Your task is to design, build, and test an amplifier for an ultrasonic receiver which can detect a signal from at least 2 meters away.

## 1. Description

Required decisions
• How much voltage gain is enough?

• Which op amp to use?

• How many amplifier stages?

Some of these questions can only be addressed with thoughtful experimentation.

• Begin with a 1 kΩ reference resistor.

• See this for instructions on using this feature: Using the Impedance Analyzer

• Estimate the resonance frequency of your device from these measurements and determine whether this is a 24 kHz or 40 kHz unit.

• Draw the schematic of a 1-stage amplifier, including the sensor.

• Use a voltage gain of 100 V/V.

• Select an opamp from the available list which can amplify this signal.

• Don’t forget to deal with DC offset voltage and input bias current issues in your circuit!

• Build this prototype amplifier.

• Measure the voltage gain at the signal frequency, it should match your prediction.

• Measure the signal output magnitude for at least one known distance from the transmitter. Use 0.5 to 1.0 meter distance for convenience.

• Compute the gain required for a reasonable output amplitude for a distance of 2 meters.

• Design a multiple-stage amplifier schematic which achieves this gain. Incorporate things you learned from the first amplifier’s testing.

• Build and measure!

### 1.1. The ultimate goal of this effort

For a given selected opamp, estimate the maximum distance achievable for at least a 100 mVp-p output amplitude.

### 2.1. Inverse square law

Figure 1. Inverse square law setup

Consider a finite amount of sound intensity (power per area, W/m2) leaving a speaker and a virtual sphere of diameter d centered at the speaker having a small patch on it which collects all energy pasing through the patch’s area. See Inverse square law setup for an illustration. A certain fraction of the total sound power emitted from the speaker will be absorbed by the patch depending on the proportion of the patch’s area to the sphere’s total surface area.

If the patch’s distance is doubled to 2d, the patch’s area must be increased by a factor of $\left(\frac{2d}{d}\right)^2$ (4x) in order to absorb the same power as before. Stated the other way, a (same area) patch at distance d2 will absorb the following power compared with a patch at distance d1:

$p_2 = p_1 \left(\frac{d_1}{d_2}\right)^2$

Since the voltage output of the microphone is proportional to the square root of the power, the relationship may also be expressed as:

$v_2 = v_1 \frac{d_1}{d_2}$

→ Doubling of distance reduces received power by 1/4 and a reduction in received voltage by a factor of 1/2.

### 2.2. 1-pole RC high-pass filter:

$H(f) = \dfrac{j\dfrac{f}{f_H}}{1 + j\dfrac{f}{f_H}} \quad\text{with}\quad f_H = \dfrac{1}{2\pi R C}$

### 2.3. 1-pole RC low-pass filter:

$H(f) = \dfrac{1}{1 + j\dfrac{f}{f_H}} \quad\text{with}\quad f_H = \dfrac{1}{2\pi R C}$

### 2.4. DC errors

If you make the following assumptions:

• The open-loop gain Av0 is large enough, which means $\gg \left(1 + \frac{R_f}{R_1}\right)$

• The opamp’s open-loop output impedance Zout is low enough, or much less than the impedance seen by the output node (for this lab = 50,047 Ω).

The total non-inverting opamp output from the input signal and DC errors VOS and (IB, IOS) ←→ (IB+, IB-) is:

$v_{OUT} = v_{in}\underbrace{\left(1 + \dfrac{R_f}{R_1}\right)}_\text{gain} + \underbrace{\biggl(V_{OS} + I_{B+} R_{eq+} - I_{B-} R_{eq-}\biggr)\left(1 + \dfrac{R_f}{R_1}\right)}_\text{offset}$
• $R_{eq+}$ and $R_{eq-}$ are the impedances seen by the + and - input terminals of the opamp.

### 2.7. Figures

Figure 2. Ultrasonic sensor equivalent circuit
Figure 3. Example variable gain non-inverting amplifier