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Linear frequency tuning in an LC-resonant system using a C–V response controllable MEMS varactor


This paper proposes a device level solution to achieve linear frequency tuning with respect to a tuning voltage (V tune ) sweep in an inductor (L)–capacitor (C) resonant system. Since the linearity of the resonant frequency vs. tuning voltage (fV) relationship in an LC-resonant system is closely related to the CV response characteristic of the varactor, we propose a CV response tunable varactor to realize the linear frequency tuning. The proposed varactor was fabricated using microelectromechanical system (MEMS) surface micromachining. The fabricated MEMS varactor has the ability to dynamically change the CV response characteristic according to a curve control voltage (V curve-control ). When V curve-control was increased from zero to 9 V, the CV response curve was changed from a linear to a concave form (i.e., the capacitance decreased quickly in the low tuning voltage region and slowly in the high tuning voltage region). This change in the CV response characteristic resulted in a change in the fV relationship, and we successfully demonstrated almost perfectly linear frequency tuning in the LC-resonant system, with a linearity factor of 99.95%.


Today’s multi-band wireless communication devices and systems use various frequencies, and therefore require frequency tunable modules, which have been the subject of continuous study. Among them, the voltage-controlled oscillator (VCO) is one of the key components for changing the output frequency in wireless communication systems. Among its various characteristics, the linearity of the VCO’s output frequency tuning is a critical figure-of-merit in most applications. For example, the linearity of the output frequency sweep is directly connected to the range accuracy of the frequency modulation continuous wave (FMCW) radar system [1]. In phase-locked loop circuits, it is known that the nonlinear frequency tuning characteristic in a VCO is one of the main sources affecting phase noise degradation [2].

In the common VCO, which uses inductor (L)–capacitor (C) resonant systems (LCVCO), a varactor is the main device for changing the output frequency [3]. Solid-state varactors based on semiconductors, however, produce a fixed CV response curve that is not suitable for a linear fV response, and it is not changeable once it is fabricated. As a result, it is difficult to linearize the output frequency response with conventional varactors. Accordingly, there have been notable efforts to linearize the fV response by employing specifically designed circuits [4,5,6]. However, these complicated circuits tend to increase the cost of the system design and consume more electrical power, which are undesirable features.

To address these problems, this study focused on the CV response of the varactor itself, to achieve linear frequency tuning in the LC resonant system without any circuit based support. First, we sought the kind of CV response characteristic necessary to achieve the linear frequency tuning. Then a MEMS varactor satisfying the required CV response characteristic was designed and fabricated using surface micromachining. At this step, the core concept for this kind of varactor was adopted from the previous paper [13]. However, since the capacitor needs to be connected with an inductor to demonstrate linear fV tuning in this paper, the design and fabrication process were fully modified to achieve the goal. The fabricated MEMS varactor was then integrated with a monolithically fabricated spiral inductor to verify its effect on frequency linearization.

Proposed concept and design

Required C–V response

Figure 1a represents the basic LCVCO circuit. Since the oscillation occurs at the resonant frequency of the connected inductance (L) and the effective capacitance (C eff ) at the node of x or y, the oscillation frequency is described as follows:

Fig. 1

Ideal linear frequency tuning in LC–VCO. a Basic components of the LC–VCO circuit, b ideal linear output frequency tuning characteristic, c required C tune– V tune response to achieve linear frequency tuning, with various parasitic capacitances, C p

$$f_{o} = \frac{1}{{2\pi \sqrt {L\,C_{eff} } }} = \frac{1}{{2\pi \sqrt {L\,(C_{tune} + C_{p} )} }}$$

where, C tune is the capacitance of the varactor and C p is the parasitic capacitance, if any, at the node of x or y.

The ideal linear output oscillation frequency response with respect to the tuning voltage (V tune ) is hypothetically plotted in Fig. 1b. The straight line in Fig. 1b is expressed as follows:

$$f_{o\_ideal} = \alpha V_{tune} + \beta$$

where, α is the gradient of f o_ideal vs. V tune graph, and β is the intercept value of the vertical axis (f o_ideal at V tune is zero). By plugging f o in (1) into f o_ideal in (2), the following relationship is deduced:

$${\text{C}}_{{tune}} = \frac{ 1}{{ 4\pi^{ 2} \left( {\alpha {\text{V}}_{{tune}} + \beta } \right)^{ 2} {\text{L}}}} - {\text{C}}_{\text{p}}$$

Next, a reasonable boundary condition is set to determine α and β. The minimum and maximum tuning voltages (V tune ) are set as 0 and 15 V, respectively, and the capacitances for each voltage are set to be 900 fF (V = 0 V) and 400 fF (V = 15 V), respectively, for operation at frequency ranges of several GHz. According to (3), the L value does not affect the CV response shape, and in this work it is set to 2 nH to obtain several GHz output oscillation frequencies.

Figure 1c represents the required C tune V tune response in (3) in order to achieve the linear output frequency tuning characteristic shown in Fig. 1b for several parasitic capacitances (C p ). As can be seen in Fig. 1c, we need a concave shape in the CV response of the varactor, and more importantly, the CV response characteristic should be changeable (tunable), since the C p might be unknown or vary from system to system.

Proposed MEMS varactor

There have been a lot of works on RF MEMS capacitors to acquire a unique property that is not easily achieved by the conventional semiconductor-based technology [7,8,9,10,11]. These works tell that the MEMS capacitors have much more flexibility in the design that determines the CV response. Therefore, we made use of the MEMS technology to satisfy the specific CV response. Although, it has integration issues, the MEMS variable capacitor provides two main benefits: a possibly high Q-factor from its high conductive metal signal path, and variable tuning characteristic even adaptive to unknown parasitic components.

Figure 2a describes the proposed MEMS varactor, which can realize the required CV response discussed in the previous section. The capacitance is formed between the top and bottom plates in the capacitor section. When the tuning voltage (V tune ) is applied to the V tune electrode, the tuning plate in the actuator section (electrically grounded) moves down due to the electrostatic force. Then, the top plate in the capacitor section is lifted up by the seesaw mechanism, making the capacitance smaller. This increasing-gap actuation can give us a linearly decreasing CV characteristic, shown as a dotted line in Fig. 2b [12].

Fig. 2

Proposed MEMS varactor employing a levering actuator. a Perspective view, b tunable C–V tune characteristic with respect to V curve-control

The proposed MEMS varactor shown in Fig. 2a has an another electrode, which is a curve-control electrode, to control the CV response curve shape. The initial position of the top plate in the capacitor section is closer to the bottom plate as we apply V curve-control . This results in an increase in the initial capacitance (at V tune  = 0 V) in the C tune V tune curve, as shown in Fig. 2b (solid line), while the final capacitance (at the maximum tuning voltage) is preserved, since the V curve-control loses its attractive effect as the control plate moves up. Therefore, as we increase V curve-control , the shape of the CV tune curve becomes a more concave form [13]. In this way, we can control the CV response characteristic of the varactor to accommodate the parasitic capacitance, to eventually achieve the linear output frequency tuning discussed in Fig. 1.

Design of the structural parameters

This section details the process of designing dimensional parameters. Therefore, the author suggests that readers who are not interested in the designing process skip this section.

To achieve the linear fV response, it is beneficial to set the initial CV response curve to be linear when the curve control voltage is zero (V curve-control  = 0 V). According to previous works, the linearity of the CV response is mainly determined by the ratio of l 2 and l 3 [12, 13] shown in Fig. 2a. When l 3 increases leaving l 2 unchanged from the optimal length showing a linear CV response, the CV curve is changed to a concave form. On the other hand, as l 3 decreases without changing l 2 , the CV response is turned into a convex shape. In addition, another variable exists in this work: The ratio of the gap height between the capacitor section and the actuator section. In other words, the length of l 3 (which determines the total length of the varactor) can be shortened by reducing the gap in the capacitor section so that it is smaller than that of the actuator section. Taking these points into account, the capacitance in the capacitor section can be described as follows:

$${\text{C}}_{{cap}} = \int\limits_{{{\text{l}}_{ 2} }}^{{{\text{l}}_{ 3} }} {\frac{{\varepsilon {\text{b}}}}{{{\text{l}}\theta + {\text{g}}_{{cap}} }}{\text{dl}} \cong \frac{{\varepsilon \left( {{\text{l}}_{ 3} - {\text{l}}_{ 2} } \right){\text{b}}}}{{\frac{{\left( {{\text{l}}_{ 3} + {\text{l}}_{ 2} } \right)}}{ 2}\theta + {\text{g}}_{{cap}} }}}$$

where, ε is permittivity and θ is a rotational angle that has positive value as the capacitor plate moves up. Since the rotational angle of the proposed device is small enough, the additionally increasing height in the capacitor section can be simply expressed as l × θ.

For simpler calculation, it can be assumed that the rotational movement of the capacitor section is a parallel plate actuation [13]. Therefore, the integral form can be re-written as the right hand side of (4). As a result, the expression for the capacitor section can be arranged by θ, as follows;

$$\theta = \left( {\frac{{\varepsilon {\text{b}}\left( {{\text{l}}_{ 3} - {\text{l}}_{ 2} } \right)}}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)\frac{ 2}{{\left( {{\text{l}}_{ 3} + {\text{l}}_{ 2} } \right)}} = {\text{f}}_{\text{I}} \left( {{\text{C}}_{{cap}} , {\text{g}}_{{cap}} , {\text{l}}_{ 2} , {\text{l}}_{ 3} } \right)$$

Next, in the actuator section, the well–known equation that models the rotational movement with respect to the input voltage is adopted, as follows [13, 14]:

$${\text{T}}_{{tune}} = \frac{{\varepsilon {\text{bV}}_{{tune}}^{ 2} }}{{ 2\theta_{ {max} }^{ 2} }}\left( {\frac{ 1}{ 2}\left( { 1- \gamma^{ 2} } \right) + \frac{ 2}{ 3}\frac{{\left( { 1- \gamma^{ 3} } \right)\theta }}{{\theta_{{max} } }} + \frac{ 3}{ 4}\frac{{\left( { 1- \gamma^{ 4} } \right)\theta^{ 2} }}{{\theta_{{max} }^{ 2} }} + \cdots + \left( {\frac{{{\text{n}} + 1}}{{{\text{n}} + 2}}} \right)\left( { 1- \gamma^{{{\text{n}} + 2}} } \right)\left( {\frac{\theta }{{\theta_{{max} } }}} \right)^{\text{n}} } \right)$$

where, T tune is an electrostatic torque generated by the tuning voltage input, γ is l 0_tune /l 1_tune , and θ max is g tune /l 1_tune . Here, we separate the variable for the gap of the tuning electrode (g tune ) from the one in the capacitor plate (g cap ) to increase the freedom of design while the previous model used one fixed variable for those two values [13]. Therefore, as a result, we can reduce total length of the capacitor by controlling the rate of g tune and g cap with high CV response tunability.

By expanding (6) with n = 2, the following equation is deduced:

$$\begin{aligned} {\text{T}}_{{tune}} &=\, \frac{{\varepsilon {\text{bV}}_{{tune}}^{ 2} {\text{l}}_{{ 1 {{\_tune}}}}^{ 2} }}{{ 2 {\text{g}}_{{tune}}^{ 2} }}\left( \frac{ 1}{ 2} - \frac{ 1}{ 2}\frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 2} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 2} }} + \frac{ 2}{ 3}\frac{{\left( { 1- \frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 3} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 3} }}} \right){\text{l}}_{{ 1 {{\_tune}}}} \theta }}{{{\text{g}}_{{tune}} }}\right. \\ & \quad +\left. \frac{ 3}{ 4}\frac{{\left( { 1- \frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 4} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 4} }}} \right){\text{l}}_{{ 1 {{\_tune}}}}^{ 2} \theta^{ 2} }}{{{\text{g}}_{{tune}}^{ 2} }} \right) \\ & =\, {\text{f}}_{\text{II}} \left( {\theta , {\text{g}}_{{tune}} , {\text{V}}_{{tune}} } \right) \end{aligned}$$

By relating (7) to the rotational restoring force generated from the levering beam, the model for the actuation section is described as follows:

$${\text{f}}_{\text{II}} \left( {\theta , {\text{ g}}_{{tune}} ,\;{\text{V}}_{{tune}} } \right) = {\text{K}}_{\text{m}} \theta$$

where, K m is a rotational mechanical spring constant determined by the geometry of the levering beam [15].

As the final step, the CV response can be expressed by plugging (5) into (8). Logically, it can be expressed as follows:

$${\text{f}}_{\text{II}} \left( {{\text{f}}_{\text{I}} \left( {{\text{C}}_{{cap}} ,\;{\text{g}}_{{cap}} ,\;{\text{l}}_{ 2} ,\;{\text{l}}_{ 3} } \right) ,\;{\text{g}}_{{tune}} , {\text{ V}}_{{tune}} } \right) = {\text{K}}_{\text{m}} {\text{f}}_{\text{I}} \left( {{\text{C}}_{{cap}} ,\;{\text{g}}_{{cap}} ,\;{\text{l}}_{ 2} ,\;{\text{l}}_{ 3} } \right)$$

The structural parameters which do not affect the linearity of CV response are pre-determined, as shown in Table 1 (parameters that are out of the solid-lined box). Regarding fabrication burden and measurable capacitance range, l 3 l 2 is fixed as 200 μm. Therefore (9) can be arranged as follows (the detailed equation is written in the appendix section at the end of this paper):

$${\text{C}}_{{cap}} = {\text{f}}_{\text{III}} \left( {{\text{V}}_{{tune}} ,\;{\text{g}}_{{tune}} ,\;{\text{g}}_{{cap}} ,\;{\text{l}}_{ 3} } \right).$$
Table 1 Dimensional parameters

Now that the CV response equation is developed, the relationship among the variables can be analyzed by plotting the coefficient of the second order term in C cap after expanding (10) in the Taylor series.

$${\text{C}}_{ 2} = \frac{ 1}{ 2}\left( {\frac{{{\text{d}}^{ 2} {\text{C}}_{{cap}} }}{{{\text{dV}}_{{tune}}^{ 2} }}} \right)\,\left| {\,{\text{V}}_{{tune}} = {\text{V}}_{\text{t}} } \right. ,\;\;{\text{V}}_{\text{t}} = \frac{{{\text{V}}_{{{{pull}} - {{in}}}} }}{ 2}$$

where, C 2 means the coefficient of the second order term of the Taylor series expansion. The linear shaped CV response is realized when C 2 is zero, otherwise, a concave or convex shape is achieved when C 2 has positive or negative values, respectively [13].

Figure 3a represents the three dimensional plot of C 2 obtained by calculating (10) using (11) with several g cap cases. Since the length of l 2 is determined once l 3 is selected, we focused on the gap difference between the capacitor (g cap ) and actuator section (g tune ). Therefore, C 2 is depicted with respect to l 3 and g ratio (g tune /g cap ). In the case of g cap  = 1 μm, it can be seen that the slope is higher than in the case of g cap  = 1.2 and 1.5 μm as l 3 changes. This means that the linearity of the CV response becomes more sensitive as the actuator length changes when the capacitor plate is closer to the bottom electrode. Another noticeable point is that all three g cap cases show the same values of g ratio and l 3 , which make C 2 zero, as shown in Fig. 3b. This tells us that the absolute value of g cap and g tune is not important, however, the ratio of the two is the determining factor for making the initial CV response (V curve-control  = 0) linear. Figure 3c shows l 3 making C 2 zero and the corresponding pull-in voltages when g cap is 1 μm. The pull-in voltage is derived from (8), and it is determined only by g tune (the other geometrical dimension is already fixed in this work, and θ or V tune are not determinant factors for the pull-in voltage). As depicted in Fig. 3c, l 3 can be shortened as g tune gets larger for the same result, which reduces the burden of fabricating a long suspended actuator beam. However, the increment of g tune results in an increase of the pull-in voltage. Therefore, a compromise point should be chosen considering the fabrication burden and available voltage level in the desired system. In this work, we chose g cap to be 1 μm and l 3 to be 560 μm, and therefore, g tune is determined to be 1.4 μm. The calculated CV response is compared with the measured result in appendix section of this paper.

Fig. 3

Analysis of parameters that determine the linearity of C–V response. a 3D plot for the coefficient of second order term in Taylor series. b Top view of (a) showing the cross points with C 2  = 0 plane are the same in every case. c l 3 and pull-in voltage in the case of g cap  = 1 μm

Results and analysis


The fabrication process for the proposed LC resonator starts with a bottom metal patterning as shown in Fig. 4, step (1). The upper Ti layer (50 Å) acts as an adhesion layer for the BCB. Since the device is supposed to handle an RF signal, the electrical pad for the DC voltage application must be isolated using a highly electrically resistive material, which is formed by leaving the bottom thin Ti film (500 Å) in this work (step 2). Then, BCB (Benzocyclobutene) is spin-coated, cured, and patterned using a polymer etcher to keep the high resistive connection from oxidizing, which would excessively increase the electrical resistance.

Fig. 4

Fabrication process of the proposed linear frequency-tuned LC resonator

A photoresist (PR) is patterned two times as a sacrificial layer. The first PR is patterned and additionally cured at high temperature for hardening (150 °C, step (4)) because the second PR is patterned on top of the first PR to form the capacitor section. The high temperature treatment changes the molecular structure of PR so that it is insensitive to UV light. In this way, a different thickness of PR sacrificial layer can be achieved, as shown in step (5). This difference realizes the desired gap ratio (g ratio ) between the capacitor section and actuator section discussed in Fig. 3. After thermally evaporated Cu is deposited as a seed layer, the levering beam is formed by electroplating in step (6). In step (7), the capacitor and inductor are formed, simultaneously. As the last step, the seed Cu layer and PR sacrificial layer are removed by respective wet etchants, followed by a critical point dryer (CPD) release process.

Figure 5 shows the final LC resonator device. From the top view, it can be seen that the capacitor and spiral inductor are serially connected through one end of the levering beam (upper image in Fig. 5a). The lower levering beam is connected to the electrode, Px that is used to measure only the CV response of the capacitor.

Fig. 5

Fabricated results. a Optical top view including electrical tipping pad below the LC device. b SEM images of the proposed device (upper), magnified view of the actuator beam showing the height of g cap and g tune (lower left), and serially connected suspended spiral inductor (lower right)

The difference in gap heights between the capacitor section and the actuator section was verified by scanning electron microscope (SEM) images, as shown in Fig. 5b. In the middle of the actuator beam, there is a specific point where the gap height is differently formed, so that it is 1 μm in the capacitor section (left direction) and 1.4 μm in the actuator section (right direction). The serially connected spiral inductor is suspended from the bottom with the same height as that of the actuator section (1.4 μm).

Measurements and analysis

First, the CV response was measured to verify that the capacitor device had the desired tuning performance. Following the measurement setup shown in Fig. 6a, the CV response could be measured solely by tipping the points, P1 and Px. Figure 6b represents the measured result. When we consider the V tune range from 6 to 18 V, which is designated the actual usage region (depicted with sky blue colored dotted vertical lines), the CV response changes from convex to concave form as the curve control voltage (V curve-control ) increases from 0 to 9 V. When V tune is applied over 18 V, the capacitance decreases abruptly due to the pull-in phenomenon in the actuator section.

Fig. 6

Capacitance measured results. a Measurement setup. b C–V response of the capacitor device with respect to tuning voltage in several control voltage cases

Having the result of the CV response tuning performance, RF measurement was conducted using a vector network analyzer, as shown in Fig. 7a. Figure 7b represents the measured result of return and insertion loss when V curve-control is 9 V. The case of V tune  = 20 V is overlapped with the case of V tune  = 22 V due to the pull-in phenomenon, which happened at V tune  = 19 V, as shown in Fig. 6b. It can be seen that the return loss of the LC resonator changes its resonant points (the lowest value of return loss) as V tune changes. By extracting the frequency that makes the return loss lowest, the fV response is finally acquired, as shown in Fig. 7c. The resonant frequency is tuned more and more linearly as V curve-control increases to 9 V in the actual usage region (V tune  = 6–18 V).

Fig. 7

Measured results. a Measurement setup. b Return loss and insertion loss with respect to frequency sweep. c Resonant frequency of the fabricated series LC resonator according to V tune

These results are well matched to the relationship between the CV response and fV response that we discussed previously. As expected from Fig. 1c, the linear fV response is achieved when the capacitor shows a concave CV response. This is the first experimental demonstration that the concave CV response results in a linear fV response in an LC resonator module. In view of Q-factor, the proposed device did not show a high Q due to the fact that the RF signal should pass though the relatively narrow levering beam. This can be solved by changing the signal path by splitting the bottom electrode [16]. Additionally, increasing the thickness of the bottom plate would be an effective solution for the high Q-factor [17].

Two commercial products were compared in terms of the linearity of the fV response and their tuning range. Figure 8 shows the plots of the frequency tuning with respect to tuning voltage (Fig. 8a) and a summary table of their performance (Fig. 8b) [18]. As shown in Fig. 8a, the two commercial semiconductor based VCO have convex fV responses. It is assumed that the MOS varactors used in those products have severely concave CV responses [19, 20]. Consequently, to achieve linear frequency tuning, those two VCOs must inevitably be connected to other extra circuits referred in the introduction section [4,5,6].

Fig. 8

Performance comparison. a fV response of commercial VCOs and proposed LC module. b Data for linearity, frequency tuning range, and tuning voltage

In contrast, the proposed LC resonator has the capacity to control the shape of the CV response within the module, so that the resonant frequency can be linearly tuned without any external circuital supports. Since the oscillation frequency of the VCOs is the resonant frequency of the inductor and varactor connected in parallel as discussed in the ‘proposed concept and design’ section, this high fV linearity is obviously expected to be shown even in the form of VCOs.

In addition to managing the frequency linearity, the proposed device is able to cover much wider range of resonant frequencies. This results in a much higher figure of merit, as shown in Fig. 8b.


In this paper, a device level approach was conducted to achieve linear output frequency tuning in an LC resonant system. The proposed concept was successfully demonstrated using a developed CV response tunable MEMS varactor and a monolithically-integrated spiral inductor. The fabricated LC module exhibited almost perfect linear frequency tuning characteristics, from 3.125 to 4.175 GHz with a linearity factor of 99.95%. Integration of the proposed MEMS capacitor with the VCO circuits could require an extra fabrication cost. However, since this device-level solution can reduce the complexity of RF circuits as well as relative power consumption, this development can be favorably considerable in multi-band RF system design.


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Authors’ contributions

CHH carried out overall design, fabrication, measurement and analysis of the results, and drafted the manuscript. YHY supported fabrication process and measurement. SDK analyzed failure issue of fabrication and made solutions especially sacrificial layer process. MHS suggested fabrication plan, method, and material selection. All authors participated in editing process of the final manuscript. All authors read and approved the final manuscript.


We also thanks to members of our laboratory (3D micro-nano structures lab.) for sincere comments on this research.

Competing interests

The authors declare that they have no competing interests.

Availability of data and materials

The datasets supporting the conclusions of this article are included within the article and its additional files.


This research was supported by the Center for Integrated Smart Sensors funded by the Ministry of Science, ICT & Future Planning as “Global Frontier Project” (No. CISS-2012054187), and by Samsung Research Funding Center of Samsung Electronics under Project Number SRFC-IT1402-06.

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Corresponding author

Correspondence to Jun-Bo Yoon.



Equation (9) is expressed as (12). After applying the structural parameter values shown in table I, except the parameters surrounded by the solid line (l 2 is expressed with l 3 according to the pre-determined relationship that l 3 l 2  = 200 μm) (12) is simply re-written with only four variables as (13).

$$\begin{aligned} \frac{{\varepsilon {\text{bV}}_{{tune}}^{ 2} {\text{l}}_{{ 1 {{\_tune}}}}^{ 2} }}{{ 2 {\text{g}}_{{tune}}^{ 2} }} & \left( {\frac{ 1}{ 2} - \frac{ 1}{ 2}\frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 2} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 2} }} + \frac{ 4}{ 3}\frac{{\left( { 1- \frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 3} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 3} }}} \right)\left( {\frac{{\varepsilon {\text{b}}\left( {{\text{l}}_{ 3} - {\text{l}}_{ 2} } \right)}}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right){\text{l}}_{{ 1 {{\_tune}}}} }}{{\left( {{\text{l}}_{ 3} + {\text{l}}_{ 2} } \right){\text{g}}_{{tune}} }} + \frac{{ 3\left( { 1- \frac{{{\text{l}}_{{ 0 {{\_tune}}}}^{ 4} }}{{{\text{l}}_{{ 1 {{\_tune}}}}^{ 4} }}} \right)\left( {\frac{{\varepsilon {\text{b}}\left( {{\text{l}}_{ 3} - {\text{l}}_{ 2} } \right)}}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)^{ 2} {\text{l}}_{{ 1 {{\_tune}}}}^{ 2} }}{{\left( {{\text{l}}_{ 3} + {\text{l}}_{ 2} } \right)^{ 2} {\text{g}}_{{tune}}^{ 2} }}} \right) \\ & = \frac{{ 2 {\text{K}}_{\text{m}} \left( {\frac{{\varepsilon {\text{b}}\left( {{\text{l}}_{ 3} - {\text{l}}_{ 2} } \right)}}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)}}{{{\text{l}}_{ 3} + {\text{l}}_{ 2} }} \\ \end{aligned}$$
$$\begin{aligned} \frac{{ 1. 4 2\times 1 0^{ - 23} {\text{V}}_{{tune}}^{ 2} }}{{{\text{g}}_{{tune}}^{ 2} }} & \left( { 0. 4 9+ \frac{{ 1. 0 6\times 1 0^{ - 4} \left( {\frac{{ 8. 8 5\times 1 0^{ - 19} }}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)}}{{\left( { 2 {\text{l}}_{ 3} - 2\times 1 0^{ - 4} } \right){\text{g}}_{{tune}} }} + \frac{{ 1. 9 2\times 1 0^{ - 8} \left( {\frac{{ 8. 8 5\times 1 0^{ - 19} }}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)^{ 2} }}{{\left( { 2 {\text{l}}_{ 3} - 2\times 1 0^{ - 4} } \right)^{ 2} {\text{g}}_{{tune}}^{ 2} }}} \right) \\ & = \frac{{ 6. 7 4\times 1 0^{ - 7} \left( {\frac{{ 8. 8 5\times 1 0^{ - 19} }}{{{\text{C}}_{{cap}} }} - {\text{g}}_{{cap}} } \right)}}{{ 2 {\text{l}}_{ 3} - 2\times 1 0^{ - 4} }} \\ \end{aligned}$$

For the Taylor expansion, (13) should be arranged by C cap as shown in (14). According to (14), C cap is categorized in the capacitor section (g cap ), actuator section (H: g tune and V tune ), and actuator beam length related parameter (B: l 3 ).

$$\begin{aligned} & {\text{C}}_{{cap}} = {\text{f}}_{\text{III}} \left( {{\text{V}}_{{tune}} ,\;{\text{g}}_{{tune}} ,\;{\text{g}}_{{cap}} ,\;{\text{l}}_{ 3} } \right) = \frac{{ 8. 8 5\times 1 0^{ - 1 9} }}{{{\text{BH}} + {\text{g}}_{{cap}} }} \hfill \\& {\text{H}} = \,{\text{g}}_{{tune}} \left\{ {\left( {\frac{{ 1. 2 6\times 1 0^{ 2 4} \,{\text{g}}_{{tune}}^{ 3} }}{{\,{\text{V}}_{{tune}}^{ 2} }} - 2. 7 9\times 1 0^{ 3} } \right) - \sqrt {\left( {\frac{{ 1. 2 6\times 1 0^{ 2 4} \,{\text{g}}_{{tune}}^{ 3} }}{{\,{\text{V}}_{{tune}}^{ 2} }} - 2. 7 9\times 1 0^{ 3} } \right)^{ 2} - 2. 5 8\times 1 0^{ 7} } } \right\} \hfill \\& {\text{B}} = {\text{l}}_{ 3} + {\text{l}}_{ 2} = 2 {\text{l}}_{ 3} - 2 0 0\times 1 0^{ - 6} \hfill \\ \end{aligned}$$

According to Fig. 9, g tune determines the sensitivity of H to V tune . As g tune becomes smaller, the rate of change of H becomes greater with respect to the change of V tune . It is also notable that l 3 affects the sensitivity of H; C cap decreases radically according to the increase in V tune when beam length (l 3 ) is longer. Otherwise, g cap controls the effect of BH to C cap , which means that when g cap is high, the rate of capacitance change (C cap ) to V tune is low. This analytical model properly describes our structural intuition.

Fig. 9

3D plot of H with respect to V tune and g tune

The proposed device is expected to have the mechanical resonance at several kHz. Figure 10 shows the simulated result of the proposed device, which represent the mechanical resonant frequency is 2.545 kHz. This result roughly means that it takes about 0.4 ms to move the capacitance back and forth between two values.

Fig. 10

CoventorWare™ simulation result of mechanical resonance. It occurs at 2.545 kHz

Figure 11a represents the calculated result of the proposed capacitor’s CV response using the model in the ‘design of the structural parameters’ section and the measured result. By adding the parasitic capacitance component on the capacitor plate and the tuning electrode, those two values are well matched as shown in Fig. 11b. To compensate the initial capacitance (value at V tune  = 0 V), the structural parameter, b was increased by 45 μm representing the parasitic capacitance. In terms of the tuning plate, since there exists two triangle shaped stopper at the end of the plate, the attractive force between the tuning plate and the tuning electrode is increased due to the increased parasitic capacitance. Therefore, we increased l 1_tune by 6.5 μm. Although this modification is well matched in the initial and the end of the tuning region, there is still a mismatched region in the middle. We consider this is because the levering beam sinks down a little when the control and the tuning voltages are applied simultaneously, which is not included in the model.

Fig. 11

Comparison of the calculated and measured C–V response of the proposed capacitor. a Pure comparison. b Modified comparison result including optimization factor (parasitic capacitance)

See Figs. 9, 10 and 11.

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Han, CH., Yoon, YH., Ko, SD. et al. Linear frequency tuning in an LC-resonant system using a C–V response controllable MEMS varactor. Micro and Nano Syst Lett 5, 25 (2017).

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  • CV response
  • Linearity
  • LC-resonant system
  • Microelectromechanical systems (MEMS) varactor
  • Resonant frequency
  • Voltage-controlled oscillator (VCO)