the metal-to-metal capacitor,i.e.Ec.and the other inductor is equally divided into k units (i>oo); is in the metal-to-substrate capacitor,i.emand it According to assumption 2),the voltage ith unit is V, can be derived as =。m-石(ag.) (6) where, 2 ((m)(m)Therefore.the electrical energy stored in the capacitor between the ith unit metal and the substrate can be expressed as Therefore,the equivalent capacitor of the inductor can 50-号co-(af-cm-j(の be expressed as where Cms represents the capacitance per unit area between the mth half turn and the substrate.The Cog=Cm_+Cm_ (4) electrical energy stored in the equivalent capacitance between the metal tracks of the Lms(m)and the substrate 2.1 Assumptions and definitions can be derived from [5] To accurately quantify Cm and Cms in inductors,the proposed DCM can analytically calculate them rather E_(m)=Cw(mV(m)+V(m)+V(m)V(m)(8) than qualitatively approximate [2].The fundamental 6 assumptions of DCM can be derived from the voltage Whatever structure inductor is,we can get entire distribution over the inductor,which is called voltage E.by adding up all E.m(m). profile in [3].For the conveniences of calculation and analysis,the following assumptions are made. Two terminal voltages of the differentially driven 1)The same layer metal traces of inductor have the same resistivity p,current,and metal track width w(at least inductor with symmetrically planar configuration(DSPD), in the same half turn),metal thickness t. ae+与'and-ly respectively.Hence 2)Voltage distribution is proportional to the lengths of 2 the metal tracks [3]. 11 (9) 3)The kth unit voltage difference between the adjacent 2m邮 half turns is regarded as constant and it is determined by averaging the beginning voltage and the ending voltage (10) of the half turns,regardless of the type of the inductor. Cm.=12m k (Note:This assumption only using in quantifing Cmm, where Cis equivalent capacitance between the not in quantifing Cm s) metal track and the substrate of the D.SPI The length of each half turn can be defined The signal terminal voltage of the single-ended driven as,,,..(n is turn number,sequential m planar inductor (SEPD)is Is and that of another represent current flow direction in inductor),and the terminal is 0,hence total length is defined asl=++.+).AC 11 1 signal voltage of one terminal of inductor is e while Ec.ms_dug= (11) 23 2 that of the other is Vend.According assumptions,AC signal voltage at the end terminal of the mth half turn C=CmWl (12) 3 inductor nd(m)]can be expressed as where Cis the equivalent capacitance between the metal track and the substrate of the SEPI (5) According to equation (10)and (12),under the same AC signal voltage at the beginning terminal of the mth equivalent area between the metal track and the substrate, half turn inductor [bee(m)]equal to the Vend(m-1). we can get the following equation Vbeg(0)=Vbeg. CC (13) 2.2.Equivalent Capacitance C Formula The DSPI can be regarded as two SEPI that have The lowest layer metal track of the mth half turn identical substrate parasitics at signal ports,which is halfthe metal-to-metal capacitor, i.e. ECm m , _ and the other is in the metal-to-substrate capacitor, i.e., ECm s , _ and it can be derived as ( ) 2 , ,_ ,_ 22 2 _ _ __ 1 2 111 (3) 222 C total eq s C m s C m m mm s ms s mm ms s E CV E E C V CV C C V = =+ = + =⋅ + ⋅ Therefore, the equivalent capacitor of the inductor can be expressed as CC C eq m m m s = + _ _ (4) 2.1 Assumptions and definitions To accurately quantify Cm_m and Cm_s in inductors, the proposed DCM can analytically calculate them rather than qualitatively approximate [2]. The fundamental assumptions of DCM can be derived from the voltage distribution over the inductor, which is called voltage profile in [3]. For the conveniences of calculation and analysis, the following assumptions are made. 1) The same layer metal traces of inductor have the same resistivity ρ , current, and metal track width w (at least in the same half turn), metal thickness t. 2) Voltage distribution is proportional to the lengths of the metal tracks [3]. 3) The kth unit voltage difference between the adjacent half turns is regarded as constant and it is determined by averaging the beginning voltage and the ending voltage of the half turns, regardless of the type of the inductor. (Note: This assumption only using in quantifing Cm_m, not in quantifing Cm_s) The length of each half turn can be defined as, 12 2 , ,... , m n ll l l (n is turn number, sequential m represent current flow direction in inductor), and the total length is defined as 12 2 ( ... ) tot n l ll l =+++ . AC signal voltage of one terminal of inductor is Vbeg, while that of the other is Vend. According assumptions, AC signal voltage at the end terminal of the mth half turn inductor [Vend(m)] can be expressed as ( ) 1 1 ( ) m j j end beg end tot l Vm V V l + = = − ∑ (5) AC signal voltage at the beginning terminal of the mth half turn inductor [Vbeg(m)] equal to the Vend(m-1). Vbeg(0)= Vbeg. 2.2. Equivalent CapacitanceCm s _ Formula The lowest layer metal track of the mth half turn inductor is equally divided into k units ( i → ∞ ); According to assumption 2), the voltage ith unit is Vi, i beg m ( ) () ( ) i V V m Vi k = − ⋅∆ (6) where, 1 ( ) ( ( ) ( )) k Vi V m V m m beg end k →∞ ∆≡ − . Therefore, the electrical energy stored in the capacitor between the ith unit metal and the substrate can be expressed as ( ) ( ) 2 2 , 1 11 () () ( ) ( ) 2 2 c ms m ms beg end wl E i Ci V C V m V m k k ⎛ ⎞⎛ ⎞ ∆ = ⋅ ⋅∆ = ⋅ ⋅ − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (7) where Cms represents the capacitance per unit area between the mth half turn and the substrate. The electrical energy stored in the equivalent capacitance between the metal tracks of the Lms(m) and the substrate can be derived from [5] ( ) 2 2 , _ 1 () () () () () () 6 beg E m C wl m V m V m V m V m c m s ms end beg end = + +⋅ (8) Whatever structure inductor is, we can get entire Ecm s , _ by adding up all , _ ( ) E m cm s . Two terminal voltages of the differentially driven inductor with symmetrically planar configuration (DSPI), are 1 2 + VS and 1 2 − Vs , respectively. Hence 2 2 , _ __ 11 1 2 12 2 E C wl V C V c ms diff ms tot s m s diff s ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ i ii i (9) _ _ 1 12 tot m s diff ms wl C C k = (10) where Cm s diff _ _ is equivalent capacitance between the metal track and the substrate of the DSPI. The signal terminal voltage of the single-ended driven planar inductor (SEPI) is VS and that of another terminal is 0, hence 2 2 , _ __ 11 1 23 2 E C wl V C V c ms diff ms tot s m s se s ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ i ii i (11) _ _ 1 3 C C wl m s se ms = (12) where Cm s se _ _ is the equivalent capacitance between the metal track and the substrate of the SEPI. According to equation (10) and (12), under the same equivalent area between the metal track and the substrate, we can get the following equation __ __ 1 4 C C m s diff m s se = (13) The DSPI can be regarded as two SEPI that have identical substrate parasitics at signal ports, which is half