same value. This circuit has been introduced originally [31] with four grounded resistors of different values. Changing the magnitude of G3 at node 5 to become G4 [31] adds more degrees of freedom in the tuning of the oscillator circuit and the NAM equation will be modified to:

Fig. 6a

Three CCII oscillator circuit I [31,32].

This is similar to Eq. (15) except for the signs of G2 and G4. In this case the condition of oscillation is controlled by varying G1 or G3 and the frequency of oscillation is controlled by varying G2 or G4 without affecting the condition of oscillation. Table 3 summarizes these results.

Table 3

Summary of the class III oscillator circuits.

Circuit figure number	Active element	Number of C	Number of G	Condition of oscillation	ωO	Ref.
6(a)	2 CCII+, 1 CCII−	2 Grounded	4 Grounded	G3 = G1	G2G4C1C2	[31,32]
6(b)	2 CCII+, 1 CCII−	2 Grounded	4 Grounded	G3 = G1	G2G4C1C2	[32]
7(a)	DOCCII, CCII−	2 Grounded	3 Grounded	G3 = G1	G2G3C1C2	[34]
7(b)	BOCCII, CCII+	2 Grounded	3 Grounded	G3 = G1	G2G3C1C2	[34]
8(a)	2 CCII+, 1 CCII−	2 Grounded	4 Grounded	G3 = G1	G2G4C1C2	[31]
8(b)	2 CCII+, 1 CCII−	2 Grounded	4 Grounded	G3 = G1	G2G4C1C2	[32]

The second CCII oscillator circuit considered in this section is shown in Fig. 6b [31,32] and is generated from Fig. 1d or its equivalent CFOA oscillator circuit shown in Fig. 5b both are represented by Eq. (6) and using NAM expansion as follows.

Fig. 6b

Three CCII oscillator circuit II [32].

Starting from Eq. (6) and add a third blank row and column and then connect a nullator between columns 1 and 3 and a current mirror (CM) between rows 1 and 3 in order to move −G1 from the 1, 1 position to the diagonal position 3, 3 to become G1 therefore:

Adding a fourth blank row and column to the above equation and then connect a nullator between columns 1 and 4 and a norator between rows 2 and 4 in order to move G2 from the 2, 1 position to the diagonal position 4, 4 the following NAM is obtained:

Adding a fifth blank row and column to the above equation and then connect a nullator between columns 2 and 5 and a CM between rows 1 and 5 in order to move −G3 from the 1, 2 position to the diagonal position 5, 5 the following NAM is obtained:

The above equation is realized as shown in Fig. 6b using two CCII+ and one CCII−. Changing the magnitude of G3 at node 5 to become G4 adds more degrees of freedom in the tuning the oscillator circuit and the NAM equation in this case will be modified to:

This is similar to Eq. (15) except for the signs of G1 and G3. Table 3 summarizes these results.

It is worth noting that the two circuits of Fig. 6a and b can also be generated from Fig. 1b as given in Soliman [33]. They can also be generated from the circuits given in Soliman [34] as explained in the following section.

Two-output CCII based oscillators

The first type of the two-output CCII is the double output CCII++ which is defined by:

The oscillator circuit shown in Fig. 7a using a DOCCII++ and a CCII− was first introduced by Soliman [34]; its NAM equation is given by:

Fig. 7a

Grounded passive elements oscillator circuit I [34].

The second type of the two-output CCII is the balanced output CCII which is defined by:

The oscillator circuit shown in Fig. 7b using a BOCCII and a CCII+ was first introduced by Soliman [34]; its NAM equation is given by:

Fig. 7b

Grounded passive elements oscillator circuit II [34].

Replacing the DOCCII++ by its equivalent two single output CCII+ as demonstrated by Soliman [32] results in the circuit of Fig. 8a. Similarly the circuit of Fig. 8b can be generated from Fig. 7b by replacing the BOCCII by its equivalent single output CCII+ and single output CCII−. Note that G3 at node 5 has been taken as G4 to increase degrees of freedom in the tuning of the oscillator circuit. The results of this circuit are summarized in Table 3.

Fig. 8a

Single output CCII equivalent circuit to Fig. 7a [31].

Fig. 8b

Single output CCII equivalent circuit to Fig. 7b. [32].

Transconductance amplifier based oscillators

The TA is a very powerful active building block [35-40] in realizing oscillator circuits. Three different types of TA are considered in this section.

Single input single output TA based oscillators

There are two types of the single input single output TA (SISO TA) defined as follows. The TA+ has current pointing outwards and the TA− has current pointing inwards. It is well known that the CCII+ with a conductance connected to its X terminal realizes a TA+ with the same magnitude as the conductance. Similarly a CCII− with a conductance connected to its X terminal realizes a TA−.

The first circuit considered in this section is shown in Fig. 9a and is generated from Fig. 6a by replacing the three CCII by three TA. The conductance G1 is realized by the TA− shown to the left of Fig. 9a. The four SISO TA circuit is shown in Fig. 9a and has the same equation as that of Fig. 6a and Eq. (27) applies to it as well. It is worth noting that the TA of transconductance G3 realizes a negative grounded conductance at node 1 of magnitude −G3. This −G3 was realized by the CCII+ number 1 in Fig. 6a acting as a Negative Impedance Converter (NIC), with G3 acting as its load.

Fig. 9a

Four TA grounded capacitors oscillator generated from Fig. 6a.

The second SISO TA circuit considered in this section is shown in Fig. 9b and is generated from Fig. 6b; and Eq. (31) applies to this circuit. A TA circuit with the same topology has been reported in [40,41].

Fig. 9b

Four TA grounded capacitors oscillator generated from Fig. 6b.

Single input two-output TA based oscillators

The circuit shown in Fig. 10a is generated from Fig. 7a and it employs a Double Output TA with two positive outputs (DOTA++) and two-SISO TA. Its NAM equation is the same as given by Eq. (33).

Fig. 10a

Three TA oscillator circuit I.

The condition of oscillation is controlled by the TA (G1) without affecting the frequency of oscillation and the frequency of oscillation is controlled by the TA (G2) without affecting the condition of oscillation.

The circuit shown in Fig. 10b is generated from Fig. 7b and it employs a Balanced Output TA (BOTA) and two-SISO TA. Its NAM equation is the same as given by Eq. (35).

Fig. 10b

Three TA oscillator circuit II.

The circuit has the same condition and the radian frequency of oscillation as the previous circuit and is given in Table 4. A TA circuit with the same topology has been reported by Swamy et al. [42].

Table 4

Summary of the class IV oscillator circuits.

Circuit figure number	Active element	Number of C	Condition of oscillation	ωO	Ref.
9(a)	4 SISO TA	2 Grounded	G3 = G1	G2G4C1C2	_
9(b)	4 SISO TA	2 Grounded	G3 = G1	G2G4C1C2	_
10(a)	2 SISO TA 1 DOTA++	2 Grounded	G3 = G1	G2G3C1C2	_
10(b)	2 SISO TA 1 BOTA	2 Grounded	G3 = G1	G2G3C1C2	_
11	2 SISO TA 1 TISO TA	2 Grounded	G3 = G1	G2G3C1C2	[42–44]

Two-input single-output TA based oscillator

The circuit shown in Fig. 11 employs a two input TA and two SISO TA and its NAM equation is given by:

Fig. 11

Three TA oscillator with one two input TA [42–44].

This circuit has been reported [42-44] and is the adjoint [45,46] of the circuit shown in Fig. 10b.

Nonlinearity

It is well known that oscillators are nonlinear circuits, and the reported oscillators are based on a two integrator loop. Regarding amplitude control it should be noted that most oscillators rely on output voltage saturation of the active devices as a nonlinear mechanism for amplitude control. Regarding the amplitude control and stability of limit cycle this topic has been demonstrated on a current conveyor oscillator published by Soliman and Elwakil [47].

Discussion of non-idealities

Although the main objective of the paper is to review quadrature oscillators using various active building blocks and to show how they are related to each other, it may be useful to discuss the non-idealities of the active building blocks in the following sections.

Op Amp circuits

The NAM equations of the Op Amp oscillators shown in Fig. 1a–d are based on assuming ideal Op Amps having infinite gain are used. In the actual case the single pole model of the Op Amp gain should be used. This model is represented by the following equation [3]:where ωt is the unity gain bandwidth of the Op Amp; the use of this model in the oscillator circuit analysis affects the NAM equation [5] and hence both the condition of oscillation and the frequency of oscillation are changed from their ideal values.

OTRA circuits

The main parameter which is affecting the performance of the oscillator circuits of Fig. 2a and b is the finite and frequency dependent nature of the transresistance Rm which is represented by:where Ro is the DC value of the transresistance and ωο is the pole radian frequency of the OTRA.

DCVC circuits

The non-ideality of the DCVC is mainly due to the parasitic elements that are represented by the two X terminal resistances RX1, RX2 and the Z terminal capacitance CZ. For the circuit of Fig. 3a all the parasitic elements can be absorbed in the circuit parameters except RX1 of the first DCVC which affects the circuit operation. On the other hand for the circuit of Fig. 3b all the parasitic elements can be absorbed in the circuit parameters.

CFOA circuits

The CFOA parasitic elements are mainly represented by the X terminal resistance RX and the Z terminal capacitance CZ and the O terminal resistance RO. For the circuit of Fig. 5a the resistor R2 value should be adjusted to accommodate the added parasitic resistance RX2, the resistor R3 value should be adjusted to accommodate the added resistance (RX1 + RO2). Similarly C1 and C2 should be adjusted to accommodate the parasitic capacitances CZ1 and CZ2 respectively. For the circuit of Fig. 5b the resistor R1 value should be adjusted to accommodate the added parasitic resistance RX1, the resistor R2 value should be adjusted to accommodate the added parasitic resistance RX2 and the resistor R3 value should be adjusted to accommodate the added output terminal resistance RO2. Similarly C1 and C2 should be adjusted to accommodate the parasitic capacitances CZ1 and CZ2 respectively. Thus it is seen that the two circuits of Fig. 5a and b have the advantage of absorbing all parasitic elements within the design values of the circuit components.

CCII circuits

The CCII parasitic elements are mainly represented by the X terminal resistance RX and the Z terminal capacitance CZ. For the circuit of Fig. 6a the resistor R3 should be adjusted to accommodate the added parasitic resistance RX1, the resistor R2 value should be adjusted to accommodate the parasitic resistance RX2 and the resistor R4 should be adjusted to accommodate the parasitic resistance RX3. Similarly C1 and C2 should be adjusted to accommodate the parasitic capacitances (CZ1 + CZ3) and CZ2 respectively. Similarly for all other CCII oscillator circuits presented in Figs. 6b–8b. Thus it is seen that the CCII oscillator circuits presented in this paper have the advantage of absorbing all parasitic elements within the design values of the circuit components.

TA circuits

The main parameter which is affecting the performance of the TA oscillator circuits is the finite and frequency dependent nature of the transconductance G which is represented by:where Go is the DC value of the transconductance Gm and ωο is the pole radian frequency of the TA.

Conclusions

Quadrature oscillators are reviewed in this paper using several active building blocks namely; operational amplifier, operational transresistance amplifier, differential current voltage conveyor, unity gain cells, current feedback operational amplifier, current conveyors and transconductance amplifiers. Generation methods including nodal admittance matrix expansion shows that many oscillators that are reviewed in this paper can be obtained from each other. Simulation results or experimental for most of the oscillator circuits reviewed are available in the proper references indicated below and are not included to limit the paper length. It is worth noting that the chip size and the power dissipation for each of the reported oscillators will depend on the circuit design and the CMOS analog circuit realizing the active building block used.