Fundamental of Nonlinearity Analysis

In this chapter it is investigated how weakly nonlinear behavior of analog integrated circuit can be computed. As we use simulation tool to get the response of distortion, simulation results can not present information from which designers can derive which circuit parameters or circuit elements should be modified to meet the required specifications. Volterra series can help us get such information [17].

LO+ LO-RF

RL

VDD

RL

IF+

IF-M2

M1

RF

Fig. 4.3. Single-balanced bulk-driven mixer.

Volterra series describe the output of a nonlinear system as the sum of the response of a first-order operator, a second-order one, a third-order one and so on.

Every operator is described either in the time domain or in the frequency domain with a kind of transfer function, called Volterra kernel. Actually, Volterra series describes a nonlinear system in a way which is equivalent to the way Taylor series approximate an analytic function. The higher input amplitude, the more terms of that series need to be taken into account in order to describe the system behavior properly. For very high amplitude, the series diverges, just as Taylor series. The most difference from Taylor series and Volterra series is ability to deal with a circuit with memory. Volterra series can analyze the memory circuit where the inductors and capacitors play a role because it retains phase information, but Tailor series can not. Taylor series can deal with a memory-less circuit only.

For a one-port system, such as low noise amplifier (LNA), Volterra kernel can be computed to get the output response for any input signal. However, for multiple-input circuits, Volterra kernel becomes tensors. Calculations with tensors are quite cumbersome in a multiple-input system. It is called a direct calculation method (DCM), variant Volterra series approach, which directly calculates the required response repeatedly so that it does not make use of tensors. Briefly, the DCM calculates the nonlinear response at specific frequency directly and if you want to

obtain another response at other frequency, you need to calculate repeatedly. Using DCM or Volterra kernels will get the same nonlinear response. For mixer circuits, such as Gilbert mixer and the bulk-driven mixer, they are a two-port system with two inputs of RF signal and LO signal. Therefore, the DCM is more appropriate for the nonlinearity analysis. Consequently, we focus on the DCM in this section.

The DCM is explained with the circuit shown in Fig. 4.4. This is a half circuit of the proposed bulk-driven mixer in this chapter later. Fig. 4.5 shows its equivalent circuit for nonlinear analysis.

The circuits is excited at two different input ports by two sinusoidal signals, vRF(t) and vLO(t) at frequencies wRF and wLO, respectively.

) Re(

)

( _RF ^{jw t}

RF t V e ^RF

v = (4-1) )

Re(

)

( _LO ^{jw t}

LO t V e ^LO

v = (4-2) Under steady-state conditions, every node voltage vx(t) consists of a sum of harmonics and intermodulation of RF signal and LO signal as indicated in (4-3). RF port has two RF signals at frequencies wRF1 and wRF2, respectively, with little frequency offset for a two test in an IIP3 calculation.

Z

RF LO

Fig. 4.4. The half circuit of the bulk-driven mixer for the nonlinearity analysis.

)

where Vx,m,n,k denotes the voltage at the node x at the frequency of

LO RF

RF nw kw

mw ₁ + ₂ + . The goal of this section is to explain the method to compute the complex phasor Vx,m,n,k.

The drain current is a function of VGS, VDS, and VBS where source node is the voltage reference so that it can be expanded by a three dimensional power series as shown in equation of (4-4). This power series will take nonlinearities of gm , gmb , and

go into account.

Fig. 4.5. The equivalent circuit for the nonlinear analysis.

where the nonlinear parameters are defined in the following equations:

(i) First-Order Response

Fist-order response is a linear response which can be obtained by Kirchoff’s laws.

We use a technique of superposition to get the individual linear response to the inputs of vRF1, vRF2, and vLO, that is, as we calculate the response to the vRF1 input, we set vRF2, and vLO to zero. Take the first-order response to the vRF1 input for instance. The ac equivalent circuit is then in Fig. 4.6. We can get three equations at the nodes marked ○¹ ,○² , and ○³ by using Kirchoff’s current and voltage laws, After making an arrangement into matrix, we can write down a matrix equation:

⎥⎥ We can the above equation in a general form

0

Y(w) is a transconductance matrix and U1,0,0 is a matrix of the voltage response.

IN1,1,0,0 is a vector whose only nonzero components are terms in the network equations that contain VRF1. Then the linear response can be gained by the inverse operation.

Fig. 4.6. The equivalent circuit for the first-order response.

can be found by the following equations:

(ii) Second-Order Nonlinear Response

With the information from the previous step the second-order response can be computed. Again the same linearized network must be solved as the one that has been used to compute the first-order response. However, the inputs are different now:

instead of the external excitation, so-called nonlinear current source of order two must be applied. There is one nonlinear current source for each nonlinearity in the circuit and the source is applied in parallel with the linear current source. This nonlinear current source of order two can be obtained by the following method. We simplify our question by considering only one nonlinear current source, related to K2gm nonlinear parameter. The equivalent circuit for the analysis of the second-order response is shown in Fig. 4.7. We write the nodal equation at node 2 and get the following equation.

And from the equation (4-3), v1 can be written as:

Because we consider the second-order nonlinear response now, we only see the response at the frequency which is the sum of frequencies of wRF1 and wRF2. The responses at the frequencies of the combination of wRF1 and wRF2 frequencies can be found by the same method. After substituting (4-12) into (4-11) we can get the following equation:

t

Together with nodal equation (4-5) and (4-6), we can arrange the three equations in

Z

Fig. 4.7. The equivalent circuit for the second-order nonlinear response.

⎟⎟

Hence, we get the second-order nonlinear current source related to K2gm. The contribution of the nonlinear current source related to K2gm to the second-order nonlinear response can be calculated by the (4-13). Other nonlinear current sources can be gotten by the same method. Therefore, the matrix equation related to the second-order response is summarized in the following equation.

⎟⎟

where iNL2 is the total second-order nonlinear current sources. It is written as:

& ....

We summarize the nonlinear current source at Table 4.1 for the responses at the frequencies of 2w1 and |w1± w2| where w1 and w2 can be wRF1, wRF2, or wLO. The interpretation of the values of the nonlinear current sources from Table 4.1 is as follows: the second-order nonlinearity of every one-dimensional conductance or capacitance combines the first-order response of its controlling voltage due to VRF1

only, with the first-order response of its controlling voltage due to VRF2 only, to a

second-order signal. This signal, at frequency |w1± w2|, then propagates through the rest of the circuit. When considering this propagation, only the linearized elements need to be taken into account. Indeed, any interaction of the second-order signal with another one yields a response of order higher than two.

0

(iii) Third-Order Nonlinear Response

We can obtain the third-order nonlinear response by the same procedure. The third-order nonlinear current sources are summarized in Table 4.2 and 4.3. The matrices for the third-order nonlinear response are as followed:

⎟⎟ Type of nonlinearity Nonlinear current source for

response at |w1±w2|.

Nonlinear current source for response at 2w1

(trans)conductance (one cross-terms)

0

Table 4.2. Third-order nonlinear current for response at 2w1±^w2. Type of

nonlinearity

Nonlinear current source for response at |2w1±^w2|.

(trans)conductance (one cross-terms)

0 (only cross terms)

0

Table 4.3. Third-order nonlinear current for response at 3w2

Type of nonlinearity Nonlinear current source for response at 3w2.

(trans)conductance (one cross-terms)

4 ] (only cross terms)

0

(iv) Fourth-Order Response

In general, nonlinear current sources are calculated up to third-order because it is enough to obtain the desired information. These nonlinear currents sources up to third-order can be found at the textbook. However, for the bulk-driven mixer circuit, we should make efforts to obtain the fourth-order nonlinear current sources. The details why we need to calculate up to the fourth-order will be discussed in the next section. The fourth-order nonlinear current sources can not be found in the text. We can use a mathematic tool to help us find out the fourth-order nonlinear current source.

These fourth-order nonlinear current sources are summarized in Table 4.4 and 4.5.

The matrix for the IM3 response is as followed.

⎟⎟

(v) Calculation

Before we start to calculate the nonlinear response, those nonlinear parameters need to be known. We extract these parameters from the DC simulation of the circuit shown in Fig. 4.4 using Angilent ADS. With the above equations and the extracted parameters we can calculate the each order nonlinear response directly.

Table 4.4. Fourth-order nonlinear current for response at |2w1+w2+ w3| (one cross-terms)

1 (only cross terms)

1

0

nonlinearity

Nonlinear current source for response at 3w2±w2|

(trans)conductance ₃

0

Capacitor

4 ]

Two-dimensional conductance (one cross-terms)

] conductance (only cross terms)

]