Response to small perturbations of circuit's excitation

Suppose we are interested in how much a circuit's operating point will change if we slightly change the circuit's excitation. The system of nonlinear equations describing the circut can be written as

Function f is a vector valued function. Every component of its return value corresponds to the LHS value of one nonlinear equation from the circuit's system of equations. Its argument (x) is a vector whose components are the unknowns we are trying to compute (i.e. nodal voltages and selected branch currents) that represent the solution of the circuit's equations. Function f can be split in two parts. One part (y) corresponds to the contributions of the independent sources. The sign of the components of y is chosen in such manner that y is equal to the RHS contributions of these independent sources to the system of equations of a linear circuit. Vector y is a constant vector. The remaining part of f will be denoted by g which is a vector-valued function of x.

The system of equations can now be written as

A small perturbation of the circuit's excitations (i.e. independent sources represented by y) results in a small perturbation of the circuit's solution x. Mathematically we can formulate this as

By taking the first two terms of the Taylor series for g and neglecting the rest we can write the system of equations as

where G is the Jacobian (i.e. the matrix of first derivatives) of g. Because g and f differ only by a constant term it is also equal to the Jacobian of f (i. e. the LHS part of the circuit's equations). The Jacobian itself depends only on the solution of the unperturbed circuit (x). Note that we can neglect higher order terms in the Taylor series because we assume the perturbation is small. Because the initial circuit equation (the one without perturbations) must still be satisfied the first LHS term and the first RHS term cancel each other out and we are left with

We obtained a linear system of equations which corresponds to some linear circuit. By solving it we can express the perturbation of the circuit's operating point with the perturbation of the circuit's excitation. The process of formulating this system of linear equations is also referred to as circuit linearization.

Example: perturbation analysis of a MOSFET-based amplifier

Fig. 1: A MOSFET-based amplifier (top) and the same amplifier with perturbed excitations (bottom).

For instance, take a simple circuit with an NMOS transistor depicted in Fig. 1. After solving the corresponding system of equations we obtain the operating point of the circuit specified by v₁, v₂, v₃, i_GG, and i_DD (Fig. 1, top). The operating point depends on the values of the two independent voltage sources U_GG and U_DD. Now suppose we slightly perturb these two sources by ΔU_GG and ΔU_DD, respectively. This causes the operating point of the circuit to slightly change by Δv₁, Δv₂, Δv₃, Δi_GG, and Δi_DD, respectively (Fig. 1, bottom). Let us write down the nonlinear circuit equations and linearize them to obtain the equations for computing the perturbation of the operating point.

The circuit has 4 nodes and 2 voltage sources. The list of unknowns includes the node potentials of nodes 1, 2, and 3, and the currents flowing into the two independent voltage sources. We assume the MOSFET transistor is operating in the saturation region (u_GS≥0 and u_DS≥u_GS-U_T) the transistor currents can be expressed as

The three KCL equations and the two constitutive relations of the independent voltage sources are

After solving this system of equations we obtain the operating point of the circuit. Next, we perturb the two independent voltage sources. The operating point of the circuit changes to v₁+Δv₁, v₂+Δv₂, v₃+Δv₃, i_GG+Δi_GG, and i_DD+Δi_DD. We can compute the perturbations of the operating point (Δv₁, Δv₂, Δv₃, Δi_GG, and Δi_DD) from the linearized system of equations

Where g₂₁ and g₂₂ are expressed as (see previous lecture)

The matrix of coefficients of this linearized system of equations is in fact the circuit's Jacobian matrix computed at the circuit's operating point. The RHS comprises the contributions of the perturbed independent voltage and current sources.

Fig. 2: Small-signal model of the circuit in Fig. 1 for computing the operating point perturbations. The circuit was reconstructed from the linearized system of equations.

From the obtained linear system of equations we can reconstruct a linear circuit depicted in Fig. 2. This circuit is also referred to as the small-signal model of the circuit. Conductance g₂₂ and transconductance g₂₁ represent the small-signal model of the NMOS transistor. If we compare it to the linearized model from the previous lecture (used for solving a nonlinear circuit's equations by means of the NR algorithm) we can see that the small-signal model lacks the independent curent source.

We can see that the small-signal model of a circuit is similar to the original nonlinear circuit. All linear elements remain unchanged. The values of the independent sources are replaced with their respective perturbations. Nodal voltages and branch currents are also replaced with their respective perturbations. Finally, nonlinear elements are replaced with their lienarized models which are identical to those we constructed in the previous lecture, with the exception that independent current sources are omitted in the small-signal model.

If an independent voltage source is not perturbed (i.e. ΔU=0) that source acts like a short circuit in the small-signal model of a circuit. On the other hand, unperturbed independent current sources (i.e. ΔI=0) behave as open circuits in the small-signal model of a circuit.

Small-signal models of nonlinear elements

We can build small-signal models of semiconductor devices in a manner similar to the one we used in the previou lecture. Suppose we have a semiconductor diode with the following constitutive relation

A perturbation of the diode voltage results in a perturbation of the diode current. If the perturbation is small, we can approximate the diode current with the first two terms of the Taylor series expansion computed at the circuit's operating point.

After simplification we get

where

The simple relation between Δu_D and Δi_D corresponds to a resistor (Fig. 3, bottom left). When constructing the small-signal model of a circuit that contains a diode the diode is replaced by a resistor with conductance g_D computed at the circuit's operating point. In a similar manner we can derive the small-signal models of other semiconductor devices (Fig. 3).

Fig. 3: Small-signal models (bottom) of nonlinear elements (top) for diode (left), NMOS transistor (center), and NPN bipolar transistor (right).

Parameters of the small-signal model of an NMOS transistor are computed at the operating point defined by U_GS and U_DS as

Parameters of the small-signal model of a NPN bipolar transistor are computed at operating point defined by U_BE and U_CE as

The transfer function, the input impedance, and the output impedance of a nonlinear circuit

Consider the circuit in Fig. 1. Independent voltage source U_GG can be considered as the input signal. As its value changes the output signal u₂ also changes (Fig. 4). The dependence of u₂ on U_GG is generally nonlinear. For the circuit in Fig. 4 this dependency is depicted in Fig. 5 with a thin line.

Fig. 4: Input and output of a MOSFET-based amplifier.

The operating point of the circuit corresponds to the point marked with a circle in the u₂(u₁) characteristic. If we linearize the characteristic at this point (Fig. 5) and compute the slope of the tangent line the obtained slope is also referred to as the transfer function (A). The transfer function of a nonlinear circuit depends on the circuit's operating point.

Fig. 5: Dependence of output voltage on the input voltage for the circuit in Fig. 4. The thick line represents the linearization of this dependence at the circuit's operating point where the input and output voltage are u₁=U_GG and U₂, respectively. The slope of the tangent is also reffered to as the transfer function at the circuit's operating point.

We can compute A numerically if we analyze the circuit's response to a small perturbation of U_GG while U_DD is kept unperturbed. The slope can then be computed as the quotient A=Δu₂/Δu₁=Δv₂/ΔU_GG. If we set the ΔU_GG perturbation to 1 (with all of the remaining independent source perturbations set to 0) the transfer function is A=Δv₂/ΔU_GG=Δv₂. The system of equations we need to solve is therefore

In our example the obtained value of A would be negative. The transfer function of an amplifier is also referred to as the amplifier's gain. As the input voltage changes so does the input current i₁. The dependence of the input current on the input voltage is generally nonlinear. In our particular example this dependence is linear. Fig. 6 depicts a possible dependence of a circuit's input current on the input voltage.

Fig. 6: A nonlinear dependence of input current on the input voltage. The thick line represents the linearization of this dependence at the circuit's operating point where the input voltage is u₁=U₁. The inverse slope of the tangent is the input impedance at the circuit's operating point. Note that the input current dependence of the circuit in Fig. 4 is much simpler, i.e. linear.

The slope of the tangent line at the circuit's operating point is the inverse of the input impedance. To obtain this slope for the circuit in Fig. 4 we simply set the input voltage perturbation ΔU_GG to 1 (with all of the remaining independent source perturbations set to 0) and solve the following system of equations

The slope of the tangent line is equal to Δi₁/Δu₁=-Δi_GG/ΔU_GG=-Δi_GG. The input impedance is therefore Z_in=-1/Δi_GG. Note that for the circuit in Fig. 4 the input impedance can be expressed as Z_in=R_G and does not depend on parameters g₂₁ and g₂₂ which are operating point dependent. Thus the slope of the tangent and the value of the input impedance do not depend on the operating point. Consequently the dependence of i₁ on u₁ is linear.

Fig. 7: The circuit in Fig. 4 with an independent current source injecting current at its output.

Now, consider we are interested in the circuit's response to the changes of the current injected at the circuit's output by an independent current source I₂ (Fig. 7). The nonlinear system of equations for computing the operating point of the circuit in Fig. 7 is

When I₂=0 the circuit in Fig. 7 has the same solution (operating point) as the circuit in Fig 4 (i.e. an independent current source with zero current behaves as an open circuit). As the injected current I₂ changes the output voltage u₂=v₂ also changes. This dependence is depicted in Fig. 8.

Fig. 8: The nonlinear dependence of output voltage u₂ on the injected current I₂ at the output. The thick line represents the linearization of this dependence at the circuit's operating point where I₂=0. The slope of the tangent is the output impedance at the circuit's operating point.

The slope of the tangent line at the circuit's operating point is the output impedance. It can be computed by setting ΔI₂ to 1 (with all of the remaining independent source perturbations set to 0) and solving the system of equations

DC small-signal analysis in SPICE OPUS

In SPICE OPUS the DC small-signal analysis is referred to as TF analysis. One has to specify two things at its invocation: the output signal (e.g. node potential, voltage between two nodes, or current of a voltage source) and the independent source (voltage or current) that provides the small-signal perturbation. The analysis then computes three things. The small-signal gain (also referred to as the transfer function) whose type depends on the choice of the input and the output. There are four possibilities: voltage gain, current gain, transconductance, and transimpedance. The analysis also computes input impedance at the nodes of the circuit where the independent source providing the excitation is located and the output impedance at the circuit's output. The latter is computed correctly only if the output is a node potential or a voltage between two nodes.

Most of the time in TF analysis is spent for computing the circuit's operating point. Once it is computed the Jacobian matrix describing the small-signal model circuit and its LU decomposition are already available. The simulator only constructs a right-hand side vector according to the specified input excitation and performs a forward and backward substitution to obtain the solution. From this solution the transfer function and the input impedance are computed. For computing the output impedance the simulator constructs the appropriate right-hand side vector and performs another forward+backward substitution. The output impedance is computed from the obtained result.