How can they believe in the one of whom they have not heard? And how can they hear if the microphone amplifier’s gain is too low?

— Romans 10:14b (modified)

1. Introduction

This activity is to design an amplifier for a microphone which meets performance specifications. Doing so requires accounting for the major non-ideal DC and AC characteristics of opamps.

Documenting a design, including capturing the process used for making design decisions (numbers!), is what separates a professional engineering process from an interesting school activity.

2. Background

The referenced Figures are in this PDF

2.1. Condenser type microphones

A condenser microphone operates on the principle of a variable capacitor[1] where one plate is stationary and the other, the diaphram, is allowed to move towards or away from the fixed plate (dz) in response to air pressure variations. Diaphram movement changes the capacitance of the structure. A constant charge is applied to this capacititor, usually by a DC voltage source through a very large (10 MΩ to 1+ GΩ) resistor. Therefore the potential difference (a.k.a. voltage) across this capacitor varies in response to air pressure changes by the following chain of reasoning:

\[\begin{align} C_m(t) =& \dfrac{\epsilon_0 A}{\left(d_{\mathrm{sep}} - k \cdot p_{\mathrm{snd}}(t)\right)}\\ Q_{\mathrm{fixed}} =& C_m(t) V_m(t) \\ V_m(t) =& \dfrac{Q_{\mathrm{fixed}}}{C_m(t)} = \dfrac{Q_{\mathrm{fixed}} \left(d_{\mathrm{sep}} - k \cdot p_{\mathrm{snd}}(t)\right)}{\epsilon_0 A} \end{align}\]

Since we don’t indend to measure atmospheric pressure but are only interested in the variations in response to changes in air pressure, the audio-related portion of this voltage is:

\[\begin{align} V_{\mathrm{mic}}(t) =& -\dfrac{Q_{\mathrm{fixed}} \cdot k}{\epsilon_0 A} \cdot p_{\mathrm{snd}}(t)\\ V_{\mathrm{mic}}(t) =& -K_{\mathrm{mic}} \cdot p_{\mathrm{snd}}(t) \end{align}\]

Where \(K_{\mathrm{mic}}\) collects all of the geometry and material parameters.

2.2. Electret condenser microphones

Electret condenser microphones (ECM) do not require a DC polarization voltage to maintian a constant charge. The moving diaphram is made of a dielectric material (plastic) with charges embedded in it. Such elements remain effectively permenantly charged.

The voltage changes \(V_{\mathrm{mic}}(t)\) resulting from sound pressure variations may be modelled as a Thevenin source with an output impedance equal to the capacitance of the microphone capsule, shown in Figure 1. Because this capacitance is quite small, on the order of N pF, a very large load resistance is required to ensure that the resulting high-pass filter’s corner frequency reaches low enough for audio purposes. For example, a 1 pF capsule capacitance requires at least a 1.6 GΩ load resistance to give a -3 dB frequency of 100 Hz, which is barely adequate for high-quality speech.

Due to this severe loading constraint, most electret microphones include a JFET amplifier mounted inside the capsule to buffer the signal, typically a common-source configuration. Figure 2 shows the standard configuration of a common-source amplifier. Also shown in the figure is the use of an external resistor (typically 500 Ω to 5 kΩ) to provide drain current (around 0.5 mA) for the transistor and an impedance across which to develop the small-signal output voltage (as \(v_{out} = g_m R_{\mathrm{bias}} v_{\mathrm{mic}}\)). Electret condenser microphone units increasingly integrate more circuitry into the capsule enclosure for enhanced performance. See Toward More-Compact Digital Microphones from an Analog Dialog article for a good perspective on these changes.

The ECM’s output is a superposition of DC voltage and AC signal due to this biasing arrangement. The DC component may be several volts, while the signal is only a few millivolts in magnitude. A coupling capacitor is placed in series in many circuits to block this DC bias value and pass only the small microphone signal, as shown in Figure 2. Figure 3 shows the Thevenin-equivalent circuit of a biased electret microphone signal.

The output resistance of the circuit is the parallel combination of the JFET’s ro and Rbias, and is approximately equal to Rbias alone because the tranistor’s small-signal ro is usually much larger. When finding the frequency response of a circuit using this structure, this Thevenin-equivalent output impedance must be taken into account.

3. Specifications

Your system design should meet the following specifications

  • |VOUT| (DC) ≤ 10 mV.

  • Gain of ≥ 100 V/V (40 dB).

  • Upper -3 dB frequency ≥20 kHz.

  • Lower -3 dB frequency ≤100 Hz.

  • Gain variation between the -3 dB points of ≤|±3 dB|.

Your design should be guaranteed to meet these specifications even with worst-case parameters according to the datasheet of your chosen opamp. Use the list of choices from the day30_opamps_table (hw13) spreadsheet. Datasheets for these devices are in the usual GDrive folder.

The datasheet numbers you extracted for the hw13 table were typical values for each part — this design uses different numbers!

You need to guarantee that every manufactured unit of this design will meet the specifications. Therefore, you need to use the worst parameters that are guaranteed by the opamp manufacturer, not the statistically average (typical) values.

→ Also assume you are using ±5% tolerance resistors and ±10% tolerance capacitors.

For the purposes of measurement, replace the microphone and Rbias resistor with a signal generator in series with the Rbias, leaving the coupling capacitor in place. The output DC offset voltage is measured with the signal generator turned off. Gain is measured at an input frequency of 2 kHz.

It will be necessary to deal with both the amplifier’s input offset voltage and input offset current as sources of offset. Compensating for the opamp’s input bias current IB may also be necessary. See Figures 4, 5, 6, and 7 for partial circuit ideas on how to both achieve this large gain in multiple stages and to compensate for opamp offset.

4. Design documentation

As part of verifying your design, also come up with a procedure for measuring your circuit’s performance on these 3 metrics.

In the documentation for your decision, include:

  • A list of all the opamps which meet the design’s requirements.

  • Identify which of that list are the “better” choices. As part of this decision, you must clearly say what is “better” and why that ranking is reasonable for making decisions with a product design.

  • Calculations showing that the design meets the specification even the resistor and capacitor values are at the edge of their allowable tolerance.

  • Simulation results that show the frequency response and DC offset are within bounds. This requires several simulations in order to demonstrate the two bounds.

  • Describe which specific situations are “worst case” where the measurements are the closest to the specification boundaries.

5. Optional

Replace the signal generator with the microphone as shown in Figure 4. Monitor the amplifier output on an oscilloscope on a time scale of about 5 ms/div and speak directly into the capsule to observe the time-domain waveforms of speech. Also, use the FFT feature and display the spectrum of your speech (span/center to 5 kHz / 2.5 kHz, adjust the time scale to trade-off spectral resolution against update rate).

1. “Condenser” is an old name for a capacitor, hence the microphone name.