PWM motor controller with slow ramp-up
OK, here's my design. I think it will do what you want.
It's not simple, and not really a beginner's project. It's up to you whether you want to tackle it. If not, don't worry that I've designed this for nothing... it will be useful as a reference for other similar projects, at least.
Here's a fairly detailed description of the design. I will use words like oscillator and transistor that are familiar to most users of this forum; if you need to, look them up on Wikipedia. I will only give detailed descriptions of the unusual aspects of the design.
The circuit uses two NE555 timer ICs (integrated circuits) configured as oscillators. The first oscillator produces the slow ramp-up and fast ramp-down that will control the average output voltage (and therefore, the stirrer speed) and the second produces a roughly triangular wave at a frequency of about 450 Hz (hertz), which is used in the PWM (pulse width modulation) signal generation.
The first oscillator produces a control voltage at the point marked RAMP. The voltage ramps up from 4V to 8V over a period of about four minutes, then ramps back down to 4V over a period of about 15 seconds, and the cycle repeats.
These times can be changed by adjusting R1 and/or R2. R2 determines the ramp-down time at the end of the cycle, and should be 14.3 kilohms multiplied by the desired ramp-down time in seconds. For example if you want a ramp-down time of 30 seconds, R2 should be 430 kilohms.
The ramp-up time is determined by the sum of R1 and R2, using the same formula: 14.3 kilohms multiplied by the desired time in seconds. So if you want a ramp-up time of 2.5 minutes (150 seconds), R1+R2 should be 2145 kilohms, or 2.145 megohms. If R2 is 430 kilohms, R1 needs to be 1715 kilohms, or 1.715 megohms; the closest standard value is 1.8M.
The ramp produced at the RAMP point is not linear; it is slightly curved. But it is a reasonable approximation.
This ramp is buffered by Q1 and Q2, which are connected as a Darlington transistor, used as an emitter follower stage. The voltage at Q2's emitter follows the ramp, with a positive offset of about 1.4V (due to the two base-emitter junctions). This voltage has a further 1.4V added by D1 and D2, which provide a roughly constant voltage drop of about 0.7V. R5 provides the emitter load for Q2 and pulls the voltage up towards +12V.
So the signal at the node marked SRAMP is a shifted-up version of the slow ramp that's generated by the first oscillator.
The second oscillator operates continuously and independently of the first. It runs at a frequency of about 450 Hz (set by C3, R4 and R3) and produces a roughly triangular wave at the TRI node; this is shown in the blue trace on the waveform graph. The top and bottom of this waveform are 8V and 4V respectively. This signal is buffered by Q3, another emitter follower. This time, the voltage offset is negative, so Q3's emitter is about 0.7V lower than the TRI node voltage.
The two signals are combined in Q4, which operates as a crude voltage comparator. Q4 will conduct when its base-emitter junction is forward-biased; this occurs when its base is brought more positive than its emitter by about 0.7V. The base-emitter voltages of Q3 and Q4 cancel out; the result is that Q4 will conduct when the SRAMP voltage is higher than the TRI voltage.
So Q4 compares the shifted slow ramp from the first oscillator against the 450 Hz triangular wave from the second oscillator, and conducts when the former is higher than the latter. These times correspond to the times when the load should be energised. If you don't know why that works, look up pulse width modulation on Wikipedia or Google.
This comparison generates a pulse-width-modulated control signal, which is buffered and cleaned up by Q5 and produces the GATE signal (shown in green on the graph), which drives Q6, which controls the current to the load.
The action of Q4 does actually cause some cross-interference between the two signals. Specifically, when TRI goes below SRAMP and Q4 conducts, the base-emitter diode in Q4 drags the SRAMP voltage down, so that it actually follows the TRI node for that part of the waveform. This is visible in the waveform view on the red trace, which shows the SRAMP voltage. This effect causes Q1 and Q2 to lose their operating current during those parts of the PWM cycle, but that isn't a problem.
During roughly the second half of the slow ramp, the up-shifted voltage is so high that Q4 remains ON constantly. This produces the period of continuous full speed operation.
Changes in the relationship between the ramp and the PWM can be made in various ways; increasing or reducing the number of diodes in the D1-D2 circuit is a simple one. If you want a change, describe exactly how the current circuit behaves, and exactly what you want instead.
Component suggestions
Resistors are all standard - 1/4 watt (or higher), 5% (or better) tolerance. Capacitors should be ceramic or multi-layer ceramic except C1 which should be an aluminium electrolytic, rated at 16V or 25V, from a good manufacturer (Nichicon, Rubycon, UCC or Panasonic) and C3, which should be a good quality capacitor (because it's a frequency-setting component) such as polyester film:
http://www.digikey.com/product-detail/en/R82EC2470AA60J/399-5873-ND/2571308.
The BC557B transistors can be replaced with 2N3906, and the BC547B transistors can be replaced with 2N3904. All types are readily available but the 2N series is more widely used in America. (The BC series is of European origin.)
Although the schematic shows two separate 555s, you can use a single NE556, which contains two 555s in a single 14-pin IC. These are available in several different technologies and from several manufacturers, with different prefixes: LM556, NE556, SA556, NA556, TLC556, KA556, TS556 and ICM7556 are all suitable in this application. Get a DIP (aka DIL) package, with through-hole leads, not a surface-mount part. The 556 contains two independent 555s with commoned VCC and GND pins. You will need to match up the pin descriptions from the 555s shown on the schematic to the 556.
You will need data sheets for the 555 and 556, and the transistors (so you can tell which pin is which). The components and data sheets are all available through the Digikey web site
http://www.digikey.com.
The output MOSFET, Q6, can be any N-channel MOSFET with suitable voltage and current ratings. The MTD3055 I've suggested is readily available but there are many alternatives.
D3 is needed if the load (the stirrer motor) is inductive; I assume that it is. I've specified a 1N5817 which is a Schottky diode, but a plain old 1N4001 would be fine at this relatively low frequency. I have assumed that your stirrer draws 1A of current at 12V. If it draws more current than that, D3 will need to be up-rated to a 1N5401 (3A rating) or more.
All of these components are available as through-hole components; that is, components with wire leads that you can mount onto a piece of stripboard. I have not designed a stripboard layout; you will have to do that. If you're not experienced with circuit assembly, perhaps you have a friend who can help? If you're the adventurous type, there are plenty of tutorials available on the web, and of course the folks here can answer any specific questions you may have.