## 1. Introduction

Sinusoidal oscillators are used in almost all systems of receiving and transmitting information, as well as in measuring instruments and systems. One of the most rapidly developing class of sinusoidal oscillators is the class of the voltage-controlled oscillator (VCO). Voltage-controlled oscillators are commonly used in digital frequency synthesizers, which are one of the main subsystems of modern communications systems. The rapid development of modern communications and instrumentation systems has created a high demand for low noise VCOs [

1,

2,

3]. Modern microwaves sinusoidal oscillators use bipolar junction transistors (BJT), heterojunction bipolar transistors (HBT), a low-noise high-electron-mobility-transistors (HEMT) and pseudomorphic HEMT (pHEMT) as active devices for achieving low phase-noise performance [

4,

5,

6]. The appearance on the market of a ultra-low-noise, high-speed, broadband operational amplifier (OPA) creates the possibility of their use in the sinusoidal oscillators in the ultra-high frequency range. For example, the LMH6629MF (Texas Instruments) OPA has an input noise voltage of 0.69 nV/√Hz at the corner frequency of 4 kHz, a slew rate of 1600 V/μs and small-signal −3 dB bandwidth of 900 MHz [

7]. Low-noise, high-speed OPA oscillators can offer a practical alternative to transistor oscillators, the performance of which to some extent depends on the variability of the transistor small-signal parameters. The use of a low-noise, high-speed OPA as an active device in the oscillator circuits has some advantages [

8,

9]. An operational amplifier is an amplifier with very high input impedance and very low output impedance; it is easy to introduce required positive feedback around the OPA; the oscillator design is simple due to the lack of the bias circuit; the oscillator circuit does not require any adjustment during fabrication.

As is well known [

8,

10,

11], oscillators based on the Colpitts and Hartley topology require a relatively high voltage gain to start-up oscillation. High voltage gains reduce the maximum frequency of oscillation because of the limited gain-bandwidth product of OPA. Besides, the maximum frequency of generated sinusoidal oscillations at the output of the OPA in the Colpitts and Hartley topologies is limited by the ratio of the slew rate to the amplitude of voltage oscillations multiplied by two pi [

12]. Thus, with the required voltage amplitude of, say, three volts, and the use of the LMH6629 OPA, the maximum achievable frequency will be less than 85 MHz, which is very far from the microwave frequency range. By simple calculations, we can estimate what should be the slew rate of an OPA to achieve a frequency of 1 GHz with a 3 V amplitude of oscillations. It should be about 19,000 V/μs. Currently, such a slew rate in OPA is unattainable.

The VCO topology proposed in [

13] uses the OPA circuit with negative input inductance observed at the noninverting input of the OPA. The disadvantage of this circuit is the use of two resistors, one of which presents the positive feedback circuit, and the second connects the inverting input of the OPA to the ground. These resistors are the source of thermal noise.

In this study, we propose several new VCO topologies that use the idea of a negative impedance converter. However, unlike the well-known studies [

14,

15,

16,

17], the converter circuit does not include resistors. In contrary to the Colpitts and Hartley oscillators requiring a sufficiently high voltage gain for self-excitation, the proposed oscillator circuits operate with a voltage gain of less than unity; this feature significantly increases the operating frequency, which can exceed the unity-gain bandwidth of the OPA. This particularity of the proposed VCO circuits is due to the location of the tank circuit not at the output of the OPA, but at the noninverting input. Mathematical modeling, simulation, and prototype implementation of the proposed oscillators are given.

## 2. Architecture of Oscillators

Figure 1 shows the general VCO electronic diagram. The circuit inside the dashed rectangular is the negative impedance converter. The input impedance observed at the noninverting terminal of the OPA is as follows [

16]:

where

Z_{in} is the input impedance seen by the noninverting terminal of the OPA,

Z_{0} is the impedance between the inverting input and output of the OPA,

Z_{1} is the impedance between the noninverting input and output of the OPA, and

Z_{2} is the impedance between the inverting terminal of the OPA and ground. The step-by-step derivation of Equation (1) is given in [

18] (pp. 349–350).

In Reference [

15,

16,

17], one of the impedances

Z_{0},

Z_{1}, or

Z_{2} is capacitive, and the other two are resistive.

The presence of significant resistances in the circuit of any oscillator leads to an increase in thermal noise [

19]; therefore, in this study, the impedances

Z_{0},

Z_{1}, and

Z_{2} are capacitive or inductive.

In the circuit of

Figure 1, inductor

L and two contrary connected varactors

VR_{1} and

VR_{2} present the tank circuit of the VCO. Resistor

R_{dc} isolates the dc control voltage line from the VCO tank.

As can be seen in

Figure 1, the tank circuit is connected to the noninverting input of the OPA rather than its output. This feature of the proposed VCO allows extending operation frequency range beyond the unity-gain bandwidth of OPA.

Figure 2 shows VCO circuits based on impedance converters with two inductors and one capacitor. The circuits in

Figure 2a,b introduce positive feedback through inductor

L_{1} and capacitor

C_{1}, respectively.

Figure 3 presents VCO circuits on the base of impedance converters with three inductors (a) and two capacitors and one inductor (b).

The oscillator circuits based on impedance converters in which a capacitor is used between the inverting input of the OPA and ground are not considered.

Further, in the article, we will analyze in detail the VCO circuit presented in

Figure 2a, and where necessary, we will refer to other VCO configurations.

## 4. VCO Analysis

The electronic part of the VCO introduces an alternating current (ac) source

i_{OPA}(

t) into the tank circuit, as shown in

Figure 5, where

C_{VCO} and

L_{VCO} are, respectively, the VCO total capacitance and inductance, and

R_{par} is the equivalent parallel resistance of the tank circuit at the fundamental frequency. Essentially the current

i_{OPA} (

t) is the feedback current flowing through inductor

L_{1}.

The VCO total capacitance includes the capacitance of the contrary connected varactors

VR_{1} and

VR_{2}, the input capacitance of the OPA, and the parasitic capacitance of the printed circuit board (PCB). Therefore,

As we can see in Equation (2), the VCO capacitance is a function of voltage V_{dc} because the varactor capacitance depends on this voltage. Onwards, inductor L_{1} provides positive shunt–shunt feedback.

By applying the y-parameter analysis to the positive feedback network, we find that y_{11} parameter incorporates into the tank circuit. Since y_{11} = 1/jω_{fun}L_{1}, inductance L_{1} appears in parallel with tank inductance L, where ω_{fun} is the fundamental angular frequency of oscillation.

Therefore, the total VCO tank circuit inductance is given by

The current i_{OPA}(t) flows through the tank circuit. However, only the first harmonic of the current flow creates a significant voltage drop across the tank. The second, third, and subsequent current harmonics create insignificant voltage drops that can be neglected.

We should also note that the tank circuit is connected to the noninverting input of the OPA, where the input impedance is extremely high. Therefore, the OPA does not practically short-out the tank circuit.

This property of the proposed VCO allows obtaining high voltage amplitude of the generated voltage

v_{out}(

t), which is not limited by the OPA output. Indeed, we can describe the VCO output voltage by the following equation:

where |

I_{OPA}_{,1}(

V_{dc})|,

ω_{fun}(

V_{dc}), and

φ_{1}(

V_{dc}) are, respectively, the amplitude, frequency and initial phase of the first harmonic of current

i_{OPA}(

t). Therefore, we can see from (4) that the amplitude of the generated sinusoidal voltage is proportional to |

I_{OPA}_{,1}| and

R_{par}. In a virtual case of lossless tank circuit, i.e., when

R_{par} → ∞, the amplitude of the output voltage would tend to infinity.

This is a unique property of the proposed OPA oscillator because, for the Colpitts and Hartley OPA oscillators [

8], the output amplitude satisfies the following inequality:

where

V_{sat} is the OPA saturation voltage, and

R_{C} is the resistance of the load resistor connected between the OPA output and tank circuit.

Since the OPA saturation voltage in (5) is less than the power supply voltage

V_{cc}, the oscillation amplitude is also less.

Figure 6a illustrates the behavior of the OPA output voltage (curve 1) and the oscillator output voltage (curve 2) for the Colpitts oscillator operating at frequency of 53 MHz with

V_{cc} = ±5 V and voltage peak amplitude of 3.7 V. We can also observe in

Figure 6a that the shape of the voltage at the output of the OPA, in this particular case, is trapezoidal, which means the slew rate of the OPA is not high enough to ensure a rectangular shape.

With a further increase in the frequency of the generated oscillations in the Colpitts oscillator, the voltage shape at the output of the OPA becomes triangular with the amplitude less than V_{sat}. In this case, also, the amplitude of sinusoidal oscillations in the Colpitts oscillator will always be less than the magnitude of triangular oscillations.

A completely different relationship exists between the amplitudes of the voltages at the output of the OPA (node 1) and the oscillator (node 2) in the circuit in

Figure 2a.

Figure 6b illustrates the behavior of the OPA output voltage (curve 1) at node 1 and the oscillator output voltage (curve 2) at node 2 for the proposed oscillator operating at frequency of 667 MHz with the same OPA, power supply voltages and output peak amplitude of 4 V. As can be seen in

Figure 6b, the amplitude of the sinusoidal voltage can be much larger than the voltage amplitude at the output of the OPA (node 1).

Therefore, the proposed VCO can operate at frequencies significantly exceeding the unity-gain bandwidth of OPA.

It should also be noted that by choosing a low-noise OPA, the phase noise of the VCO will be determined mainly by the noise of the tank circuit.

## 5. Amplitude of Oscillations

Let us determine the amplitude of the sinusoidal voltage at the output of the VCO (node 2). Assume the voltage at node 1 be triangular, as shown in

Figure 6b (curve 1). For the triangle wave, we can describe the voltage at node 1 by the complex exponential Fourier series as follows:

where

V_{μ} is the complex amplitude of the harmonic number

μ of voltage

v_{OPA}(

t).

The complex amplitude

V_{μ} can be represented as

where |

V_{μ}| and

θ_{μ} are the amplitude and phase of the voltage harmonic number

μ, respectively.

As is well known [

20], the amplitude of the voltage harmonic

μ for the triangle wave is

where

V_{triangle} is the amplitude of the triangular voltage at the OPA output.

The feedback current

i_{OPA}(

t), which flows from node 1 into the tank circuit, we also present by the complex exponential Fourier series

where

I_{OPA,μ} is the complex amplitude of the current harmonic number

μ.

The complex amplitude

I_{OPA,μ} we write in polar form as follows:

where |

I_{OPA,μ}| and

φ_{μ} are the amplitude and phase of the current harmonic number

μ, respectively.

In the ac equivalent circuit of the VCO shown in

Figure 5, the feedback inductor

L_{1} is in parallel with tank inductor

L. The feedback current

i_{OPA}(

t) flows through inductor

L_{1} and tank circuit due to the applied voltage

v_{OPA}(

t). Considering the forward transfer admittance of the feedback network

y_{12} = −1/

jω_{fun}L_{1} and parallel connection of inductors

L_{1} and

L in the equivalent circuit of

Figure 5, by Ohm’s law, we have

By substitution of (7) and (8) into (11), we get

The first harmonic of current

i_{OPA}(

t) is of interest because for the higher harmonics, the equivalent resistance of the tank circuit is negligible, and they do not create a significant voltage drop. By substitution

μ = 1 into (12) gives

We determine the first harmonic of the voltage across the tank by multiplying the complex amplitude

I_{OPA}_{,1} and the equivalent resistance of the tank at resonance

R_{par} as follows:

By comparing (7) when

μ = 1 and (14), we can state that the first harmonic of voltage across the tank leads the first harmonic of triangular voltage at the OPA output by 90°, which is confirmed by

Figure 6b.

From (14) it follows that the amplitude of the voltage across the tank is

Analyzing (15), we can observe that the amplitude of the VCO output voltage |V_{out}_{,1}| is directly proportional to resistance R_{par} and inversely proportional to inductance L_{1}‖L. As already indicated in the analysis of Equation (4), when using a tank circuit with low losses, the voltage amplitude at the output of the VCO can be significant.

Let us simplify (15) by considering the case when

L_{1} ˃˃

L. In this case,

L_{1}‖

L ≈

L and we can write Equation (15) in the following form:

At resonance we have

where

ρ is the tank circuit characteristic impedance.

As is well known [

21] (p. 909), the equivalent resistance of the parallel tank at resonance is

where

r_{s} is the series loss resistance of the tank circuit.

Substituting (17) and (18) into (16) gives

Since the ratio of

ρ to

r_{s} in (19) is equal to the quality factor of the parallel tank circuit (

Q), then we write (19) in the following form:

We derived Equation (20) under the condition of an ideal OPA with infinite input resistance, which does not load the tank circuit. However, in the real OPA, the input resistance is not infinite; therefore, in (20), we should use the loaded quality factor. Therefore,

where

Q_{L} is the quality factor of the loaded VCO tank circuit.

For the VCO shown in

Figure 2b, following the same analysis as for the circuit of

Figure 2a, we obtain that the amplitude of oscillations is

From the analysis of Equation (22), it follows that the capacitance of the positive feedback capacitor C_{1} should be much less than the capacitance C_{VCO}; in this case, the capacitance C_{1} will not affect the frequency of generated oscillations.

Given that at the resonant frequency

We transform Equation (22) as follows:

Considering that R_{par}/ρ = Q in (24), we obtain again Equation (20), which transforms to (21) due to the loaded tank circuit.

The amplitude of the generated sinusoidal voltages in the VCO circuits in

Figure 3a,b is also calculated by Equation (21).

## 6. VCO Start-Up Conditions

According to the Barkhausen criteria [

22], to provide the sustained oscillations, the following conditions must have a place:

where |

A| and

|β|, and

φ_{A} and

φ_{β} are, respectively, the gains and initial phases of impedance converter and feedback network.

As shown in

Figure 4b, the voltage gain of the impedance converter is less than unity. We determine the feedback network ratio as follows:

where |

V_{1}| is the voltage amplitude of the first harmonic of triangular voltage at the output of the OPA (node 1).

Substituting (8) at

μ = 1 and (21) into (27) gives

We can conclude from (25) and (28) that at the steady-state the gain of the impedance converter is

To start the oscillations, the product of |A| and |β| is made higher than unity, and when the amplitude reaches a constant value |V_{out,}_{1}|, this product becomes equal to unity.

From relations (28) and (29), it follows that in the proposed oscillators, the feedback network coefficient |β| is more than unity, and the gain of the active network |A| is less than unity; this is a unique property because in the Colpitts and Hartley oscillators the opposite is true, i.e., |A| ˃ 1 and |β| < 1. Due to this property, the proposed VCO can operate at frequencies significantly exceeding the unity-gain bandwidth of an OPA with sufficiently large oscillation amplitudes.

## 7. Simulation and Discussion

The circuit of

Figure 2a was simulated with the help of Multisim (ed. 14.1) using SPICE models of a real OPA, varactors, and RF coil to confirm the operation efficiency of the proposed VCO topology. We selected ultra-low noise, high-speed OPA LMH6629MF with −3 dB bandwidth of 900 MHz, slew rate of 1600 V/μs, input voltage noise of 0.69 nV/√Hz, and power supply voltage ±5 V. We also selected UHF varactors BB215 and a 3.3 nH RF coil. The other component values are as follows:

L_{1} =

L_{2} = 1 μH,

C = 100 nF, and

R_{dc} = 1 kΩ.

Figure 7 shows the simulated VCO schematic with chosen component values.

Figure 8 shows the VCO start-up voltage in the interval 0–720 ns at

V_{dc} = 1 V (a) and

V_{dc} = 11 V (b). As we can see in

Figure 8, the oscillations reach the steady-state condition within 500 ns at

V_{dc} = 1 V and 50 ns at

V_{dc} = 11 V. Thus, the starting time of VCO is speedy.

Figure 9 shows the steady-state voltage waveforms at nodes 1 and 2 when

V_{dc} = 1 V (a) and

V_{dc} = 11 V (b). As can be seen in

Figure 9, the amplitude of the triangular voltage at the output of OPA (node 1) is much less than the amplitude of the sinusoidal voltage at the VCO output (node 2).

From the simulation results shown in

Figure 9, it is observed that the THD is 1.5% at

V_{dc} = 1 V and 1.7% at

V_{dc} = 11 V, which corresponds to −36.5 dB and −35.4 dB, respectively.

Since in the microwave frequency range, the THD value of −30 dB is considered quite well [

23] (p. 291), [

24], we can argue that the designed VCO is a low-distortion oscillator.

Let us check Equation (21). From

Figure 9a it can be observed that

V_{triangle} = 0.37 V and |

V_{out,}_{1}| = 3.19 V. To find the loaded quality factor

Q_{L}, we use the property of the parallel tank circuit that the current circulating in the tank circuit

I_{tank} in

Q_{L} times higher than the current in the general circuit

I_{res} [

21] (p. 911).

Figure 10a illustrates the location of currents

I_{tank} and

I_{res} in the VCO tank circuit.

Therefore, the loaded quality factor can be calculated by the following equation:

By simulation, we find that I_{tank} = 127 mA and I_{res} = 15 mA. Substituting the currents into (30) gives that Q_{L} = 8.5. Further, by substitution of V_{triangle} and Q_{L} into Equation (21), we calculate that |V_{out,}_{1}| = 2.76 V. So, the relative error of amplitude calculation by Equation (21) is only 13.5%.

Figure 10b shows the VCO tuning characteristics. As we can see in

Figure 10b, the VCO operates from 830 MHz to 1.429 GHz, i.e., the tunable band is 599 MHz. Thus, the designed circuit is a wideband VCO, which operates in the frequency range, significantly exceeding the bandwidth of the OPA. Indeed, the bandwidth of the LMH6629MF is 900 MHz, but the highest VCO frequency is 1.429 GHz. Using the equation for maximum signal frequency [

12], we can find that with the OPA LMH6629MF in the Colpitts or Hartley VCO, the 4 V amplitude of the sinusoidal voltage corresponds to a frequency that is less than 64 MHz.

Thus, by using the proposed VCO topology, we can increase the maximum operating frequency by more than 22 times.

Figure 11 shows the simulated power spectrum of the designed VCO at the lowest frequency (a) and the highest frequency (b). As we can see in

Figure 11a, only the second and third harmonics contribute to the THD because the fourth and subsequent harmonics attenuated for at least 90 dB compared to the first harmonic. By analyzing

Figure 11b, we find that the second and upper harmonics decrease slowly. However, the second and other harmonics attenuated by at least 36 dB.

One of the essential characteristics of an oscillator operating at microwave frequencies is phase noise. To calculate the phase noise of the simulated VCO, we use the improved Leeson’s formula [

25] (p. 128)

where

PN is the phase noise (dBc/Hz),

F is the noise figure of the oscillator active device (dB),

$k\approx 1.38\times {10}^{-23}$ is the Boltzmann constant (J/K),

T is the temperature in Kelvin,

P_{out} is the oscillator output power,

f_{fun} is the frequency of oscillations (Hz),

f_{c} is the 1/

f corner frequency of active device (Hz), and

f_{m} is the offset frequency (Hz).

According to Reference [

7], the noise figure of OPA LMH6629MF is 8 dB.

Figure 11a,b shows that

P_{out} = 9 dBm when

V_{dc} = 1 V and

P_{out} = 11 dBm when

V_{dc} = 11 V.

Figure 10b indicates that the minimum frequency of oscillations is 830 MHz, and the maximum is 1429 MHz. From Reference [

7] (p.15, Figure 28) we find that

f_{c} = 4 kHz. The loaded quality factor,

Q_{L}, is 8.5 and 2.6 for

V_{dc} = 1 V and 11 V, respectively.

Figure 12 shows the dependence of the phase noise versus offset frequency for the VCO shown in

Figure 7. As can be seen in

Figure 12, the VCO phase noise changes from −153.4 dBc/Hz to −139.3 dBc/Hz at 100 kHz offset frequency when the control voltage varies from 1 V to 11 V. Thus, the maximum in-band phase noise is −139.3 dBc/Hz at

f_{m} = 100 kHz.

Let us compare the characteristics of the simulated VCO with the best VCO designs operating in the microwave frequency range. The commonly used FoM includes phase noise (

PN), the ratio of

f_{fun} to

f_{m}, and the power consumption (

P_{c}) [

26].

In Equation (32), the second term allows one to compare the phase noise of the oscillators determined at different frequencies and various frequency offsets. The third term in (32) has a positive sign if P_{c} ˃ 1 mW and negative if P_{c} < 1 mW. Thus, the smaller the value of FoM, the higher efficiency of the oscillator.

Table 1 shows a comparison of the designed VCO with microwave oscillators that have been reported in journals and conference proceedings.

We can see in

Table 1 that the designed VCO has the best FoM among oscillators fabricated in GaN HEMT, SiGe, HEMT, GaAs pHEMT, and SiGe BJT technologies where, in general, power consumption is high enough. Moreover, as shown in

Table 1, the designed VCO can even compete with the latest achievements in CMOS VCOs; this is due to low phase noise and in spite of higher power dissipation.

## 9. Conclusions

This article has proposed a new type of VCO based on resistorless negative impedance converter circuits. A distinctive feature of each proposed VCO circuit is that its output is located at the noninverting input of the OPA; this allows us to significantly increase the frequency and amplitude of the generated oscillations. Mathematical modeling of the amplitude of the oscillations and the start-up conditions were carried out. We have shown that, unlike existing oscillators of the Colpitts and Hartley topology, in the proposed VCO, the gain is less than unity, and the feedback coefficient is greater than unity. Therefore, the proposed oscillators can operate at frequencies exceeding the unity-gain bandwidth. The advantage of the proposed VCO circuit is also the fact that the amplitude of the generated sinusoidal oscillations can be dozens of times higher than the magnitude of the nonsinusoidal oscillations at the output of the OPA. The designed and simulated VCO uses ±5 V supply voltage and operates in a wide frequency range from 830 MHz to 1429 MHz showing a maximum in-band THD of 1.7% while the OPA bandwidth is only 900 MHz; the amplitude of oscillations varies from 3.2 V to 4 V. It has a maximum in-band phase noise of −139.3 dBc/Hz at 100 kHz offset frequency and has an outstanding value of FoM of −198.6 dBc/Hz. The conducted comparison of the designed VCO with VCOs in previously published studies concerning standard FoM has shown that the proposed oscillator is about 8–15 dB more efficient than the GaN HEMT, GaN pHEMT, HEMT, and SiGe VCOs and close to the best CMOS VCOs. We have shown that for the same OPA, the operating frequency of the proposed VCO is 22 times higher than that of the Colpitts and Hartley oscillators. The fabricated prototype-oscillator based on OPA LMH6624 operates at a frequency of 583.1 MHz with a power level of 0 dBm.

The proposed OPA VCO circuits are easy to design. They do not require settings during the manufacturing process. These oscillators can be used from high-frequency to the microwave frequency range. Since the power consumption in the proposed VCO is higher than in the CMOS oscillators, the field of application extends to communications, navigation, and radar equipment with an uninterruptible power supply line. As the proposed schemes do not require tuning, they can be useful in conducting laboratory works in the departments of electronics at universities.

Our future work will include mathematical modeling of phase noise in the proposed oscillators using the method of impulse sensitivity functions.