In Part 3, we started to explore how to analyze the results of Monte Carlo analysis. In Part 4, we will consider the question, what is the relationship between process variation and the circuit’s performance variation? The tool for exploring the relationship process variation and circuit performance variation is mismatch analysis in the tool Virtuoso^{®} Variation Option (VVO).

Let’s start by looking at a simple example that shows the sources of offset voltage of a two-pole operational amplifier, see Figure 1.

Figure 1: Two Pole Operational Amplifier

Looking at the design, we would expect that mismatch of the p-channel input transistors are the primary source of offset voltage. First, let’s look at the Monte Carlo simulation results for the op-amp, see Figure 2.

Figure 2: Monte Carlo Analysis Results

The results show that the offset voltage is ~7.3mV. While Monte Carlo analysis tells us how much offset voltage there is, it does not tell us anything about the source of the offset voltage or how much improvement can be achieved. So, what are the sources of the offset voltage? After Monte Carlo analysis, we can plot the relationship between threshold voltage of input p-channel transistors, M17 and PM5, and the n-channel transistors in the first stage load current mirror. The scatter plots in Figure 3 show that there is no correlation between threshold voltage and the offset voltage of the operational amplifier since the correlation between offset voltage and the device threshold voltages is effectively 0.

Figure 3: Scatter Plots, Threshold Voltage versus Offset Voltage

Now let’s try using contribution analysis, see Figure 4.

Figure 4: Mismatch Analysis Results

Mismatch analysis shows the relationship between the threshold voltage and the offset voltage. The reasons that the scatter plot showed no correlation was because it looks for linear correlation. Mismatch analysis reports that the dependency is second order, the label shows R^2, The results show that most of the variation, 99.997%, can be explained by the threshold variation of the M17, PM5, NM4, and NM6. The results also show that ~70% of the offset voltage variation is due to the p-channel variation, the contribution from M17 is 34%, and the contribution from PM5 is 34%. The other source of offset voltage variation is the n-channel threshold voltage contribution of 30%.

Let’s use this information and see if we can improve the design. Since the p-channel contributes most of the offset voltage, we will try an experiment. We will increase the p-channel transistor area by 16x, length by 4x and width by 4x, keeping the W/L ratio constant. Increasing the device size should decrease the effect of p-channel mismatch by a factor of four.

Figure 5: Monte Carlo Analysis with 16x P-Channel

The effect of scaling the p-channel transistors on the offset voltage of the op-amp is to reduce the offset voltage from 7.2mV to 3.7mV. Doing some math, the p-channel offset contribution is ~6.4mV and the n-channel contribution is ~3.3mV. Verifying the offset voltage, the initial offset voltage is (6.4^{2}) + (3.3^{2}) = 7.2mV. After device sizing, the offset voltage is ((6.4/4)^{2}) + (3.3^{2}) = 3.7mV.

This example shows how mismatch analysis can be used to understand the effect of process variation on circuit performance. While we understand qualitatively that input transistors are the primary contributor to offset voltage, mismatch analysis provides us a tool for qualitative analysis of variation. In the next blog, we will apply mismatch analysis to additional circuits.