While this is not my design, this is my analysis and derivations.

I used this in my sound reactive lighting project.  I had only one op-amp to spare for a triangle wave generator, so I couldn’t use the popular two op-amp design.

For those who are not interested in the details, you can find the final formulas in an example here.

Overview

view this circuit simulation here

This single op-amp circuit uses positive feedback with hysteresis to create a square wave, which charges and discharges an RC circuit, which roughly produces a triangle wave.

Before we begin, assume the op-amp is acting like an ideal comparator.  The highest voltage the comparator can output is V_CC and the lowest is 0V.  These high and low output voltages are determined by the power supply connected to the op amp (which isn’t shown in the schematic).

Start at V_{1 .}  For now, let’s ignore R₃ .  Currently, V₁ is equal to V_CC  – this voltage comes  from the comparator output.  The output is V_CC  because the noninverting input is greater than the inverting input.

Connected to the comparator output is R₄ and C₁ .  Together, they act as an integrator.  As the voltage on the capacitor ramps up, it produces the up ramp of a triangle wave.  This triangle wave appears at V₂.

As V₂ is increasing, it eventually becomes greater than V₃.  When this occurs, the comparator sees the inverting input is now greater than the non-inverting input. Accordingly, the comparator switches its output from a high voltage (V_CC) to a low voltage (0V).  With a comparator output of 0V, C₁ drains through R₄ .  This produces the down ramp of a triangle wave.

Eventually, V₂ becomes less than V₃ – the comparator output goes back up to V_CC and the cycle repeats.

So far, we’ve ignored R₃ .  However, it is a very important resistor.  It allows for the cyclic action by adding hysteresis.  It affects the voltage V₃.  We’ll look at R₃ in more detail below.
 
 

Hysteresis and R₃

Left: simplified diagram when comparator output is high (V₁ = V_CC ).  Right: simplified diagram when comparator output is low ( V₁ = 0V).  View circuit here

This simplified diagram shows all inputs for V₃. In this diagram, we’re simplified the comparator output (V₁) as a voltage source.  V_CC , R₁ , and R₂ form a voltage divider where the divided voltage is V₃ .  So does V_CC , R₃ , and R₂.

When the comparator output V₁ is high, the voltage at V₃ increases.  This sets the upper threshold for the triangle wave – the voltage the comparator’s inverting input must reach in order to cause the comparator output to switch low (0V) and begin the down ramp of the triangle wave.

When the output V₁ is low, the voltage at V₃ decreases.  This sets the lower threshold for the triangle wave – the voltage the comparator’s inverting input must reach to cause the comparator output to switch high (V_CC) and begin the up ramp of the triangle wave.

In short, R₃ is adding hysteresis, making the triangle wave possible.  The amplitude of the wave depends on V_CC, R₁, R₂, R₃, and the maximum voltage output swing of the op-amp.  We’ll look at calculating this amplitude next.
 
 

Determining Amplitude

Notice the way I set up the figure from the previous section.  It’s ready for Millman’s Theorem.  It allows us to quickly write the minimum and maximum voltages of the triangle waves:

We won’t be solving directly for V_tri,max or V_tri,min with these equations.   Let’s introduce some more variables first:

• V_O,max is the maximum output of the comparator (you can find this on the data sheet under Maximum Voltage Output Swing).  It is typically within 1 volt of V_CC
• V_O,min is the minimum output of the comparator (you can find this on the data sheet under Minimum Voltage Output Swing).  It is typically within 1 volt of V_EE .
• V_tri,amp is the desired amplitude (peak-to-peak voltage) of the triangle wave.  This is a value you choose.  It is equal to:

• V_tri,max is the maximum peak voltage of the triangle wave.  While this value won’t be calculated for determining amplitude, we’ll need it to determine frequency in the next section.
• V_tri,min is the minimum voltage of the triangle wave.  Like V_tri,max we will need this value to determine frequency.

Again, our goal is to solve for R₃ . We can set the two equations from above equal to each other and get the following:

To further reduce this equation, we can set R₁ equal to R₂.  This is a fine assumption for 99% of applications.  We’ll also replace V_tri,max – V_tri,min with V_tri,amp .

After reducing, we’re left with the following formula:

where all resistances are in ohms and voltages are in volts.

You choose the value of R₁ in order to calculate R₃ .  As it is a simple voltage divider, anything between 10 kΩ and 1 MΩ is fine.  In the example at the bottom of this page, I chose 100 kΩ.
 
 

Determining Frequency

As mentioned early, the frequency is determined by the R₄ and C₁ integrator.  Our goal here will be to choose a value for C₁ then calculate R₄ .

simplified diagram of the RC section

To determine the triangle wave frequency, we’ll start with the RC time constant equation:

This is the formula for the rising section of the triangle wave.

Graphing the formula above as a function of time produces something like this:

The graph above shows that V_tri,min occurs at time t_a and V_tri,max occurs at time t_b . The time from t_a to t_b is the upramp of the triangle wave.

V_tri,min and V_tri,max will be needed to determine frequency.  Assuming R₁ is equal to R₂, we can calculate them with the following formulas:

t_a and t_b don’t need to be calculated.  However, what we need is Δt –  the time between t_a and t_b .  This Δt is one half of the period, so we can express it in terms of frequency:

Notice the 2 in the denominator.  It’s because we are calculating the up ramp of the triangle wave, which is only half of a period.

We can create two equations.  One at time t_a and another at time t_b :

Rearranging in terms of t gives:

Subtracting these two equations from each other gives:

We now substitute t_b – t_a for frequency:

Solving for R₄ , we get:

where f is the desired triangle-wave frequency.  Frequency is in Hz, capacitance is in farads, voltage in volts, and resistance in ohms.

Note – you have to choose C₁ first.  I recommend starting with 100nF.  See more details on this in the Example Design section.

C₂, R₅ , R₆ , and the output load will change the frequency.  As long as the output load is high impedance(100k+), the frequency shouldn’t change more than 1%.
 
 

Output Stage

If you want to remove DC bias from the output, then you’ll need to add  a coupling capacitor on the output.
 
 

Example Design

This is an example from my Sound Reactor project.

Step 1 – Define Criteria

For this example, we’ll use the following criteria:

• f = 1.1 kHz
• V_tri,amp  = 1 V
• V_CC  = 12 V
• V_O,max  = 11.715 V (this value came from the comparator datasheet when the comparator is powered by 12V.  I used the MC4558)
• V_O,min  = 0.285 V (this value was from the comparator datasheet)

Step 2 – Choose R₁ = R₂

R₁ and R₂ form a voltage divider.  I recommend anything in between 10 kΩ and 1 MΩ. Too low of a value will waste power.  Too high of a value will cause inaccurate results.  I chose R₁ = R₂ = 100 kΩ.

Step 3 – Calculate R₃

I calculated R₃ = 521.5 kΩ, which is close to the standard 520 kΩ.

Step 4 – Calculate V_tri,min and V_tri,max

This is an intermediate step to get values for R₄ and C₁ .

I calculated V_tri,min = 5.5 V and V_tri,max = 6.5 V.

Step 5 – Calculate R₄ and C₁

Note – you have to choose C₁ first.  The quick way is guess-and-check.  Guess a value for C₁ and check if R₄ is a reasonable value.

I recommend starting with C₁ = 100 nF.  If the resulting value of R₄ is too high (>1 MΩ), choose a lower value for C₁ and recalculate. If the resulting value of R₄ is too low (<10 kΩ) choose a higher value for C₁ and recalculate.

I started with C₁ = 100 nF.  The calculation gave R₄ = 25.9 kΩ.  I settled for an actual value of 24.9 kΩ.

Results

view this circuit here

• f = 1.1 kHz
• V_tri,amp = 1 V
• R₁ = R₂ = 100 kΩ
• R₃ = 520 kΩ
• R₄ = 24.9 kΩ
• C₁ = 100 nF

 
 

Comparison to the Two Op-Amp Design

The most common design for an op-amp triangle wave generator uses two op-amps.  It looks something like this:

The two op-amp design generators a square wave then integrates it to produce a triangle wave.   It uses an active integrator, instead of a passive RC integrator.

The two op-amp design uses 8 discrete components and 2 op-amps whereas the single op-amp design uses 5 discrete components and 1 op-amp.

I believe the two op-amp design is overall less dependent on component tolerances and op-amp specs and therefore more consistent from circuit to circuit.

Neither should be used for any circuit that needs accurate, precise, or stable frequencies or amplitudes.  Nor for circuits that need a true triangle wave instead of the RC integrated approximation.

Overall, the single op-amp triangle wave generator is certainly simple to implement and practical for real world circuits!