Exponentials simplify linear circuit analysis (part two)

The last installment of Engineering on Purpose presented harmonically-related series of sinusoids as one, though in no way the only, means of modeling periodic signals (Reference 1). Euler’s theorem,

provided a way to describe sinusoids in exponential notation, which was advantageous due to the extremely simple procedure of taking derivatives of exponentials.

In general form, this is:

Manipulating circuit equations to express quantities in the form est simplifies taking derivatives — a necessary step when analyzing circuits with reactive elements due to their native I/V relationships:

for capacitive and inductive reactances, respectively. Expressing circuit voltages and currents in terms of exponentials reduces to simple algebra the otherwise complex task of taking derivatives of arbitrarily complex functions.

If we assume exponential driving functions, that is if we let Vc(t) = Vo • est and IL(t) = Io • est, the reactances’ familiar, if cumbersome, I/V relationships reduce to

As we calculate with resistances, the impedance of a reactance is simply the ratio of voltage to current. However, in the case of reactive components, the impedance is a function of both reactance and signal frequency:

where ZC and ZL are the capacitive and inductive reactance, respectively.

As a quick check, note that these expressions predict the most fundamental attributes of capacitors and inductors operating in circuits: At DC, that is at s = 0, capacitors act as (and, in fact, are) open circuits with infinite theoretical impedance. The capacitor’s impedance is inversely proportional to both frequency and capacitance, as the expression ZC = 1/sC indicates. Conversely, an inductor’s impedance is directly proportional to frequency; the device theoretically models as a short circuit at DC.

Note also that practical devices differ from the ideal: All real devices exhibit parasitic terms in the forms of lead resistances and inductances and inter-node capacitances. In the presence of significantly larger bulk impedances, the spectral implications of parasitic terms are often ignorable. As bulk impedances fall or as signal frequencies increase, the affects of parasitics become more pronounced.

A SIMPLE BANDPASS FILTER

Constructing a simple audio interface demonstrates the usefulness of these concepts (Figure 1). The design goal is to AC couple an incoming signal to remove its DC bias, which derives from a mid-supply tap that powers a single-supply audio processor that provides our source signal. The circuit also needs to limit the interface’s output to a reasonable bandwidth that should be well below the gain-bandwidth product of the interface’s amplifier while providing about 6dB of gain in the mid-band. Specifically, the circuit’s bandwidth should cover the audio spectrum to about 15kHz for MP3 playback.

Before analyzing the circuit, take a moment to look qualitatively at its features. The input capacitor, C1, blocks DC. The feedback resistor, R2, keeps the amplifier operating in closed loop and, with no input signal, the DC output reflects the potential of the non-inverting input node, which here is tied to ground.

As the input frequency increases, so does the gain until it plateaus at its mid-band value, which is approximately –R2/R1. Note that the negative signs here and in the equations that follow denote that this is an inverting amplifier and, therefore, the output voltage is nominally 180° out of phase with respect to the input signal.

At some suitably high frequency, capacitor C2 dominates the feedback network and, because its impedance falls with increasing frequency, the circuit’s gain behaves in like fashion. The circuit’s input impedance, under the same conditions, is dominated by R1 because here the capacitor is in series with its associated resistance, not shunting it, as is the case in the feedback network.

Lumping the interface’s input and feedback impedances yields,

which gives the circuit’s transfer function — fundamentally that of an inverting operational amplifier stage — as

This expression contains several noteworthy attributes. Roots of the numerator, or zeros, mark points in the spectrum where the gain increases by a factor of 20dB/decade (about 6dB per octave). Roots of the denominator, or poles, mark points in the spectrum where the gain decreases by the same amount.

Another way of looking at the same expression is that the roots of the denominator give the circuit’s natural response — it’s behavior with zero input. A circuit can have a non-zero output with zero input, for example, when the input signal is set to zero while the circuit’s capacitors still store non-zero charges. Mathematically, however, our expression for H(s) must be infinite under those conditions. That doesn’t mean that the output is infinite, or even particularly large, but simply that it can be non-zero with a zero input. Similarly, the zeros — the roots of the numerator — give the conditions under which the circuit’s output can remain zero in the presence of a non-zero input.

An examination of the transfer function reveals the circuit’s behavior with more specificity than did the earlier qualitative description. The expression for H(s) indicates that we have one zero and two poles. The only way to set the numerator to zero is to set s0 to zero, which occurs at DC.

This is the aforementioned input signal condition under which a non-zero input produces a zero output.

By contrast, we can set the denominator to zero at two points: s1 = -1/R1C1 and s2 = -1/R2C2. From our qualitative description, we know that s1 corresponds to the circuit’s low-frequency band limit and the s2 sets the high-frequency limit.

If the two poles are mutually isolated, that is if they do not interact with one another, then we can set them directly from these simple expressions. An expansion of the denominator indicates that, in this circuit topology, the two poles do interact somewhat, but the ratio of the upper and lower frequency limits is large — roughly three orders of magnitude — which suggests that the interaction is not severe. Given this large pole separation, setting the component values directly from the roots of H(s)’s denominator should yield reasonable accuracy.

Note also that this analysis, from circuit topology to component selection, proceeded without explicit differentiation of any expression. This held true despite the presence of two complex impedances and their limited but non-zero interactions.

COMPLETING THE FIRST-PASS DESIGN

For the first pass design of this circuit, I chose to set the poles at or near 10Hz and 30kHz, well outside the required audio spectrum, because this simple topology provides soft corners (Figure 2): With poles at 10Hz and 30kHz, the -1dB gain-error points lie at about 30Hz and 10kHz and the -3dB gain-error point fall at about 20Hz and 18kHz.

To complete the task, note that the resolution in component values is far greater for resistors than for capacitors so it is often easier to solve the three simultaneous equations for the two pole locations and the gain by starting with a judicious choice of the largest capacitor. Note also that the problem of satisfying the three behavioral parameters is underspecified: We have three parametric targets and four variables.

In theory, therefore, there are an infinite number of solutions available to the three simultaneous equations. Narrow these down by considering additional criteria such as manufacturing-technology imposed limits on component sizes, noise (particularly in high gain circuits), and amplifier loading. Your design can also benefit from significant economies of scale simply by restricting yourself to component values your company is already using in large quantities for multiple products.

I selected C1 from available multi-layer ceramic capacitors with X7R dielectric — specifically a 680nF device. From the first pole location, R1 follows:

The nearest standard component to this calculated value in the EIA’s (Electronic Industry Alliance’s) 10 percent series is a 22KΩ resistor. To set the gain to the specified 6dB, R2 = 2R1. There are, however, no pairs in the EIA 10 percent resistor series in the ratio of 2:1. We can either switch to EIA 5 percent resistors, which offer three such pairs, or stay with the lower cost 10 percent parts by sacrificing mid-band gain accuracy. For this application, a mid-band gain error of 10 percent is easily tolerable, so selecting a 27KΩ resistor from the EIA 10 percent series yields acceptable results.

Finally, the value for C2 derives from the second pole location and the real nominal value (not the calculated ideal value) of R2:

The nearest standard value is 120 pF.

The plot in Figure 2 is based on the standard component values, not the calculated ideals. These component selections bring the poles in — that is they raise the lower frequency limit and lower the upper frequency limit — by about 6 percent with nominal parts. Add to this an error tolerance for individual instances of components to account for part-to-part deviations from the manufacturer’s nominal value. In the case of the resistors, recall, we chose from the 10 percent series but you can select from 5 percent or 1 percent devices if your application requires tighter tolerances on either the pole locations or the gain. Similarly, X7R ceramic capacitors are available in 5 percent, 10 percent, and 20 percent tolerances; C0G ceramic capacitors’ tolerances are commonly 2 percent, 5 percent, and 10 percent.

The performance of this topology, of course, is hardly worth the tighter tolerance parts but it is simple enough to demonstrate the concepts with which we’ve been working. It also is simple enough that you can simulate the circuit’s spectral behavior in an Excel spreadsheet, which allows you to easily try other design targets and component values.

To improve the design’s performance with regard to, for example, gain flatness and corner sharpness, consider replacing this simple topology with a slightly more complex one that provides a greater count of poles and zeros.

About the author

Joshua Israelsohn is a co-founder of JAS Technical Media where he manages the company’s technical-communication consultancy practice. You may find his contact information at www.jas-technicalmedia.com.

Click here for the illustrations:

Figure 1, Figure 2