Mutual Radiation Impedance of Uncollapsed CMUT Cells with Different Radii
Alper Ozgurluk, H. Kagan Oguz, Abdullah Atalar, Hayrettin Koymen
11 Mutual Radiation Impedance ofUncollapsed CMUT Cells with Different Radii
A. Ozgurluk , H.K. Oguz , A. Atalar , H. Koymen EECS Department, UC Berkeley, CA, USA EE Department, Stanford University, CA, USA EE Department, Bilkent University, Ankara, Turkey
Abstract —A polynomial approximation is proposed for themutual acoustic impedance between uncollapsed capacitive mi-cromachined ultrasonic transducer (CMUT) cells with differentradii in an infinite rigid baffle. The resulting approximation isemployed in simulating CMUTs with a circuit model. A verygood agreement is obtained with the corresponding finite elementsimulation (FEM) result.
I. I
NTRODUCTION
An array of transducers is usually employed for generatingpowerful and focused ultrasonic beams for imaging or therapypurposes [1]. Mutual interactions between the cells of the arrayare especially important when the mechanical impedance ofcells is relatively low. This is exactly the case for capacitivemicromachined transducers (CMUT) which provide a wideband operation in liquid immersion because of its low mechan-ical impedance [2]. Mechanics of uncollapsed CMUT cellscan be accurately modelled with flexural disks with clampededges [3]. It was shown that [4], [5] it is possible to accuratelymodel an array of CMUT cells using an electrical circuit modelcoupled to a impedance matrix composed of self and mutualradiation impedances. Self and mutual radiation impedances ofequal size CMUT cells in uncollapsed [6] and collapsed [7]mode can found in the literature.Porter [8] was the first to quantify the mutual impedance be-tween two identical flexural disks for different edge conditionsand obtained an infinite series solution. Chan [9] extendedthis work to cover the mutual impedance of flexural diskswith different radii. In both cases, the mutual impedanceturns out to be an oscillatory and slowly decaying functionof the distance between the two radiators. This implies thatthe acoustic coupling between all cells must be taken intoaccount to accurately predict the performance of an array.This demanding requirement increases the computation timesubstantially. Therefore, a sufficiently accurate but computa-tionally low-cost approximation is highly desirable to expeditethe simulations of mutual interactions.An approximation for the radiation impedance of the un-collapsed CMUT cells of equal size was presented earlier [5].This approximation is extended to that of unequal size cells.It is verified in Section III performing a finite element method(FEM) simulation for a particular case.II. A P
OLYNOMIAL A PPROXIMATION
In Chan’s work [9], the mutual impedance between twoflexural disks of clamped edges with radii a and a and Coefficient Real Part Imaginary Part p − . · − − . · − p − . · − . · − p . · − − . · − p − . · − . · − p . · − − . · − p − . · − . · − p − . · − − . · − p − . · − . · − p . · − − . · − p − . · − . · − p . · − − . · − TABLE IC
OEFFICIENTS OF THE APPROXIMATE POLYNOMIAL FOR A ( x ) center-to-center separation of d is found as ∗ Z ( ka , ka , kd ) ρcπa a = 3 (cid:40) ka ) ( ka ) × ∞ (cid:88) m =0 ∞ (cid:88) n =0 Γ( m + n + 1 / m ! n ! √ kd (cid:32) a m a n d m + n (cid:33) J m ( ka ) J n ( ka ) × [ J m + n + ( kd ) + i ( − m + n J − m − n − ( kd )] (cid:41) (1)where the reference is rms velocity, ρ is the density of themedium, c is the speed of sound and k is the wavenumber.For identical disks ( a = a = a ), an approximation for themutual radiation impedance is given as [5] Z ( ka, kd ) ρcπa ≈ A ( ka ) sin( kd ) + j cos( kd ) kd for ka < . (2)and A ( x ) can be approximated with a th order polynomial: A ( x ) ≈ (cid:88) n =0 p n x n (3)The values of p n ’s are tabulated in Table I.When the disks have different radii, finding an approx-imation is harder due to the cross-coupled ka and ka related terms in the mutual impedance expression. It can benumerically shown that the mutual impedance can be written ∗ There is an additional factor of 9 in (1), since the reference is rms velocityrather than peak velocity [9]. a r X i v : . [ phy s i c s . c l a ss - ph ] D ec ka R ea l P a r t o f B ( k a1 , k a2 ) AnalyticalApproximation ka =3ka =5 ka =1ka =0.5 Fig. 1. A comparison of the real parts of the analytical and approximateexpressions for different values of ka and ka . very accurately as a summation of a separable component in ka and ka , and an inseparable one as follows: Z ( ka , ka , kd ) ρcπa a ≈ B ( ka , ka ) sin( kd ) + j cos( kd ) kd (4)where B ( ka , ka ) = S ( ka ) S ( ka ) + jI ( ka , ka ) (5)where S ( x ) is a complex-valued function with S ( x ) = A ( x ) (6)With S r ( x ) and S i ( x ) real-valued functions, we write S ( x ) = S r ( x ) + jS i ( x ) (7)From (6) and (7), S r ( x ) and S i ( x ) are found as S r ( x ) = 1 √ (cid:32) A r ( x ) + (cid:112) A r ( x ) + A i ( x ) (cid:33) / (8) S i ( x ) = A i ( x ) √ A r ( x ) + (cid:112) A r ( x ) + A i ( x ) ) / (9)where A r ( x ) and A i ( x ) represent the real and imaginary partsof A ( x ) , respectively.The inseparable component in (5), I ( x , x ) , is approxi-mated as a polynomial in the following form I ( x , x ) ≈ ( x − x ) (cid:88) m =0 5 − m (cid:88) n =0 q mn x m x n (10)where the values of q mn ’s are given in Table II. We note thatfor ka = ka = ka , (4) reduces to (2).Figs. 1 and 2 are the plots of real and imaginary parts of B ( ka , ka ) as a function of ka for various values of ka .The analytical solution and the approximate expression agreevery well. ka I m ag . P a r t o f B ( k a1 , k a2 ) AnalyticalApproximation ka =3ka =5 ka =1ka =0.5 Fig. 2. A comparison of the imaginary parts of the analytical and approximateexpressions for different values of ka and ka .Coefficient Value Coefficient Value q − . · − q − . · − q , q . · − q , q . · − q , q − . · − q , q − . · − q , q . · − q , q . · − q , q − . · − q . · − q , q . · − q , q . · − TABLE IIC
OEFFICIENTS OF THE APPROXIMATE POLYNOMIAL FOR I ( x , x ) III. A C
OMPARISON OF THE A PPROXIMATION WITH
FEMThe accuracy and efficiency of the presented approximationis checked by employing it to couple CMUT cells in a cluster.We used (5) to model the mutual impedance between thecentral cell and peripheral cells of the 7-cell cluster depictedin Fig. 3 and we simulated this structure with an electricalcircuit simulator † capable of accepting frequency domain data.A transient analysis is performed using the equivalent circuitof CMUT cell [5]. Table III shows the geometric dimensionsand parameters used. Equivalent circuit model simulation † ADS, Agilent TechnologiesFig. 3. The geometry of CMUT element used for verification.
Frequency (MHz) | x p | ( n m ) ModelFEM
Edge Cells Center Cell
Fig. 4. The magnitude of the peak displacement for the center and edgecells for an excitation voltage of 1V peak.
Frequency (MHz) ∠ x p ( n m ) ModelFEM
Edge CellsCenter Cell
Fig. 5. The phase of the peak displacement for the center and edge cells. results are compared with FEM analysis results in Figs. 4and 5, where a very good agreement is observed. Notice theamplitude and phase differences of center cell and edge cells.For example, at 2.99MHz the center CMUT cell does not moveat all, while the maximum displacements of the edge cellsand center cell occur at 3.39MHz and 4.90MHz, respectively.At 3.75MHz, the center cell and edge cells displacementmagnitudes are equal with a 114 ◦ phase difference.IV. C ONCLUSIONS
A mutual impedance approximation is presented forclamped flexural disks having different radii. The resultingapproximation is inserted into the electrical equivalent circuitto couple CMUT cells in a cluster and a very good agreementwith the FEM results is obtained. This approximation makesit possible to design mixed-sized CMUT arrays using circuitsimulation tools.
Parameter ValueCenter Cell Radius, a µ mEdge Cell Radius, a µ mCenter-to-Edge-Cell Gap, a . µ mGap Height, t ga n mMembrane Thickness, t m µ mInsulator Thickness, t i n mYoung’s Modulus, Y GPaDensity, ρ kg/m Poisson Ratio, σ . Bias Voltage, V DC VExcitation, V AC V peakTABLE IIIP
ARAMETERS OF THE
CMUT