Fig. 1.  Simplified layout plane of a 1 : 1 on-chip stacked transformer. $d_{\rm in}$ and $d_{\rm out}$ are the inner and outer diameters of primary and secondary coils, respectively. $w$ is the width of metal coils. $L_{\rm gap}$ represent the gap distance from the port 1/2 (P1/2) to port 3/4 (P3/4). The sixth (top) metal M6 and fifth metal M5 were employed for the primary and secondary coils, respectively, whereas the first metal layer M6 was used for both the underpass and ground plane.

Fig. 2.  Topology of the proposed model. $C_{\rm t1}$ and $C_{\rm t2}$ are the turn-turn capacitor of the primary and secondary coil, respectively. $L_{i0}$ and $R_{i0}$ ($i$ $=$ 1, 2, 3) are the DC inductors and resistors of the primary and secondary coil, respectively. $L_{i1}$ and $R_{i1}$ ($i$ $=$ 1, 2, 3) represent the skin and proximity effect caused inductive and resistive parasitics of the primary and secondary coil, respectively. $C_{\rm p1}$ and $C_{\rm p2}$ represent the capacitors between the primary and secondary coil, respectively. $C_{{\rm ox}i}$ ($i$ $=$ 1, 2) in blocks D, E and F is the oxide capacitor, while $R_{{\rm sub}i}$ and $C_{{\rm sub}i}$ ($i$ $=$ 1, 2) are employed to characterize the losses caused by silicon substrate. The elements $C_{\rm ox3}$ and $R_{\rm sub3}$ in block F are introduced to capture the non-ideal parasitics between the primary coil and substrate. The $k_0$ and $k_1$ are the coupling coefficients between the primary and secondary coil. The coefficients $k_2$, $k_3$ and the elements $L_{\rm e}$ and $R_{\rm e}$ are used to characterize the substrate eddy current effect of the primary and secondary coil, respectively. The $L_{\rm e}$ and $R_{\rm e}$ are calculated using the method proposed in Ref. [10].

Fig. 3.  Simplified equivalent circuit for 1 : 1 stacked transformer model parameter extraction in two-port measurement. $C_{\rm t1}$ and $C_{\rm t2}$ are canceled. $Z_{\rm A}$, $Z_{\rm B}$, $Z_{\rm C}$, and $Z_{\rm E}$ are the $Z$-parameters of blocks A, B C and E, respectively. $Z_{\rm R}$ is the reflected impedance from P2 to P1, where $M =$ $k_0(L_{10}L_{30})^{1/2}, $ $Z_{\rm cp}$ $=$ 1/j$\omega C_{\rm p2}$.

Fig. 4.  Extracted $C_{\rm ox3}$ and $R_{\rm sub3}$ versus frequency characteristics. The average values at the range from 10 to 20 GHz are used in this work. The value of $R_{\rm sub3}$ calculated from Eq. (9) is also plotted: the value is close to 1500 $\Omega $.

Fig. 5.  Comparison of the model simulated and measured inductance of $P_1$ ($L_{11}$), $P_2$ ($L_{22}$) and the MAG characteristics.

Fig. 6.  Comparison of the model simulated and measured $S$-parameters.