Analyzing circuit sensitivity for analog circuit design
All circuits, analog or digital, have characteristics dependent on the values of their component parts. Invariably, these parts are non-ideal in many ways. Even their most fundamental characteristics, for example a resistor's resistance, will be at least slightly different from the specified value. These characteristics also can vary over time and environmental conditions.
The wise circuit designer learns how to analyze the sensitivities of circuits' critical characteristics to the variations in the components. With this knowledge, the designer can better decide how to balance any trade offs between performance and cost. Deeper use of sensitivity analysis can lead designers to modify circuit topologies, or make completely different choices in order to optimize these trade offs.
Generally, circuit sensitivity analysis is not part of the bachelors curriculum in most universities' electrical engineering programs. Fortunately, this important topic is easy to learn and use and is covered in circuit design books, especially those focusing on filter design.
Circuit sensitivity defined
A simple, if imprecise, definition of circuit sensitivity is how much a particular circuit characteristic changes as a particular component value varies. This can be any circuit characteristic: amplifier gain; bus receiver input impedance; an RF port's voltage standing wave ratio; or a digital gate threshold voltage. In this article, we use analog filters to explore sensitivity analysis as this information is often critical to good filter design. We will see that it is fairly easy to extend sensitivity analysis to other types of circuits.
Equation 1 is the mathematical definition of circuit sensitivity:
Where S is the sensitivity, X is the changing component, and Y is the circuit characteristic we wish to evaluate as X is varied.
The middle part of this equation makes intuitive sense. It is the percentage that the dependent variable changes, Δy/y, relative to the percentage that the independent variable changes, Δx/x. Taking the limit as the change in x goes to zero evaluates this ratio for minute variations.
This equation is so general that it can be used to evaluate the variation of any circuit parameter, relative to a change in any circuit component value.
Endnotes 1, 2, and 3 have detailed treatments of sensitivity and derive many of the equations we will use.
Simple circuit example
Consider the simple circuit shown in Figure 1--a voltage divider.
Equation 2 is the DC transfer function:
Use Equation 1 to calculate the sensitivity of the DC transfer function to R1 and R2:
What do these equations mean? Recall that sensitivity is the percentage that the dependent variable, in this case the DC transfer function, changes relative to the independent variable, R1 for Equation 3 and R2 for Equation 4.
These sensitivity equations are identical except for the sign. In Equation 3, the sensitivity to R1 is negative. As the negative sign implies, when R1 increases, the transfer function decreases. When R2 increases, the transfer function also increases, which is expected since Equation 4 (the sensitivity to R2) is positive.
When R1 is substantially larger than R2, the equations reduce to –R1/R1 = –1 and R1/R1 = 1. This implies that the transfer function should change by very nearly 1% for every 1% variation in either resistor under these conditions.
Take the case where R1 = 1000*R2. Here the transfer function is 1/1001 = 999e–3. If R2 is doubled, the transfer becomes 2/1002 = 1.996e–3, which is 1.998 times the earlier value, nearly double.
Similarly, if R1 is doubled, the transfer function decreases by nearly a factor of two. Doubling R1 results in a transfer function of 1/2001 = 0.4998e–3, which is 0.498 times the earlier value--nearly a factor of two less.
The other extreme, when R2 is substantially larger than R1, results in the sensitivity equations reducing to zero for R1 = 0 and R2 = ∞. For values that can be realized, the sensitivities will be near zero. Thus, the transfer function should change very little as either resistor is varied.
Where R2 = 1000*R1, the transfer function is 1000/1001 = 0.999. If R2 is doubled, this becomes 2000/2001 = 0.9995, only a 0.05% change in the transfer function for a 100% change in the component value.
Similarly, if we double R1 instead, the transfer function becomes 1000/1002 = 0.998, only a 0.1% change in the transfer function for a 100% change in the component value.
If R1 = R2, the transfer function is 0.5 and the sensitivities are –0.5 and 0.5. You would expect the transfer function to change 0.5% for every 1% change in either resistor. Let's increase R2 by 1%. Now the transfer function becomes 1/2.01 = 0.4975, which is a 0.5% reduction. Similarly increase R1 by 1% results in the transfer function being 1.01/2.01 = 5.025, or an increase of 0.5%.
This is about as far as we can go using simple circuits with resistors. Now let's include reactive elements, inductors and capacitors. This will create AC transfer functions that vary with frequency, such as filters.