# Voltage-Mode and Current-Mode Resistorless Third-Order Quadrature Oscillator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{m}. However, the CCIITA cannot control the parasitic resistance, R

_{X}, at X port. Hence, when the CCIITA is used in some circuits, it must unavoidably require some external passive components, especially with the resistors. In contrast, the introduced current-controlled current conveyor transconductance amplifier (CCCCTA) [5,6] has two electronically adjustable ports, whereas the CCIITA has only one electronically adjustable port. The parasitic resistance, R

_{X}, at current input port of the CCCCTA can be adjusted by an input bias current, which does not require a resistor in practical applications. Intuitively, the CCCCTA is a versatile active building block, which provides the possibility of utilizing its transconductance gain, g

_{m}, and its equivalent input resistance, namely parasitic resistance, R

_{X}, to create a resistorless oscillator scheme. The multiple current output terminals of the CCCCTA can be easily obtained by adding additional current mirrors at these output terminals.

## 2. Proposed Method

_{1}and I

_{4}currents as:

_{1}and I

_{2}currents as follows:

## 3. Circuit Descriptions

_{m}, and parasitic resistance, R

_{X}, are electronic adjustable. Figure 2 shows the symbol of a multiple-output CCCCTA (MO-CCCCTA) [33]. The MO-CCCCTA properties can be expressed as V

_{X}= V

_{Y}+ I

_{X}R

_{X}, I

_{Z}= I

_{ZC}

_{1}= I

_{ZC}

_{2}= I

_{X}and I

_{O}

_{1}= I

_{O}

_{2}= g

_{m}V

_{Z}, where R

_{X}is the parasitic resistance of the X-terminal, and g

_{m}is the transconductance gain of the MO-CCCCTA. The bias currents, I

_{S}and I

_{B}, of MO-CCCCTA can control the parasitic resistance, R

_{X}, and the transconductance gain, g

_{m}, respectively [33,34]. Figure 3 shows the proposed schematic of the resistorless third-order quadrature oscillator, which consists of two MO-CCCCTAs and three grounded capacitors. The only grounded capacitors without a resistor structure in the proposed circuit are easily integrated into a single chip. In Figure 3, the transfer function, B/A, realizes a current-mode lossy integrator, and the transfer functions, C/B and D/C, realize two current-mode lossless integrators.

_{X}

_{1}without disturbing the FO, and the FO can be controlled by R

_{X}

_{2}without affecting the CO. In other words, the CO and FO are controlled independently by the bias currents, I

_{S1}and I

_{S2}, respectively. As the parasitic resistances, R

_{X}

_{1}and R

_{X}

_{2}, of the MO-CCCCTA can be electronically controlled by I

_{S}

_{1}and I

_{S}

_{2}, respectively, both the CO and FO are independently and electronically controllable. In a steady state, the two voltage outputs and two current outputs in Figure 3 are:

_{X}

_{2}will affect the oscillation frequency and the magnitude ratios of the generated quadrature signals. However, this problem can be solved by using C

_{1}= C

_{2}= C

_{3}, R

_{X}

_{1}= R

_{X}

_{2}, and letting the product g

_{m}

_{1}R

_{X}

_{1}= 1. Subsequently, the amplitude of voltages and currents in Equation (11) are equal and ensure k

_{1}= k

_{2}= 1.

_{o}

_{3}, can be given as follows:

_{m}

_{2}of the second MO-CCCCTA can be independently tuned by the bias current, I

_{B}

_{2}. This means that the g

_{m}

_{2}-value can be tuned by I

_{B}

_{2}, without disturbing the CO and FO. Thus, if a modulating signal is applied to I

_{B}

_{2}, then the AM/ASK signals can be obtained from I

_{o}

_{3}.

_{o}

_{2}and V

_{o}

_{1}or I

_{o}

_{2}and I

_{o}

_{1}is dependent on the factor k

_{1}or k

_{2}. Therefore, the mechanism of amplitude limitation will depend on the factors k

_{1}and k

_{2}. For complementary metal oxide semiconductor (CMOS) implementation of CCCCTA [34], the R

_{X}and g

_{m}are written as:

_{X}is the parasitic resistance of the X-terminal and g

_{m}is the transconductance gain of the CCCCTA. Here k is the physical transconductance parameter of the MOS transistor. I

_{S}and I

_{B}are the bias current used to control the parasitic resistance and transconductance gain, respectively. In other words, the variation of CCCCTA bias current range will limit the stability of the output amplitude.

## 4. Phase Noise and Phase Error Analysis

_{o}is oscillation frequency and ω = ω

_{o}+ Δω is carrier frequency approximation.

_{I}(t) is the in-phase signal and V

_{Q}(t) is the quadrature phase signal. The phase error of the proposed circuit can be obtained by Equations (17)–(19).

## 5. Non-Ideal Discussion

^{th}MO-CCCCTA can be rewritten as V

_{Xi}= β

_{i}V

_{Yi}+ I

_{Xi}R

_{Xi}, I

_{Zi}= α

_{i}I

_{X}, I

_{ZC}

_{1i}= η

_{i}I

_{Xi}, I

_{O}

_{1i}= γ

_{i}g

_{mi}V

_{Zi}and I

_{O}

_{2i}= λ

_{i}g

_{mi}V

_{Zi}for i = 1, 2, where β

_{i}, α

_{i}. γ

_{i}, η

_{i}and λ

_{i}represent the tracking errors of the MO-CCCCTA voltage and current, respectively [20]. The characteristic equation of the proposed circuit in Figure 3 can be rewritten as:

^{th}non-ideal MO-CCCCTA are (R

_{Yi}//C

_{Yi}) of terminal Y

_{i}, (R

_{Zi}//C

_{Zi}) of terminal Z

_{i}, (R

_{ZC}

_{1i}//C

_{ZC}

_{1i}) of terminal Z

_{C}

_{1i}, (R

_{ZC}

_{2i}//C

_{ZC}

_{2i}) of terminal Z

_{C}

_{2i}, (R

_{O}

_{1i}//C

_{O}

_{1i}) of terminal O

_{1i}, and (R

_{O}

_{2i}//C

_{O}

_{2i}) of terminal O

_{2i}[20,33]. The X, Z, and O

_{1}of the first MO-CCCCTA terminals connect to the external C

_{1}, C

_{2}and C

_{3}capacitors in parallel, respectively, as shown in Figure 3. If C

_{1}>> C

_{Z}

_{2}, C

_{2}>> C

_{Z}

_{1}and C

_{3}>> (C

_{O}

_{11}+ C

_{Y}

_{2}+ C

_{ZC}

_{11}), the effects of the parasitic capacitances, C

_{Z}

_{2}, C

_{Z}

_{1}and C

_{O}

_{11}, can be ignored. Hence, to minimize the effects of the MO-CCCCTA parasitic terminal impedances, the external capacitor values should be restricted by

## 6. Simulation Results

_{DD}= −V

_{SS}= 0.9 V. In order to get the f

_{o}≅ 2.12 MHz oscillation frequency of the sinusoidal output waveforms, the values of the active and passive components have been chosen as g

_{m}

_{1}= g

_{m}

_{2}= 133.33 µA/V (I

_{B}= 42.9 µA); R

_{X}

_{1}= R

_{X}

_{2}= 7.5 kΩ (I

_{S}= 3.96 µA); C

_{1}= 10.1 pF; and C

_{2}= C

_{3}= 10 pF. The variation of the transconductance value changes I

_{B}from 1.5 μA to 210 μA as depicted in Figure 6. When the bias current is larger than 160 μA, the transconductance gain is decreased as the transistors (M20, M21) enter the linear region from a saturation region. The maximum transconductance gain is approximately 256 µA/V. In other words, when the bias current is larger than 160 μA, the internal construction transistors of the MO-CCCCTA operates in the linear region, which will result in the distortion of the output swing. The steady state output waveforms of the quadrature voltages are shown in Figure 7 and the quadrature currents are shown in Figure 8. The oscillation frequency, f

_{o}, of the simulation results is equal to 2.11 MHz, which is consistent with the theoretical analysis. The total harmonic distortion (THD) analysis of the V

_{o}

_{1}, V

_{o}

_{2}, I

_{o}

_{1}and I

_{o}

_{2}are summarized in Table 4, Table 5, Table 6 and Table 7, respectively. Furthermore, the output current, I

_{o}

_{3}, can be used as an AM or ASK signal generator, when the amplitude of the input bias current, I

_{B}

_{2}, is applied to a sinusoidal signal, a triangular signal, or a pulse signal. Thus, the proposed oscillator can generate either AM or ASK signals without additional elements. Figure 9 and Figure 10 show the simulated results of the proposed circuit serving as an AM signal generator, when the input bias current, I

_{B}

_{2}, is applied to a sinusoidal signal or a triangular signal with a 200-kHz frequency. Figure 11 shows the simulated results of the proposed circuit serving as an ASK signal generator when the input bias current, I

_{B}

_{2}, is applied to a pulse signal with a 200-kHz frequency.

_{m}-value of the MAX435 is equal to 4/R

_{g}, where R

_{g}is an external resistor [36]. In general, the applicability of such current-feedback operational amplifier (CFOA)-based oscillators is usually limited to a few hundred kilohertz [37,38]. Figure 13 shows the experimental result of the frequency range of a commercially available AD844 with ±5 V DC supply. In Figure 13, the values of the resistors were chosen as R

_{1}= R

_{2}= 1 kΩ (5 kΩ, 10 kΩ), R

_{L}= 500 Ω, C

_{L}= 10 pF, and the input power was 0 dBm. Figure 14 is the measured result of a spectrum analyzer, which shows that the measured result of the frequency range of AD844 was limited to 2 MHz. The passive component values of the circuit in Figure 12 were set as g

_{m}

_{1}= 1.428 mS (i.e., R

_{g}= 2.8 kΩ), R

_{X}

_{1}= 0.73 kΩ, R

_{X}

_{2}= 0.7 kΩ, R = 1 kΩ and C

_{1}= C

_{2}= C

_{3}= 750 pF, where R

_{X}

_{1}was designed to be larger than the theoretical value to ensure that the oscillator will start, before the centre frequency was obtained as f

_{o}= 303.15 kHz. The oscilloscope output waveforms, V

_{o}

_{1}and V

_{o}

_{2}, of the proposed oscillator are shown in Figure 15, and the X–Y plot of V

_{o}

_{1}and V

_{o}

_{2}output voltages are shown in Figure 16. The experimental oscillation frequency in Figure 15 is 295.2 kHz, which is close to the theoretical value of 303.15 kHz with a 2.62% error rate. Figure 17, Figure 18 and Figure 19 show the AM and the ASK signal outputs of the quadrature oscillator, where the modulation signal, I

_{B}

_{2}, applied a sinusoidal signal, a triangular signal, or a pulse signal with a 30-kHz frequency. The experimental results are close to the theoretical analysis of Equation (12).

_{o}

_{1}, is shown in Figure 20. The measured oscillation frequency was 296.68 kHz, which is close to the theoretical value of 303.15 kHz with a 2.13% error rate. The THD, including the first harmonic through to the ninth harmonic components, is approximately 2.35%. The experimental results are consistent with the theoretical values. For oscillators, noise is a major concern where even a small noise in an oscillator will cause dramatic changes in its frequency spectrum and timing properties. Figure 21 shows the phase noise using the Agilent phase noise measurement solution. The phase noise of the proposed oscillator is less than −88.2 dBc/Hz at 10 kHz offset. Figure 22 shows the root mean square (RMS) jitter of the proposed oscillator, which is 6 ns at an operating frequency of 295.2 kHz.

## 7. Conclusions

^{o}phase difference; two high-impedance quadrature current output signals with a 90

^{o}phase difference; and one high-impedance current output signal with an electronically controlled amplitude of the sinusoidal signal, all simultaneously. The proposed oscillator can generate either AM or ASK signals for communication systems. Furthermore, the oscillation condition and oscillation frequency are independently adjustable by two different input bias currents of the MO-CCCCTAs. Since the proposed oscillator does not require external resistors and uses only grounded capacitors, it is suitable for integrated circuit implementation. The simulation and experimental results validate the feasibility of the proposed theory.

## Author Contributions

## Conflict of Interest

## References

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**Figure 4.**Linear oscillatory system in Reference [35].

**Figure 9.**Simulated result of operation as an AM signal generator when I

_{B}

_{2}is the applied sinusoidal signal.

**Figure 10.**Simulated result of operation as an AM signal generator when I

_{B}

_{2}is the applied triangular signal.

**Figure 11.**Simulated result of operation as an ASK signal generator when I

_{B}

_{2}is the applied pulse signal.

**Figure 12.**A possible implement method of the proposed oscillator by using commercially available ICs.

**Figure 14.**The experimental evidence frequency ranges of AD844 when R

_{1}= R

_{2}= 1 kΩ (yellow line; R

_{1}= R

_{2}= 5 kΩ (cyan line); and R

_{1}= R

_{2}= 10 kΩ (purple line).

**Figure 15.**The experimental waveforms of the quadrature outputs V

_{o}

_{1}(channel 1) and V

_{o}

_{2}(channel 2).

**Figure 17.**Measured result of operation as an AM signal generator (channel 2) when I

_{B}

_{2}is the applied sinusoidal signal (channel 1).

**Figure 18.**Measured result of operation as an AM signal generator (channel 2) when I

_{B}

_{2}is the applied triangular signal (channel 1).

**Figure 19.**Measured result of operation as an AM signal generator (channel 2) when I

_{B}

_{2}is the applied pulse signal (channel 1).

**Figure 22.**The measured RMS jitter of the proposed oscillator at an operating frequency of 295.2 kHz.

**Table 1.**Number of component and number of output signal comparisons with previous third-order oscillators.

Reference | No. of Components Used | No. of Output Signals Used | ||
---|---|---|---|---|

Active Elements | C/R/Resistorless Structure | Voltage | Current | |

[22] Figure 7 | 3 OTAs | 3/0/yes | 1 | 0 |

[23] | 4 MO-CCCIIs | 3/0/yes | 0 | 4 |

[24] Figure 1 | 3 CCIIs | 3/5/no | 2 | 0 |

[25] | 2 MO-CCCCTAs | 3/0/yes | 0 | 2 |

[26] Figure 2 | 3 MO-DVCCs | 3/3/no | 5 | 2 |

[26] Figure 3 | 3 VC-DVCCs | 3/0/yes | 3 | 2 |

[27] | 3 MO-CCCIIs | 3/0/yes | 2 | 4 |

[28] | 3 MO-CDTAs | 3/0/yes | 2 | 2 |

[29] | 2 MO-CCIIs | 3/3/no | 2 | 2 |

[30] | 3 MO-DVCCs | 3/3/no | 2 | 4 |

[31] Figure 3 | 3 OTRAs | 3/5/no | 2 | 0 |

[32] Figure 4a | 2 MO-DVCCTAs | 3/1/no | 3 | 3 |

[32] Figure 4b | 2 MO-DVCCTAs | 3/2/no | 3 | 3 |

This work | 2 MO-CCCCTAs | 3/0/yes | 2 | 3 |

Reference | Electronic Tuning FO without Disturbing CO | Independent Control of CO and FO | Controllable Amplitude of AM/ASK |
---|---|---|---|

[22] Figure 7 | no | no | no |

[23] | yes | yes | no |

[24] Figure 1 | no | no | no |

[25] | yes | yes | no |

[26] Figure 2 | no | yes | no |

[26] Figure 3 | yes | yes | no |

[27] | yes | yes | no |

[28] | yes | yes | no |

[29] | no | yes | no |

[30] | no | no | no |

[31] Figure 3 | no | yes | no |

[32] Figure 4a | no | yes | no |

[32] Figure 4b | no | no | no |

This work | yes | yes | yes |

Transistors | L (µm) | W (µm) |
---|---|---|

M1–M2 | 0.35 | 7 |

M3–M4 | 0.35 | 14 |

M5–M10 | 0.5 | 20 |

M11–M17, M20, M21 | 0.5 | 10 |

M18, M19, M27 | 0.8 | 8 |

M22–M26 | 0.8 | 25 |

Harmonic Number | Frequency (MHz) | Fourier Component | Phase (degree) |
---|---|---|---|

1 | 2.122 | 131.6570 m | 152.1558 |

2 | 4.244 | 1.7963 m | −145.6957 |

3 | 6.366 | 1.3875 m | −16.7744 |

4 | 8.488 | 280.3543 u | −35.0773 |

5 | 10.610 | 59.7176 u | −158.4953 |

6 | 12.732 | 139.7939 u | −8.9033 |

7 | 14.854 | 117.2583 u | −9.5498 |

8 | 16.976 | 100.4933 u | 2.8933 |

9 | 19.098 | 101.7488 u | 8.4091 |

DC component = 3.419 × 10^{−3} | |||

Total harmonic distortion = 1.7465% |

Harmonic Number | Frequency (MHz) | Fourier Component | Phase (degree) |
---|---|---|---|

1 | 2.122 | 131.7010 m | 117.7808 |

2 | 4.244 | 557.7167 u | 107.6894 |

3 | 6.366 | 591.5883 u | 51.9561 |

4 | 8.488 | 195.7997 u | 27.5504 |

5 | 10.610 | 71.6988 u | 2.9989 |

6 | 12.732 | 64.9872 u | 16.6679 |

7 | 14.854 | 67.4512 u | 7.9859 |

8 | 16.976 | 60.2372 u | 11.2194 |

9 | 19.098 | 51.7097 u | 15.5339 |

DC component = −4.142 × 10^{−4} | |||

Total harmonic distortion = 644.0952 m% |

Harmonic Number | Frequency (MHz) | Fourier Component | Phase (degree) |
---|---|---|---|

1 | 2.122 | 17.5810 u | −27.9532 |

2 | 4.244 | 175.7963 n | −143.2494 |

3 | 6.366 | 204.1361 n | 155.1108 |

4 | 8.488 | 57.0393 n | 145.0789 |

5 | 10.610 | 12.1911 n | −93.1991 |

6 | 12.732 | 16.2208 n | −141.3161 |

7 | 14.854 | 6.6238 n | 157.6872 |

8 | 16.976 | 10.0840 n | 161.5075 |

9 | 19.098 | 13.2706 n | −179.8498 |

DC component = −2.993 × 10^{−8} | |||

Total harmonic distortion = 1.5739% |

Harmonic Number | Frequency (MHz) | Fourier Component | Phase (degree) |
---|---|---|---|

1 | 2.122 | 17.2823 u | −117.8809 |

2 | 4.244 | 72.1321 n | 106.2341 |

3 | 6.366 | 156.5673 n | 26.8057 |

4 | 8.488 | 24.1643 n | 25.9664 |

5 | 10.610 | 8.2084 n | 1.3462 |

6 | 12.732 | 7.7736 n | 16.4662 |

7 | 14.854 | 8.5334 n | 8.0360 |

8 | 16.976 | 7.7471 n | 11.0721 |

9 | 19.098 | 6.5087 n | 14.8607 |

DC component = −5.366 × 10^{−8} | |||

Total harmonic distortion = 1.0122% |

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## Share and Cite

**MDPI and ACS Style**

Chen, H.-P.; Hwang, Y.-S.; Ku, Y.-T.
Voltage-Mode and Current-Mode Resistorless Third-Order Quadrature Oscillator. *Appl. Sci.* **2017**, *7*, 179.
https://doi.org/10.3390/app7020179

**AMA Style**

Chen H-P, Hwang Y-S, Ku Y-T.
Voltage-Mode and Current-Mode Resistorless Third-Order Quadrature Oscillator. *Applied Sciences*. 2017; 7(2):179.
https://doi.org/10.3390/app7020179

**Chicago/Turabian Style**

Chen, Hua-Pin, Yuh-Shyan Hwang, and Yi-Tsen Ku.
2017. "Voltage-Mode and Current-Mode Resistorless Third-Order Quadrature Oscillator" *Applied Sciences* 7, no. 2: 179.
https://doi.org/10.3390/app7020179