SECTION II

## MODEL DERIVATION

### A. Basic Theory

The basics of PPF design are covered well in literature. In [1] this topic is treated extensively. Often, a direct synthesis approach as presented in [2], for example, is applied in polyphase filter design but this approach does not cover the case of frequency dependant feedback signals. Hence, the design method presented below will be based on system theoretic modeling.

### B. Core Circuit

For model derivation the core circuit of Fig. 1 is focused first. It resembles the single pole structure but the tapping is different. For simplicity, the single ended version is examined here. In most applications, the differential structure will be preferred so that the inverters are not required.

By using the superpostion principle with respect to ϕ_{i−} and ϕ_{q−} as well as the voltage divider law in its admittance form, the following sets of equations can be derived:
TeX Source
$$\matrix{
&\phi'_{i-} &= {G_w \over sC_1+G_q+G_c+G_w}V_{in,i}\hfill\cr
&\phi'_{q-} &= \left.{G_w \over sC_1+G_q+G_c+G_w}V_{in,q}\quad \right\vert\cdot j \cr
\noalign{\hrule width 175pt\vskip2pt}\cr
\sum&\underline{\phi'_{-}} &= {G_w \over
sC_1+G_q+G_c+G_w}\underline{V_{in}}\qquad \hbox{(1)}\cr
&\phi''_{i-} &= {-sC_1+G_q \over sC_1+G_q+G_c+G_w}V_{out,i}\hfill\cr
&\phi''_{q-} &= \left.{-sC_1+G_q \over sC_1+G_q+G_c+G_w}V_{out,q}\quad \right\vert\cdot j \cr
\noalign{\hrule width 175pt\vskip2pt}\cr
\sum&\underline{\phi''_{-}} &= {-sC_1+G_q \over sC_1+G_q+G_c+G_w}\underline{V_{out}}\qquad \hbox{(2)}\cr
&\phi'''_{i-} &= {-G_c \over sC_1+G_q+G_c+G_w}V_{out,q}\hfill\cr
&\phi'''_{q-} &= \left.{G_c \over sC_1+G_q+G_c+G_w}V_{out,i}\quad \right\vert\cdot j \cr
\noalign{\hrule width 175pt\vskip2pt}\cr
\sum&\underline{\phi'''_{-}} &= {jG_c \over sC_1+G_q+G_c+G_w}\underline{V_{out}}\qquad \hbox{(3)}\cr
&V_{out,i} &= \nu\phi_{i-}\hfill\cr
&V_{out,q} &= \nu\phi_{q-}\quad \vert\cdot j\hfill\cr
\noalign{\hrule width 175pt\vskip2pt}\cr
\sum &\underline{V_{out}} &= \nu\underline{\phi_{-}}\hfill\qquad \hbox{(4)}.
}$$In these equations, the dashes indicate complex quantities and *G*_{x} (*x* ∊ {*i*, *q*, *w*, *c*}) is the conductance according to the resistance *R*_{x}. The transfer functions themselves are all real in nature. The selection of signs in the numerators of equation set three conforms with the rule of complex multiplication. Since, a complex transfer characteristic is achieved^{1}. Equation (1) represents the outer loop feed forward transfer function. The feedback transfer characteristic is given by (2) merged with (3). Equation (4) finally denotes the innerloop feedforward transfer function. They can be combined to the system theoretic model given in Fig. 2.

The system theoretic model can also be developed in terms of equivalent circuits which is presented in [3]. By evaluation, the transfer function of the core circuit can be determined:
TeX Source
$${\underline{V_{out}}\over \underline{V_{in}}} = {\nu {G_w\over sC_1+G_q+G_c+Gw}\over 1-\nu {-sC_1G_q+j\cdot G_c\over sC_1+G_q+G_c+G_w}}.\eqno{\hbox{(5)}}$$If gain of the operational amplifiers is high enough, (5) simplifies to:
TeX Source
$${\underline{V_{out}}\over \underline{V_{in}}} \approx {G_w\over sC_1-G_q-j\cdot G_c} = {{1\over R_wC_1} \over s - {1\over R_qC_1}- j\cdot {1\over R_cC_1}}\eqno{\hbox{(6)}}$$

### C. Normalized Biquad Transfer Function

To obtain a biquad PPF, a second loop is required. A generalized form of the structure is depicted in Fig. 3.

The core circuit discussed before is used within the loop to set the inner loop feedforward transfer function *H*_{2}. Of course, this circuit has finite gain and its input can not be assumed to be virtual ground. Hence, loading of this node by the core circuit has to be taken into account. Again, the superposition principle can be applied in a similar manner as shown above, no matter which passive components are added. Thus, the outer loop feedforward and the feedback transfer function, *H*_{1} and *H*_{3}, can again be determined by voltage divider ratios in their admittance form. In consequence, the transfer functions are given by the admittance present in the respective signal path divided by the total node admittance. As *H*_{1} and *H*_{3} refer to the same node, their denominators are equal. This equality can be used to expand the generalized model of Fig. 3, where *N*_{y} (*y* ∊ {1… 3}) represents the numerator of *H*_{y} and *D*_{y} its denominator:
TeX Source
$$\eqalignno{\underline{H} &= {\underline{H_1}\underline{H_2}\over 1- \underline{H_2}\underline{H_3}}{\underline{N_1}\underline{N_2}\over \underline{D_1}\underline{D_2}-{\underline{D_1}\underline{D_2}\over \underline{D_2}\underline{D_3}}\underline{N_2}\underline{N_3}}\cr&={\underline{N_1}\underline{N_2}\over \underline{D_1}\underline{D_2}-\underline{N_2}\underline{N_3}}&\hbox{(7)}}$$*N*_{2} and *D*_{2} are determined by the core circuit and hence given by (6). They can be inserted into (7) to yield some more insight:
TeX Source
$$\underline{H} = {\underline{N_1}{1\over R_w}\over \underline{D_1}\left(sC_1-{1\over R_q}-j\cdot {1\over R_c}\right) -{1\over R_w}\underline{N_3}}\eqno{\hbox{(8)}}$$If this representation is compared with the normalized biquad polyphase filter transfer function (9), some important observations can be made.
TeX Source
$$\underline{H_n} = {Aw^2_0\over s^2 + s{w_0\over Q_0} -js2w_c-\left(w^2_c - w^2_0\right)- j {w_cw_0\over Q_0}}\eqno{\hbox{(9)}}$$

In this equation is the bandwidth of an equivalent low pass, is the center frequency, *Q*_{0} is the filter quality factor and A is the filter gain. Comparing (8) with (9) it becomes apparent, that *N*_{1} needs to be a real valued function which is usually the case. A second order polynomial will be generated by the product *D*_{1} *D*_{2} as *D*_{2} contains the total node admittance of the node being input to the core circuit. *D*_{2} must not have a complex coefficient preceeding the first order polynomial because a complex coefficient preceeding the second order polynomial of the complete denominator of *H* would arise else. *N*_{3} on the other hand does not suffer from such a restriction. This justifies the placement of crosscoupling capacitors into overall feedback as will be done below^{2}.

### D. Filter Structure

The remaining task is to find a circuit structure which allows for proper adjustment of *N*_{3}. As explained above, *N*_{3} contains the admittances in the feedback path. If we want to adjust the transfer function arbitrarily, we have to insert a feedback resistor tapping the core circuit at the inphase output, another one tapping at the quadrature output, a feedback capacitor tapping at the inphase output and finally a feedback capacitor tapping at the quadrature output into the inphase path. Signs can be varied by interchanging taps of differential paths. For the quadrature path, signs of the crosscoupling element taps have to be inverted. This will yield the feedback structure given in Fig. 4.

The denominators of *H*_{1} and *H*_{3} contain the conductance *G*_{w} due to the loading effect mentioned earlier. A circuit offering this feedback structure is shown in Fig. 5(a).

Figs. 5(b) and (c) show two slightly different structures for which reasonable parameter values could be calculated. They differ from the structure of Fig. 5(a) by interchanged feedback taps with respect to the differential paths. This measure causes changing signs in the numerator of *H*_{3} or in the denominator of *H*_{2} respectively. The transfer functions are given as follows^{3}:
TeX Source
$$\eqalignno{&{1 \over \underline{H}} = {s^2 + s \left[{1\over C_2 + C_3} \left({1\over R_i} + {1\over R_w} +{1\over R_{wo}} +{1\over R_{co}} \right) - {1\over C_1 R_q}- \right. \over {1\over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {\left.-{C_2\over C_1(C_2+C_3)} {1\over R_w}\right] -js \left[\pm {1\over R_c C_1} \pm {C_3 \over C_1 (C_2 + C_3)} {1\over R_w}\right]- \over {1 \over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {-{1\over C_1(C_2+C_3)} \left[{1\over R_q} \left({1\over R_i} + {1 \over R_w} + {1 \over R_{wo}} + {1\over R_{co}}\right) - {1\over R_w R_{wo}}\right]- \over {1 \over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {-j{1\over C_1(C_2+C_3)} \left[\pm {1\over R_c} \left({1\over R_i} + {1\over R_w} + {1 \over R_{wo}} + {1 \over R_{co}}\right) \mp {1\over R_w R_{co}}\right] \over {1 \over R_i R_w C_1 (C_2+C_3)}}&\hbox{(10)}}$$ (10) provides four selections of signs which determine the version realized. In order to obtain version 1, signs have to be chosen according to {+,+,+,−} in the order given. Version 2 will be obtained by selecting {−,+,−,+} while version 3 results from selecting {+,−,+,−}.

SECTION III

## SIMULATION RESULTS

In order to determine the parameter values, the coefficients of (10) have to be compared to those of (9):
TeX Source
$$\eqalignno{Aω_0^2 &= {1\over R_i R_w} {1\over C_1 (C_2 + C_3)}, &\hbox{(11)} \cr{ω_0 \over Q_0} &= {1 \over C_2 + C_3} \left({1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}}\right) - {1\over C_1 R_q} - \cr&\quad - {C_2 \over C_1 (C_2 + C_3)} {1 \over R_w}, &\hbox{(12)} \cr2ω_c &= \pm {1\over R_c C_1} +pm {C_3 \over C_1(C_2 + C_3)} {1\over R_w}, &\hbox{(13)} \crω_c^2 - ω_0^2 &= {1\over C_1(C_2 + C_3)} \cr&\quad \times \left[{1 \over R_q} {1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}} - {1\over R_w R_{wo}}\right], &\hbox{(14)} \cr{ω_0 ω_c \over Q_0} &= {1\over C_1(C_2 + C_3)} \cr&\quad \times \left[\pm {1 \over R_c} {1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}} \mp {1\over R_{co} R_w}\right] &\hbox{(15)}}$$

As can be seen, there is a strong dependence between (12), (13) and (15). To eliminate this dependence, the product of the right hand sides of (12) and (13) has to be equated with two times the right hand side of (15). The auxiliary equation obtained this way can be solved for *R*_{co} for example. Using the result, a set of four equations remains. The values for filter gain *A*, equivalent low pass bandwidth *f*_{0}, center frequency *f*_{c} and filter quality *Q*_{0} can be substituted into these equations according to the specifications. Furthermore, values for the capacitance *C*_{1} and the input resistor *R*_{i} can be prescribed due to area or noise considerations, respectively. Values for the remaining filter coefficients can be calculated then. Specifying *A* = 10, ω_{0} = 2 π 1.25 *MHz*, ω_{c} = 2 π 2 *MHz* and *Q*_{0} = 1.31, the values given in Table I are calculated.

Table II contains the parameter values for the same specifications but for *Q*_{0} modified to *Q*_{0} = 0.54.

If two biquad polyphase filters with filter quality factors of *Q*_{0} = 1.31 and *Q*_{0} = 0.54 are combined, a Butterworth polyphase filter response is obtained. Component values according to Tables I and II have been selected to implement the three filter versions. In the Cadence simulations, models with finite gain bandwidth product (GBW) have been used to represent the opamps. A testbench according to Fig. 6 has been applied.

As the local oscillator inputs have been stimulated by the ports at a frequency of 1.57 *GHz*, the results taken at the output port are in the *GHz* range. The gain curve progressions of the filter versions have been examined under different conditions: First nearly ideal opamps (gain *A*_{0} = 10000 and bandwidth *GHz*) have been assumed. In order to model the opamps more realistic, gain has been reduced to *A*_{0} = 400 and the bandwidth has been limited to *f*_{p} = 4 *MHz*. Due to these limitations, transfer characteristics are impaired by noticeable peaking. The latter can be avoided by appropriate filter tuning. This is shown in Fig. 7 for version 2 of the Capacitive Crosscoupling PPF (CCPPF), at which *R*_{q} and *C*_{1} have been tuned.

Owing to the use of models, the noise contribution of the opamps is neglected in the noise figure results of Fig. 8. These results have been obtained using nearly ideal models but do not differ significantly at reduced GBW. A Tow-Thomas PPF with approximately the same sum capacitance per signal branch (34 *pF*) has been implemented and used as reference. As the noise transfer functions from the opamp inputs to the filter outputs may differ, the comparison has been repeated with noise sources inserted in series to each differential amplifier input. A noise power spectral density of 6.25 · 10^{− 18} *V*^{2}/*Hz* has been assumed for each source. Fig. 9 gives the results for opamp models with *A*_{0} = 400 and *f*_{p} = 4 *MHz* as well as filter tuning.

In this paper, a design procedure to develop polyphase filter structures with capacitive crosscouplings has been presented. The component values calculated by this method have been used to implement the filter structures in Cadence. Finite GBW models have been used to represent the operational amplifiers in simulations. The influence of opamp noise has been examined using noise sources in series to the amplifier inputs. For comparison, a Tow-Thomas polyphase filter using approximately the same sum capacitance has been implemented, too. Results show, that the implemented polyphase filters suffer from a higher noise figure compared to the Tow-Thomas PPF structure if similar sum capacitances per signal branch are applied. However, version two of the CCPPF already shows good performance and it has to be pointed out, that the proposed structures allow for the reduction in both, area and power consumption due to a reduced opamp count. Thus, occupying the same chip area, the implemented filter versions might approach the noise performance of the Tow-Thomas structure closer while the advantage of reduced power consumption would be retained.