Using Small Signal Analysis in Circuit Simulations
When most folks look at the equations for electronic devices, they usually just want to plug in values to determine the behavior of a circuit. If you came of age while spending time in math and science classes, you are probably conditioned to do this. However, there are other analytical techniques that you can use to speed up circuit analysis.
In some cases, namely when you are using nonlinear circuits or multiport networks, the behavior of a circuit may not be so obvious and the procedure to calculate everything by hand becomes intractable. This is where small signal analysis becomes critical for examining the behavior of a circuit using a SPICE-based simulator.
AC Analysis and Simulation Techniques
There are many AC analysis and simulation techniques that are used to examine the behavior of a linear or nonlinear circuit. Arguably, the most common is an AC frequency sweep, where the frequency of a driver is swept through a range of values at specified increments. The goal is to examine how the circuit responds to different frequencies.
With nonlinear and active circuit elements, and especially with arbitrary voltage/current sources, a frequency sweep analysis can become complicated. In particular, the Gauss-Jordan technique used in SPICE simulations can only be used for AC analysis of linear circuits. Therefore, examining the AC behavior of nonlinear circuits requires a different technique than what is used in DC circuits.
This is where small signal analysis comes in. In this analysis, the nonlinear circuit elements are approximated as linear circuit elements. This immediately makes the nonlinear elements amenable to the same Gauss-Jordan technique for circuit analysis. Note that this analysis only applies on top of a DC analysis; the behavior of nonlinear elements at a specific DC bias must be determined first, and the behavior under AC driving close to this DC value can be determined using a linear approximation.
Why Is DC Analysis Used in Small Signal Analysis?
Why is a DC value required at all? Isn’t this an AC analysis technique? Well, this must be done because the response of a nonlinear circuit between one voltage range (say, V1 to V2) is different than the response at a different voltage range (say, V2 to V3). The point is to determine the behavior of a circuit as the AC driving voltage changes within some specific range of values. This is why the circuit must be analyzed at a specific DC voltage before examining the behavior at an AC voltage.
The key to implementing small signal analysis is to first select a DC bias value and analyze the behavior of the circuit. The goal is to determine the transresistance (the inverse of the transconductance) for your nonlinear circuit elements at this particular DC bias. Once you know the transconductance for each nonlinear element, you can rebuild the circuit as a set of linear elements with resistance equal to the inverse of the transresistance.
Here, the linear approximation for a given circuit element can be found with a Taylor series approximation. Small signal analysis relies on this approximation around your selected DC bias value; let’s call this voltage U. The current output from any device can be written as a function of the input DC voltage V and the selected DC bias U using a Taylor series:
Taylor series for the output from a nonlinear device in small signal analysis
The transresistance is then just equal to the ratio of the input voltage to the current output:
Transresistance near the DC bias U
The advantages of this should be obvious: the circuit element acts like a resistor with the transresistance calculated above for any value of input voltage V, where V is close to U. This means you can now use this equivalent resistor in a typical AC analysis. In this analysis, you will actually be simulating an AC driving voltage on top of the DC driving voltage U. This allows you to see how the current responds to the AC voltage within a certain range of values.
With more complicated circuit elements, like transistors, amplifiers, vacuum tubes, or other devices, you may need to calculate a number of other resistances to properly create an equivalent linear circuit element for use in small signal analysis. There is a great deal of literature on this subject, and different models require different parameters to provide an accurate calculation.
Other Effects: Capacitance and Transimpedance
If you’re familiar with semiconductor devices, then you know that gate and junction capacitances exist in various devices, and these devices affect the switching behavior and the AC response. As such, you will need to determine the equivalent capacitance of the device by considering the total charge moving through the device at a particular current. You can then create an equivalent linear model for the circuit element by calculating the transimpedance, which again, is just the impedance value at a particular input voltage.
For semiconducting devices, you will need to consider the forward transit time (or reverse transit time, if driven in reverse bias) of charge carriers through different regions in the device to calculate the total charge due to the flowing current. Add to this the charge stored in all other capacitive regions in the device, and you have a function for the total charge. You can then calculate the equivalent capacitance of the device by taking a derivative with respect to the input voltage.
Once you know the equivalent capacitance at your particular DC bias, you can then define the transimpedance in the usual way using the capacitance value. Once you start examining AC changes around the DC bias, the transimpedance will affect the phase difference between the voltage and current in the circuit.
Small signal analysis requires a bit of work to determine the simulation parameters, but you can expedite your simulation with the right SPICE-based simulation package. The OrCAD PSpice Simulator from Cadence helps you find the DC bias point of a circuit quickly without you having to do any manual calculations. This helps set you on the path to perform AC analysis on top of the DC conditions found by PSpice. This unique package is adapted to complex PCB designs and interfaces directly with your schematic data.
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