Introduction
This circuit is based on the article Percussion Instrument Synthesizer by Forrest M. Mims, Popular Electronics, March, 1976, pgs. 90 - 102. which describes an audio amplifier with a twin-T notch filter in the feedback loop,
The R-C network gives a sharp peak at a single audio frequency at which feedback occurs. Disturbing the touch plate introduces a transient which starts a damped oscillation "ringing". I've added a Hall effect sensor on the input instead of a touch plate to inject the transient, and an audio power amplifer on the output.
Analyzing the RC Network
Let's analyze the network in isolation and add the amplifier and input later.
Kirchhoff's Laws
Kirchoff's Laws can be
derived from Maxwell's equations
for discrete circuit elements.
Kirchoff's current law assumes that charge is constant in time at a given node:
Recall the convention that currents flowing into the node are taken as positive.
First an aside, recall the Laplace transform of a function
Recall that the voltage across a resistor is
Now we apply Kirchhoff's voltage law on the meshes 1-4.
But first we apply the Laplace transform to all voltage terms in the loop using
linearity and assuming initial currents and
voltages are zero.
For simplicity of notation, let's use
Solving the Equations
Now we'll solve the equations by substitution. Mesh equation 1 gives
Analyzing the Amplifier Plus RC Network
The Transfer Function H(s)
Returning to our complete circuit, we draw it using the twin-T notch
filter block just analyzed, an ideal operational amplifer, and a voltage source.
The ideal OpAmp has
By Kirchhoff's voltage law, recalling we are using the Laplace transforms of
voltage and current here, the mesh equation for the input is
Ringing the Bell
To ring the bell in operation, one applies a sharp impulse to the input.
To approximate this, set
Time Domain Solution
Inverting the Laplace Transform
From complex analysis, for
Cubic Equation Solution
From
classical algebra,
the roots of the cubic
If
If
Conditions for Damped Oscillations
Since our cubic has real coefficients, if
Resonant Frequency at the Edge of Oscillation
Recall Newton's formulas for the coefficients in terms of the roots,
Then multiplying out the roots gives
On the cusp of steady oscillation,
Combining the equations gives
The frequency of the oscillating solution is
For this design, pick the capacitances to satisfy
Circuit Schematic
NOTES:
- The dual 100K resistors connected to pin 3 of LM308 provide a floating ground for this op amp.
- You'll get excessive current drain unless the 330pF capacitor is connected to pin 2 of LM380N. The chip will actually heat up! Parasitic oscillation in the ultrasound range, perhaps?
- The 150K resistor at the output of the LM308 isolates the two amplifiers. Again, hissing and high current drain happened when the two amplifiers were directly connected.
- High current drain also occurred when pin 12 of the LM380N was grounded. I don't know the reason.
- Quiescent power drain is about 7mA
Operation
- Turn the coarse and fine duration adjustment potentiometers clockwise as far as they will go. There will be a continuous tone.
- Adjust the pitch.
- Adjust the loudness. Not too loud or the mini speaker will saturate and the tone will sound harsh.
- Rotate the coarse adjust clockwise until oscillation stops, the clockwise until it barely begins again.
- Rotate the fine adjust counterclockwise to set the duration of the bell tone, while ringing the bell with the push button switch.
- You might have to tweak all the pots for best results.
- Use a fresh 9V battery.
Construction
The electronic bell simulator is enclosed in a small aluminum chassis box 4cm X 5cm X 10cm to fit comfortably in the hand.
Mount the circuit on a single-sided fiberglass-epoxy circuit board. Make the ground paths wide for circuit stability and ground to the chassis. Wiring is #26 stranded Cu wire. I plotted speaker holes on polar coordinate graph paper which I used as a template for drilling.
The moving parts of the Hall effect sensor switch came from brass stock and a disassebled push button switch. Lubricate with vacuum grease or plumber's grease.
Lay out the PC board pattern on a sheet of graph paper in colored pencil. Tape to the copper foil side of the circuit board and use an awl to make indentations at the positions of the holes. Draw circuit layout by hand on the Cu foil with a resist pen. Correct excess resist with a dissecting needle.
Keep all connections short as the circuit is very sensitive.
Lettering is rub-on stencil, protected by clear-seal transparent plastic adhesive film. The box is polished with fine sandpaper and steel wool.