An FR4-Based Oscillator Loading an Additional High-Q Cavity for Phase Noise Reduction Using SISL Technology

Abstract

An FR4-based X-band low phase noise oscillator loading an additional high-Q cavity resonator was designed in this study using substrate-integrated suspended line (SISL) technology. The additional resonator was coupled to an oscillator by the transmission line (coupling TL). The impact of the additional resonator on startup conditions, Q factor enhancement, and phase noise reduction was thoroughly investigated. Three oscillators loading an additional high-Q cavity resonator, loading an additional high-Q cavity resonator and performing partial dielectric extraction, and loading an original parallel feedback oscillator for comparison were presented. The experimental results showed that the proposed oscillator had a low phase noise of −131.79 dBc/Hz at 1 MHz offset from the carrier frequency of 10.088 GHz, and the FOM was −197.79 dBc/Hz. The phase noise was reduced by 1.66 dB through loading the additional resonator and further reduced by 1.87 dB through partially excising the substrate. To the best of our knowledge, the proposed oscillator showed the lowest phase noise and FOM compared with other all-FR4-based oscillators. The cost of fabrication was markedly reduced. The proposed oscillator also has the advantages of compact size and self-packaging properties.

1. Introduction

X-band fixed-frequency oscillators are widely used in radar systems and microwave communications as stable local oscillators (LOs) and in applications that demand excellent frequency stability and low phase noise [1,2]. Previous studies explored reducing phase noise in oscillators to meet this demand [3]. A common strategy used to achieve a low-phase noise in oscillators is based on Leeson’s model [4]:

$$L(\Delta \omega )={\displaystyle \frac{2FkT}{P_{\mathrm{sig}}}}(1+{({\displaystyle \frac{{\omega }_0}{2Q\Delta \omega }})}^2)$$

(1)

where the phase noise is related to the signal power Psig, the Q factor of the resonator and the excess noise from the active devices F. Using the high Q resonator to reduce phase noise is an effective method and has been widely reported [5,6,7,8,9,10,11]. SIW tunable resonator [5], air-filled SIW resonator [6], active resonator [7,8], air cavity resonator based on SISL [9,10], and dielectric resonator [11] are used in oscillator design and exhibit good phase noise performance. However, an optimal Q factor is not achieved due to limitations such as the cost of circuit fabrication, resonator materials, and limited processing technology. The phase noise can also be reduced by improving the circuit topology. Narrowband bandpass filters (BPFs) with high group delay have been widely used in oscillator design in the recent past. Elliptic-response planar BPF [12], dual-mode BPF [13], eighth-mode SIW BPF [14], bandpass response power divide [15], high-order mode SIW filter [16], bandpass filter with multiple transmission zeros [17] have been applied in oscillators and have shown good results. These methods require multiple resonators and appropriate coupling relationships to obtain sharper roll-off and higher group delay.

In addition to designing narrow-band BPF as the frequency-selective element, some designs connected a high-Q factor network to the oscillator’s output port or as an additional feedback loop [18,19,20,21]. The advantage of this method is that it requires no modifications to the original oscillator. Phase noise optimization can be achieved solely by introducing an additional loop or additional resonator. Ref. [18] is fabricated on chip, the Q factor is relatively lower compared to SISL technology. Ref. [19] designed a push–push oscillator introducing a second resonator, forming an additional feedback loop. The λ/2 microstrip line resonator was used as the frequency selective element, which also has a low Q factor compared with the SISL air cavities used in our design. The additional network in [20] contained slow-wave structures and extended to a high-Q dielectric resonators. Theoretical analysis investigated the impact of the additional slow-wave structure on the Q factor, oscillator stability, and noise characteristics. Ref. [21] theoretically derives that by externally adding a high-Q resonator, phase noise can be reduced, and the stability of the oscillator can be increased.

In this study, an FR4-based oscillator with an additional resonator was fabricated. The basic concept of the oscillator is shown in Figure 1. The original oscillator is a fully designed and properly functioning parallel-feedback oscillator. To further reduce its phase noise, we introduced an additional high-Q resonator. This additional resonator is coupled to the original oscillator through a transmission line. Both the resonator in the original oscillator and the added one are implemented based on substrate-integrated suspended line (SISL) technology. The impact of the additional resonator on the startup conditions, the Q factor, and phase noise is analyzed. Additionally, placing the two resonators one above the other did not increase any footprint area.

The proposed oscillator exhibited a low measured phase noise of −131.79 dBc/Hz at 1 MHz offset from an operating frequency of 10.088 GHz. Additionally, we used cheaper FR4 materials to make the circuit and achieved comparable performance to the circuit using Roggers5880 material. The price difference between the two was approximately 30 times [22]. To the best of our knowledge, the proposed oscillator showed the lowest phase noise and FOM compared with other all-FR4-based oscillators [23,24].

The structure of this paper is as follows. In Section 2, the impact on startup conditions, Q factor enhancement, and phase noise reduction when an additional high-Q resonator is introduced are analyzed. The theoretical reduction in phase noise when the second resonator is introduced to the oscillator is also derived in this section. In Section 3, we introduce the design method and technical details of the oscillator with the additional resonator. The experimental results and a comparison between the proposed oscillator and the other reported ones are presented in Section 4. The conclusion is presented in Section 5.

2. Analysis of Topology and Phase Noise Reduction

2.1. The Startup and Steady-State Conditions for the Introduction of the Additional Resonator

As shown in Figure 2, the original oscillator is a traditional parallel feedback oscillator using a high-Q cavity resonator based on SISL [10]. The additional resonator is coupled to the oscillator through a transmission line (coupling TL). The phase noise can be reduced by introducing an additional resonator with a high-Q network [21]. However, the added circuit elements also introduce some changes, which may change the original frequency, give rise to instability, or even cannot meet the startup conditions.

The original oscillator must fulfill the startup conditions, which can be expressed as

$$\mathrm{Re}[Z_{\mathrm{T}}({\omega }_0)]<0$$

(2)

$$\mathrm{Im}[Z_{\mathrm{T}}({\omega }_0)]=0$$

(3)

$${\displaystyle \frac{d\{\mathrm{Im}[Z_{\mathrm{T}}({\omega }_0)]\}}{d\omega }}>0$$

(4)

The steady-state condition can be expressed as

$$Z_{\mathrm{T}}({\omega }_0)=0$$

(5)

where Z_T is the close-loop impedance of the original circuit, and ω₀ is the oscillation frequency. To avoid a big perturbation in the original oscillator’s startup and steady state, the additional resonator (Z_add) should have the same resonant frequency as the resonator in the original oscillator, which can be expressed as

$$\mathrm{Im}(Z_{\mathrm{add}}({\omega }_0))=0$$

(6)

The additional resonator and the transmission coupling line are connected to the original oscillator in series, which forms a new oscillator. This new oscillator must also satisfy the stable oscillation conditions as

$$Z_{\mathrm{T}}(\omega )+Z_{\mathrm{add}}(\omega )=0$$

(7)

As the additional network should not significantly affect the original oscillation frequency, we can expand (7) in a first-order Taylor series around ω₀, which is

$$Z_{\mathrm{T}}({\omega }_0)+{\displaystyle \frac{d{Z}_{\mathrm{T}}({\omega }_0)}{d\omega }}\cdot (\omega -{\omega }_0)+Z_{\mathrm{add}}(\omega )=0$$

(8)

Substitute (5) and splitting (8) into real and imaginary parts, we can obtain

$$\mathrm{Im}[Z_{\mathrm{add}}(\omega )]=-\mathrm{Im}[{\displaystyle \frac{d{Z}_{\mathrm{T}}({\omega }_0)}{d\omega }}\cdot (\omega -{\omega }_0)]$$

(9)

In practical designs, due to the presence of parasitic effects in the circuit, the new oscillation frequency is usually slightly lower than that of the original oscillator. From (4) and (9), we obtain

$$\mathrm{Im}[Z_{\mathrm{add}}(\omega )]>0 (\omega <{\omega }_0)$$

(10)

As shown in Figure 2a, the additional network is an one port cavity resonator coupled to the original oscillator through a transmission line. It can be modeled with a lumped parallel resonator in series with the transmission line, as shown in Figure 2b.

The input impedance can be expressed as

$$Z_{\mathrm{add}}=Z_0{\displaystyle \frac{Z_{\mathrm{R}2}+j{Z}_0\tan \theta }{Z_0+j{Z}_{\mathrm{R}2}\tan \theta }}$$

(11)

where Z_R2 is the input impedance of the resonator, and θ is the electrical length of the transmission line.

Since the tangent function has π as its smallest positive period, Z_add also exhibits periodic variations with a period of π around the resonant frequency, as shown in Figure 3. Around the resonance frequency, when the electrical length of the coupling TL differs by π, the imaginary part of Z_add essentially overlaps.

Figure 3a shows the imaginary part of Z_add of the ideal RLC tank in one period and Figure 3b shows the imaginary part of Z_add of the SISL cavity resonator. For the ideal RLC circuit, Im (Z_{add_RLC}) shows a predictable linear transition from positive to negative as the frequency crosses the resonant frequency ω₀, and (10) is satisfied when 0 ≤ θ < π/2. The SISL cavity resonator demonstrates a similar qualitative trend but with more complex reactance variations. When 0 ≤ θ < π/2 and 3π/4 < θ ≤ π, (10) is satisfied. However, in practical design, we generally make it as short as possible to save footprint area and avoid introducing additional instability into the circuit. Therefore, we typically consider cases where the 0 ≤ θ < π/2.

Figure 4a shows that different characteristic impedances have a negligible effect on the resonant frequency, with a maximum deviation of 0.11%. However, the characteristic impedance significantly impacts the return loss and must be matched to the input resistance of the resonator. Figure 4b–d illustrate the impact of different characteristic impedances on the imaginary part for a given θ. As long as θ is fixed, varying the characteristic impedance does not alter the relative magnitude of the imaginary part near ω₀ with respect to zero. Therefore, satisfying the condition in (10) only requires the consideration of θ.

Figure 5 shows the simulated negative resistance in oscillators with the additional cavity resonator. It can be observed that the oscillator with the additional cavity meets the startup condition when the electrical length of the coupling TL is from 0 to π/2.

When π/2 ≤ θ < 3π/4, the negative resistance in the circuit disappears, preventing oscillation. When 3π/4 < θ ≤ π, the oscillator meets the startup conditions again. The simulation results align well with the predictions of (10). Therefore, a key preliminary design guideline is as follows: when introducing an additional resonator to enhance oscillator performance, θ should preferably not exceed π/2. It is worth noting that satisfying this condition only ensures that the new oscillator meets the oscillation criteria; it does not guarantee an improvement in oscillator performance.

2.2. Q Factor When Introducing the Additional Resonator

The additional network can be treated as a transmission line of electrical length θ loaded by a parallel resonant circuit. Assuming Rp = Z₀, at the resonant frequency, the unloaded Q factor was given by [25] as

$${\tau }_{\mathrm{add}}={\displaystyle \frac{d{\phi }_{\mathrm{add}}}{d\omega }}=2R_{\mathrm{p}2}{C}_2\cos (2\theta )$$

(12)

and

$$Q_{\mathrm{add}}={\displaystyle \frac{{\omega }_0}{2}}\left|{\displaystyle \frac{d{\phi }_{\mathrm{add}}}{d\omega }}\right|={\omega }_0{R}_{\mathrm{p}2}{C}_2\left|\cos (2\theta )\right|=Q_2\left|\cos (2\theta )\right|$$

(13)

where τ_add is the group delay, and Q₂ is the unloaded Q factor of the additional RLC resonator. In (13), it is shown that the unloaded Q factor of the additional network is highly dependent on the length of the coupling TL. Due to the properties of the cosine function, the Q factor of the additional network varies periodically with a π/2 cycle, and reaches its maximum when the electrical length θ is 0, π/2, and π, conversely, when θ = π/4 or θ = 5π/4, the Q_add becomes zero. When 0 < θ < π/4, Q_add decreases as the value of θ increases. In practical design, since θ = 0 is not feasible, a short transmission line is typically used to couple the resonator. Moreover, the smaller θ, the higher the Q_add.

Figure 6 shows the simulated group delay using the circuit model in Figure 2b with a different θ. Consistent with the prediction of (13), curves with the same absolute cosine value intersect at 10 GHz, indicating that they have the same Q factor with the same cos (2θ).

The above analysis did not consider the effect of the load, focusing solely on the unloaded Q factor. When evaluating the impact of the additional network on the phase noise improvement, it is necessary to analyze the loaded Q factor. Thus, the problem is simplified to assessing how the known Q factor of the additional resonator enhances the loaded Q of the original resonator. First, the loaded Q of the original oscillator is analyzed, with the simplified circuit shown in Figure 7. The unloaded Q of the resonator in the original oscillator is Q₁, and the active device is connected in parallel with the resonator, which can be regarded as the load of the resonator.

By definition, the loaded Q factor is rigorously expressed at ω₀ and can be expressed as [26]

$$Q_{\mathrm{L}}={\omega }_0\cdot {\displaystyle \frac{W_{\mathrm{T}}}{P_{\mathrm{loss}}}}$$

(14)

where W_T is the reactive energy stored in the resonator and the P_loss is the power dissipated in the total resistance in the circuit.

The reactive energy stored in the original oscillator can be calculated as

$$W_1={\displaystyle \frac{1}{2}}\cdot C_1\cdot {\left|V_{in}({\omega }_0)\right|}^2$$

(15)

The total resistance in the original oscillator at ω₀ can be expressed as

$$R_{T0}({\omega }_0)=R_A({\omega }_0)\parallel R_{p1}$$

(16)

where R_A (ω₀) is the input impedance of the amplifier at oscillation frequency.

The power dissipated in the original oscillator can be calculated as

$$P_{loss}={\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{{\left|V_{in}({\omega }_0)\right|}^2}{R_{T0}({\omega }_0)}}$$

(17)

Based on (14), the loaded Q factor in the original oscillator can be expressed as

$$Q_{L0}={\omega }_0\cdot C_1\cdot R_{\mathrm{T}0}({\omega }_0)$$

(18)

when introducing an additional resonator, the energy stored in the original oscillator resonator and the additional resonator can be calculated as

$$W_{res}={\displaystyle \frac{1}{2}}\cdot (C_1+C_{\mathrm{add}})\cdot {\left|V_{in}({\omega }_0)\right|}^2$$

(19)

where

$$C_{\mathrm{add}}=C_2\cdot \cos (2\theta )$$

(20)

And the total resistance in the oscillator with the additional resonator can be expressed as

$$R_{\mathrm{T}}({\omega }_0)=R_{\mathrm{T}0}({\omega }_0)\parallel R_{\mathrm{P}2}$$

(21)

The power dissipated in the oscillator with the additional resonator can be calculated as

$$P_{\mathrm{loss}}={\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{{\left|V_{in}({\omega }_0)\right|}^2}{R_{\mathrm{T}}({\omega }_0)}}$$

(22)

Finally, the loaded Q factor in the proposed oscillator can be written as

$$Q_{\mathrm{L}}={\omega }_0\cdot [C_1+C_2\cdot \cos (2\theta )]\cdot R_{\mathrm{T}}({\omega }_0)$$

(23)

According to (16) and (21), Q_L should increase monotonically with θ, which can be written as

$$Q_{\mathrm{L}}={\displaystyle \frac{(R_{\mathrm{A}}({\omega }_0)+R_{\mathrm{p}1})(R_{\mathrm{p}2}+R_{\mathrm{p}1}\frac{Q_2}{Q_1}\cdot \left|\cos 2\theta \right|)}{R_{\mathrm{A}}({\omega }_0)R_{\mathrm{p}1}+R_{\mathrm{A}}({\omega }_0)R_{\mathrm{p}2}+R_{\mathrm{p}1}{R}_{\mathrm{p}2}}}\cdot Q_{\mathrm{L}0}$$

(24)

For simplicity of calculation, assuming

$$R_{\mathrm{p}1}=R_{\mathrm{p}2}=R_{\mathrm{p}} \mathrm{and} \alpha ={\displaystyle \frac{R_{\mathrm{p}1}}{R_{\mathrm{A}}({\omega }_0)}}$$

(25)

We derive

$$Q_{\mathrm{L}}=Q_{\mathrm{L}0}\cdot (1+\left|\cos 2\theta \right|)\cdot {\displaystyle \frac{1+\alpha }{2+\alpha }}$$

(26)

Due to the characteristics of the cosine function, the smaller the value of θ, the better. In the case when 0 < θ < π/4, we discuss three scenarios:

(a)

α→0 (Transistor as a heavy load for the resonator)

At this point, R_A (ω₀) is infinitely large, which means that the resonator of the original oscillator is in an under-coupled state, and the loaded Q factor has already approached the unloaded Q factor. Consequently, the additional resonator will not improve the loaded Q of the original circuit, and naturally, it will not reduce the phase noise of the original circuit either. Figure 8 shows the Q factor changes with different θ in three scenarios. We use the circuit model shown in Figure 7. The parameters are set as f₁ = f₂ = 10 GHz, Q₁ = Q₂ = 500, R_p1 = R_p2 = 50 Ohm. As shown in Figure 8, the curve for θ = 0 completely overlaps with the curve of the original circuit. Subsequently, an increase in θ will reduce the Q factor in the circuit, thereby worsening the phase noise.

(b)

α→∞ (Transistor as a light load for the resonator)

In this case, R_A (ω₀) is infinitesimally small, meaning that the resonator of the original oscillator is in an over-coupled state, and the loaded Q factor will be far less than the unloaded Q factor. Under these circumstances, the additional resonator can increase the loaded Q of the original circuit up to a maximum of twice its value, which would theoretically reduce phase noise by nearly 6 dB. However, this scenario is an extreme idealization; in practice, active devices cannot make R_A (ω₀) infinitesimally small. Moreover, since the loaded Q of the resonator in an over-coupled state is very low, this condition is not suitable for designing low noise oscillators. As shown in Figure 8b, despite using the same resonator with the unloaded Q of 500, when R_A (ω₀) is minimal (set to 0.5 Ohm in the simulation), the loaded Q of the original circuit is only 4.95, while the additional resonator can enhance the Q factor to twice its value, reaching 9.8. At the center frequency (10 GHz), as θ increases, the enhancing effect of the additional resonator on the Q factor gradually diminishes. When θ = π/4, the Q factor curve of the circuit intersects with the original circuit’s Q factor curve at the center frequency, indicating that the Q factor no longer increases.

(c)

α = 1 (Transistor as a match load for the resonator)

If we consider the transistor circuit as the load for the resonator, to minimize power loss, we typically aim for the transistor circuit to be matched with the resonator. In this scenario, under the ideal condition where θ = 0, the Q factor of the additional resonator can be increased by up to 33%, reducing the phase noise by approximately 2.8 dB according to Leeson’s formula. As shown in Figure 8c, we use 50 Ohm terminals for the resonator. When θ = 0, the loaded Q factor is increased from 165 to 248. As θ increases, the effectiveness of the Q factor enhancement diminishes, resulting in a lesser improvement in phase noise reduction. When θ = π/4, at the resonator frequency, the introduction of an additional resonator will not enhance the Q factor of the resonator.

2.3. Analysis of Phase Noise Reduction by Introducing the Additional Resonator

The linear time invariant (LTI) system is used to analysis phase noise in this section. The simplified model for this analysis is shown in Figure 9. The noise contributions from various components are combined into a single noise current source at the base of the BJT.

For the original oscillator, the parameterized ideal tank impedance in terms of Q of the RLC tank can be expressed as shown below:

$$Z_{\tan \mathrm{k}1}({\displaystyle △}\omega )\approx -{\displaystyle \frac{j}{2}}\cdot {\displaystyle \frac{R_{\mathrm{p}1}}{Q_1}}\cdot ({\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }})$$

(27)

where Q₁ is the unloaded Q factor of the tank1, and Q₁ = ω₀C₁R_p1 = R_p1/ω₀L₁, and ∆ω ≪ ω₀.

The total noise can be expressed as follows:

$${\displaystyle \frac{\overline{{V_{out}}^2}}{{\displaystyle △}\omega }}={\displaystyle \frac{\overline{{i_{\mathrm{noise}}}^2}}{{\displaystyle △}\omega }}\cdot {\left|Z_{\tan \mathrm{k}1}({\displaystyle △}\omega )\right|}^2={\displaystyle \frac{\overline{{i_{\mathrm{noise}}}^2}}{{\displaystyle △}\omega }}\cdot {\left({\displaystyle \frac{R_{\mathrm{p}1}}{2Q_1}}\cdot {\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }}\right)}^2$$

(28)

The output noise spectral density can be expressed as shown below:

$$P_{\mathrm{noise}}={\displaystyle \frac{\overline{{i_{\mathrm{noise}}}^2}}{{\displaystyle △}\omega }}\cdot {\left({\displaystyle \frac{1}{2Q_1}}\cdot {\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }}\right)}^2\cdot R_{\mathrm{p}1}$$

(29)

when an additional resonator is introduced, it effectively performs secondary filtering on the noise current, resulting in reduced noise at the output. To calculated the noise spectral density at the output under these conditions, we first determine the additional tank impedance of the circuit, which is given by

$$Z_{\mathrm{tank}2}({\displaystyle △}\omega )\approx -{\displaystyle \frac{j}{2}}\cdot {\displaystyle \frac{R_{p2}}{Q_2\cdot \cos (2\theta )}}\cdot ({\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }})$$

(30)

And the total impedance of the tank is given by

$$Z_{\tan \mathrm{k}}=Z_{\tan \mathrm{k}1}\parallel Z_{\mathrm{tank}2}\approx -{\displaystyle \frac{j}{2}}\cdot {\displaystyle \frac{R_p/2}{Q_1[1+\cos (2\theta )]}}\cdot \left({\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }}\right)$$

(31)

where Q₁ = Q₂, R_p1 = R_p2 = R_p.

The total output noise spectral density in the oscillator with the additional resonator can be expressed as

$${\displaystyle \frac{\overline{{V_{out}}^2}}{{\displaystyle △}\omega }}={\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{\overline{{i_{\mathrm{noise}}}^2}}{{\displaystyle △}\omega }}\cdot {\left({\displaystyle \frac{R_{\mathrm{p}}/2}{2Q_1[1+\cos (2\theta )]}}\cdot {\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }}\right)}^2$$

(32)

The output noise spectral density with the additional resonator can be expressed as

$$P_{\mathrm{noise}\_\mathrm{add}}={\displaystyle \frac{\overline{{i_{\mathrm{noise}}}^2}}{{\displaystyle △}\omega }}\cdot {\left({\displaystyle \frac{1}{2Q_1[1+\cos (2\theta )]}}\cdot {\displaystyle \frac{{\omega }_0}{{\displaystyle △}\omega }}\right)}^2\cdot \left(R_{\mathrm{p}}/2\right)$$

(33)

In order to ensure the convergence of (33), we must have

$$\cos (2\theta )>0$$

(34)

Namely

$$0<\theta <\pi /4 \mathrm{or} 3\pi /4<\theta <\pi$$

(35)

Under the conditions satisfying (35), for the same power of carrier, the phase noise with the additional resonator can be reduced by

$$P{N}_{\mathrm{add}}=P{N}_{\mathrm{original}}-20\cdot \log [1+\cos (2\theta )]+3$$

(36)

As can be seen from (36), in the ideal case where θ = 0, the phase noise can be reduced by up to 3 dB. As θ increases, the optimization effect on phase noise diminishes. The case where θ = π/4 is a special situation. Although it cannot be calculated using (35), according to (13), the Q_add of the additional resonator is zero in this scenario. This implies that the introduced resonator lacks frequency filtering capability and, consequently, cannot enhance the Q factor and phase noise performance. This observation is corroborated by the simulations shown in Figure 10.

Figure 10 shows the simulated phase noise with different θ. Taking the phase noise at the offset of 1 MHz as an example, without the introduction of the additional resonator, the original oscillator’s phase noise is −138.1 dBc/Hz. Under ideal conditions, where θ = 0, the phase noise reaches a minimum value of −141.2 dBc/Hz. For 0 < θ < π/4, the phase noise ranges between −141.2 dBc/Hz and −138.1 dBc/Hz. When θ = π/4, the phase noise returns to −138.7 dBc/Hz. According to the analysis in Section 1, for π/2 < θ < 3π/4, the additional resonator disrupts the oscillation condition of the original oscillator, preventing the oscillator from startup.

Table 1. Simulated and calculated phase noise vs. different θ.

θ	0°	15°	25°	35°	45°	Original
Simulated	−141.4	−140.6	−139.6	−138.9	−138.7	−138.1
Calculated	−141.1	−140.5	−139.4	−138.6	−138.1	/

presents the simulated phase noise for 0 < θ < π/4 alongside the calculated results from (36). In this context, PN_original is also obtained from simulations. Moreover, (36) is used to calculated the noise reduction attributable from the additional resonator.

Figure 10 also illustrates the variation of output power with respect to θ. Introducing an additional resonator does not cause significant fluctuations in output power. Especially when θ is small, these fluctuations can be considered negligible.

3. Implementation and Analysis of the Circuit

3.1. SISL Technology

Figure 11a shows the 3D view of the proposed oscillator. A typical SISL circuit has five PCB layers with double-sided metal. The substrates in this circuit are labeled Sub1 to Sub5 from top to bottom, whereas the metal layers are labeled G1 to G10 from top to bottom. Air cavities are formed by excavating Sub2 and Sub4, and the via holes are used to create metal walls for electromagnetic shielding. The main circuits are distributed on G5 and G6. The five layers are fixed and pressed together using rivets after they are assembled in sequence. The SISL circuit exhibits a self-packaging property unlike the traditional suspended line circuits. In addition, the SISL cavity resonator has a higher Q compared with the planer resonator because most of the energy is distributed in the air cavity.

We used FR4 substrates with a relative dielectric constant of ε_r = 4.4 to minimize the cost of the circuits. The thickness of the five layers was 0.6 mm, 0.6 mm, 0.127 mm, 0.6 mm, and 0.6 mm, respectively. The cavity height, which is determined by the thickness of Sub2 and Sub4, must consider the transistor’s packaging height. Increasing the cavity height should be avoided because it may cause high-order resonant modes. The appropriate thickness in the standard FR4 specifications is selected according to these factors. The thickness of Sub1 and Sub5 does not affect the circuit performance.

3.2. Basic Structure of the Original Oscillator and the Additional Resonator

The resonator above is a bandpass SISL cavity resonator (Figure 11b). Sub1 was used as a cover, Sub2 was excavated to form the air cavity, and Sub3 was used for placing the core circuit: the excitation probes (P1 and P2) were located on G5. The resonator above and the subsequent circuit together form the original oscillator.

The resonator below is the additional resonator, which is a one-port SISL cavity resonator (Figure 11d). Sub5 was used for cover, Sub4 was excavated to form the air cavity, and the excitation probe (P3) was located on G6.

The two resonators were stacked up and down to save the footprint area. We used a λ/4 coupled line as the DC block at P1, P2, and P3. The coupling line at P3 was also used as the transition from G6 to G5 (Figure 11c). Other parts of the core circuit were all located on G5.

3.3. Implementation of the Resonator

The two cavity resonators should be set to the same resonant frequency and mode. The TE₁₀₁ mode resonant frequency of two resonators can be determined as described previously [26] (pp. 284–288) using the expression below:

$$f_{101}={\displaystyle \frac{c}{2\pi \sqrt{{\mu }_r{\epsilon }_r}}}\sqrt{{({\displaystyle \frac{\pi }{L\mathrm{c}}})}^2+{({\displaystyle \frac{\pi }{Wc}})}^2}$$

(37)

where Lc and Wc represent the length and width of the cavity, μr and εr denote the dielectric constant and the permeability of the air, respectively. We set Lc = Wc for ease of calculation, and an initial value of 21.2 mm was obtained. The electric field distribution is shown in Figure 12b. The actual cavity and feeding probe sizes were optimized using ANSYS HFSS 18.0. The final exact sizes are shown in the caption of Figure 12.

The side view of the proposed resonators is shown in Figure 12a, and the configuration of the G5 and G6 of the two resonators are shown in Figure 12c,d. Port 1 and Port 2 are introduced on the G5 to excite resonator 1 formed by Sub 2, and Port 3 is added on the G6 to excite cavity 2 formed by Sub 4.

The feeding line of the additional resonator was located in G6, and it was necessary to transit Port 3 from G6 to G5. Therefore, a λ/4 coupling line was designed for transition and a DC block, as shown in Figure 13. The variation of key DC block parameters within typical fabrication tolerance (±0.1 mm) shows a negligible impact on circuit performance, indicating acceptable design robustness.

The independent simulated responses are shown in Figure 14. The two resonators had the same resonant frequency of 10.34 GHz. S11 is the return loss for resonator 1, and S21 is the transmission response for the resonator in the original oscillator. Whereas S33 is the return loss for an additional resonator. The coupling TL was not introduced at this moment, and the two resonators were independent from each other.

3.4. Implementation of the Proposed Oscillator

A schematic representation of the circuit of the oscillator with an additional resonator is presented in Figure 15. The active device was BFU910f obtained from NXP company with a NFmin of 0.65 dB and a maximum power gain of 13 dB at 10.7 GHz. The device was biased at V_CE = 2 V and Icc = 12.6 mA. The input of the transistor was minimum noise matched to 50 Ohm, and the output was conjugate matched to 50 Ohm. The λ/4 coupled lines were designed to achieve DC blocking. The RF choke comprised a λ/4 high resistance line and a λ/4 radial stub. The Wilkinson power divider was used as the output-match network. A 50-ohm conductor-backed coplanar waveguide (CBCPW) line was used to hold the SMA connector during the testing process. The SISL to CBCPW transition was designed to match the impedance and reduce the discontinuity of the dielectric [11], as shown in Figure 15.

The simulated gain of the matched transistor was approximately 9.92 dB. We connected all the parts in the original oscillator with a 50 Ohm phase compensation line to ensure the loop phase met the start-up condition. The transmission line used for the additional resonator coupling was introduced after the original oscillator was fully designed. The length of coupling TL in our design was about 23° due to the layout limitation. This length reduces the phase noise by approximately 2 dB as shown in previous analysis.

The layout of Sub3 of the oscillator with the additional resonator from the top view is presented in Figure 16a, and the partial dielectric excision version to further reduce dielectric loss is shown in Figure 16b. The layout of Sub3 of the original oscillator for comparison is shown in Figure 16c.

Photographs of each layer of the proposed oscillators are presented in Figure 17. Most of the parameters were the same for the three oscillators. Only the phase compensation line length was slightly different to meet the phase condition for the oscillator start-up.

The accurate length of the coupling TL between the cavities was not required when designing, because the length is dependent on the limitation of the layout. For example, the final value for the coupling TL θ in this study was 23°. This is the result of evaluating the phase noise and footprint area of the circuit, and the limitation of the layout.

4. Experimental Results

All oscillator measurements were conducted at room temperature. The proposed oscillator is fabricated on FR4, a substrate material known for its relatively low sensitivity to temperature variations. The typical temperature coefficient of FR4 is approximately –50 to –200 ppm/°C. A temperature change of ±30 °C would cause only a minor variation in its dielectric constant (which is ±0.0132), resulting in a resonant frequency shift of less than 0.01 GHz (about 0.096%), which can be considered negligible in practical applications.

Although the detailed temperature-dependent modeling of the transistor is not feasible, general trends reported in previous studies [27,28] suggest that increased temperature typically reduces gain, while raising thermal noise, potentially degrading the phase noise or startup conditions. Conversely, lower temperatures may alter bias points, possibly affecting the oscillation behavior. Despite these known effects, the oscillator remained stable under room-temperature testing, and the impact of modest temperature variations is expected to be limited.

To suppress the power supply noise and ripple that may degrade the oscillator performance, three bypass capacitors (1 nF, 100 nF, and 1 μF) were soldered in parallel at the power supply pin. These capacitors can filter noise across a wide frequency range and improve the overall stability and phase noise performance of the oscillator.

The measured spectrums of the oscillator loading the additional resonator with partial dielectric excision are shown in Figure 18. The second harmonic suppression was more than 65.36 dBc.

The measured phase noises of three oscillators are shown in Figure 19. The phase noise of the oscillator introducing the additional resonator improved by 1.66 dB. The loss of the circuit is reduced by cutting the substrate, and the phase noise was reduced by 1.87 dB. The results showed that the oscillator loading the additional resonator can effectively reduce the phase noise. This improvement was mainly observed in the 1/f₂ region. In the 1/f₃ region, where the phase noise is influenced by the thermal noise, the flicker noise of active devices, and the power supply noise, the enhancement is less significant. Notably, since dielectric excision can reduce the noise figure of active devices [29], oscillators with dielectric excision exhibit a lower noise floor.

The figure of merit (FOM) was introduced for a comprehensive evaluation of the oscillators. FOM can be calculated using the expression below [15]:

$$FOM=L({\displaystyle △}f)-20\log ({\displaystyle \frac{f_0}{{\displaystyle △}f}})+10\log ({\displaystyle \frac{P_{DC}}{1\mathrm{mW}}})$$

(38)

where L(Δf) represents the phase noise at the offset Δf, and f₀ denotes the oscillation frequency. P_DC represents the DC power consumption, which is 25.2 mW in this design. The measured FOM values at 1 MHz offset of the oscillators loading the additional resonator with partial dielectric excision, loading the additional resonator and original parallel feedback oscillator were −197.79 dBc/Hz, −195.92 dBc/Hz, and −194.3 dBc/Hz, respectively. A detailed comparison between the oscillator designed in this study and other reported oscillators is shown in

Table 2. Comparisons between proposed and reported oscillators.

Ref.	f0 (GHz)	Frequency Selector	Material and Cost	L (1 MHz) (dBc/Hz)	Pdc (mW)	Pout (dBm)	FOM (1 MHz) (dBc/Hz)	Frequency Selector Size (λg × λg)	Self-Packaged
[5]	9.5	SIW resonator	Duroid 6002: High	−117	30	7.5	−184	0.7 × 0.7 #	No
[6]	8	Active filter	Rogers RO3035: High	−150	200	10	−205	0.5 × 0.5 #	No
[7]	11.86	SISL DMCR	Roggers 5880 + FR4: High	−133.9	16	3.44	−203.3	0.7 × 0.7 #	Yes
[12]	8.06	Elliptic filter	Roggers 5880: High	−143.5	22	3.5	−204	0.5 × 0.5 #	No
[13]	11.57	Dual-mode SIW BPF	Taconic TLY: High	−135.5	11.4	−2.3	−206.2	0.6 × 0.6 #	No
[14]	9.97	Eight mode SIW BPF	Roggers 3003: High	−126.13	150	3.17	−184.34 #	0.75 × 0.75 #	No
[16]	3.52	SMSIW	Roggers 5880: High	−132.1	16.2	−1.5	−191	0.267 × 0.109	No
[17]	1.98	QBP-FSN-MTZ	Roggers 5880: High	−149.4 *	32.4 #	9.61	−200.2 #	Not given	No
[16]	9.81	λ/2 microstrip line resonator	Teflon: High	−123.5	600 #	13.3	−175.55	Not given	No
[20]	5.69	λ/2 SISL resonator	FR4: Low	−122.81	10.9	2	−183.9	1.4 × 2.2 #	Yes
[21]	3.76	SISL transformer	FR4: Low	−119.45	37.1	−8.25	−183.5	0.1 × 0.1	Yes
This work	10.12	Original oscillator	FR4: Low	−128.3	25.2	1.30	−194.3	0.7 × 0.7	Yes
	10.13	Additional resonator	FR4: Low	−129.92	25.2	0.57	−195.92	0.7 × 0.7	Yes
	10.09	Additional resonator and dielectric excision	FR4: Low	−131.79	25.2	1.01	−197.79	0.7 × 0.7	Yes

^#: Calculated from the data given in the paper. *: From the data given in the paper.

. The oscillator loading the additional resonator and partial dielectric excision effectively reduced the phase noise compared with the original parallel feedback oscillator. The proposed design shows comparable FOM and phase noise with the oscillators using the material with a higher cost. Additionally, the proposed oscillators are characterized by their compact size, self-packaging properties, and low cost of fabrication.

5. Conclusions

An FR4-based low-phase noise oscillator loading the additional resonator with partial dielectric excision was designed in this study. The loaded Q factor was increased by loading the additional resonator with the same resonant frequency. The phase noise was reduced by 1.66 dB through by the additional resonator and reduced by 1.87 dB through partially excising the substrate observed from the experimental results. The effect of the different electric lengths of the coupling TL on the start-up condition, Q factor increasement, and the phase noise reduction was investigated, and a trade-off between footprint area and phase noise reduction was made to implement the low phase noise oscillator with a compact size. The oscillator was fabricated using FR4 substrates for low cost. It exhibited a competitive phase noise performance of −131.79 dBc/Hz at 1 MHz offset from the carrier frequency of 10.088 GHz, and the FOM at 1 MHz offset was −197.79 dBc/Hz. The proposed oscillator is based on the SISL technology, which also has the characteristics such as compact size and self-packaging properties. The SISL cavity used in this work also offers tuning potential, which will be explored in future work towards developing tunable-frequency oscillators.