Although Form Factor reflects the overall shape or structure of a waveform, it is mathematically defined as the ratio of the RMS (root mean square) value to the average value.
Form Factor is a math tool. It analyzes periodic waveforms. It helps understand a waveform’s shape, structure, and quality. Sometimes, it shows how “peaky” a sine wave is if it has harmonic distortions.
The meaning of “form factor” depends on the waveform. But it always provides a useful visual of the waveform itself. This visual gives insights into the waveform’s electrical traits. It also shows how well the waveform uses AC power supply.
Electrical specialists utilize waveform shape to evaluate circuits. This shape provides insight into frequency composition and harmonic presence, crucial for power systems like rectifiers.
The waveform’s distinctive pattern acts as its unique identifier, revealing essential characteristics for effective analysis and design.
Understanding this unique fingerprint is vital for engineers working with high-voltage applications or intricate electrical networks.
The concept of form factor enables us to evaluate the variation in amplitude between different waveforms.
A waveform with steep rises and falls exhibits a higher form factor.
Consider a square wave with a constant amplitude, representing a baseline form factor of 1.0. In contrast, a triangular wave with its pointed peak could reach a form factor as high as 3 or 4, reflecting its greater peakiness.
Various electrical signals come with their own unique shapes. This is because the waveform of each signal has a distinct form factor.
We group these signals by their typical forms, which are standard shapes.
Common Form Factors by Waveform:
- Sine waves rise gently, then descend. It’s a smooth, constant curve. Their special traits are frequency, height, and position.
- Square waves remain level before suddenly shifting between high and low points.
- Triangle waves climb steadily, reach a peak, then descend at an equal slope.
- Sawtooth waves leap upwards rapidly, then slope down gradually, resembling jagged teeth.
- Pulse waves consist of separate spikes varying in width and amplitude.
Periodic Waveform Families
Sinusoidal Waveform
Sinusoidal waveforms (or variations like cosine waves) are the most common periodic waveforms used in electrical and electronic engineering.
Properties of Sine Waves:
An alternating sine wave of voltage or current can be described by its three key values:
- Maximum value (peak value)
- Average value (DC value)
- Root-Mean-Square (RMS) value
A waveform’s maximum amplitude is also known as its peak value or maximum value. The sine wave has a positive peak value (+Amax) which is the highest positive value, and a negative peak value (-Amax) which is the lowest negative value.
The difference between the positive peak value and the negative peak value is the waveform’s peak-to-peak value. Since a sinusoidal waveform is symmetrical between its positive and negative peaks, as shown, the peak-to-peak value will always be twice the peak value. This translates to 2 * Amax.
The mathematical average value of a sine wave is calculated over one half cycle typically from 0 to π. Because for a full cycle (0 to 2π), the average value is zero. Since a sine wave alternates positive and negative values, these opposing parts cancel each other out in a full cycle.
For a sinusoidal wave, the average value (also called mean value) is written as:
Aavg = 2 * Amax / π
This can also be written as (2/π) * Amax or approximately 0.637 * Amax.
The RMS (root-mean-square) value of an alternating wave represents its effective value. For a sine wave, the RMS value is the equivalent of a direct current (DC) value that would produce the same heating effect in a resistor.
The RMS value of a sine wave is calculated over a full cycle and is given as:
ARMS = Amax / √2
This can also be written as (1/√2) * Amax or approximately 0.7071 * Amax.
There’s a definite relationship between the peak value (Amax), average value (Aavg), and RMS value (ARMS) of an alternating waveform. This relationship can be expressed using either the form factor or crest factor.
The form factor of any waveform is defined as:
Formula to Calculate Fundamental Form Factor
Form Factor = RMS Value / Average value.
Therefore, we can express the form factor for a pure sine waveform as:
- FF = RMS Value / Average Value
- = (1 / √2 AMAX ) / (2AMAX / π)
- = 0.7071 AMAX / 0.637 AMAX
- = π / 2√2
- = 1.11
The form factor is defined as the ratio between the waveform’s RMS value (effective value) and its average value.
For a perfect sine wave, the form factor will always be equal to 1.11, regardless of the peak value. This means, the form factor of a perfect sinusoid can never be higher or lower than 1.11.
Solving a Form Factor Problem
Let’s imagine a sinusoidal waveform with a maximum peak value of 310 volts. Calculate its form factor.
- FF = VRMS / VAVE
- = 0.7071 * VMAX / 0.637 * VMAX
- = 0.7071 * 310 / 0.637 * 310
- = 1.11
Solving another form factor Problem
Let’s imagine an alternating voltage with a average value of 165 volts and a form factor of 1.35. Find out its rms value, and calculate its maximum peak value.
- FF = VRMS / VAVE
- VRMS = FF * VAVE
- VRMS = 1.35 * 165 = 222. 75 V
The maximum or the peak value of the sinusoidal voltage can be calculated as:
VMAX = √2VRMS = 1.414 * 222.75 = 315 V
In the above explanation we assumed a purely sinusoidal waveform. However the form factor can still be calculated for other shapes, like waveforms produced by diode rectification.
Single-Phase Half-Wave Rectified Waveforms:
Half-wave rectifiers use only one diode. This means they only allow half of the incoming AC power supply to pass through, as shown below.
Waveform after half wave rectification
In a half-wave rectified sinusoidal voltage waveform, the average DC voltage (VDC) could be defined as Vmax / π (which is approximately 0.318 * Vmax). The effective RMS voltage (VRMS) is given as Vmax / √2.
Therefore the form factor for a half wave rectified output can be calculated using the formula:
- FF(HW) = VRMS / VDC
- = (VMAX / 2) / (VMAX / π)
- = π / 2
- = 1.57
The above calculations show that the form factor for a single-phase half-wave rectifier fed by a pure sinusoidal supply is around 1.57.
Full-Wave Rectified Waveforms using Single Phase Supply:
Unlike half-wave rectifiers, full-wave rectifiers use both halves of the incoming sinusoidal AC supply. These full-wave rectifiers can come in different configurations such as center-tapped, bi-phase, or bridge circuits, as shown in the image.
Full-wave rectifiers come in various forms: center-tapped, bi-phase, or bridge circuits.
In a full-wave rectified waveform, the average DC voltage (VDC) is two times more than a half-wave rectifier’s VDC. This is because it uses both halves of the AC cycle. This allows current flow in the same direction during both halves.
Therefore VDC is: 2 * Vmax / π (about 0.637 * Vmax).
The effective RMS voltage (VRMS) remains the same: Vmax / √2.
The full-wave rectifier’s form factor is calculated using a formula:
- FF(FW) = VRMS / VDC
- = (VMAX / √2) / (2VMAX / π)
- = π / 2√2
- = 1.11
Full-wave rectifier form factor: Surprisingly, the calculated form factor for a single-phase full-wave rectifier is 1.11. This is the same as a perfect sine wave.
This makes sense because:
- The average DC value (0.637 * Vmax) is calculated over one half-cycle.
- The RMS value (0.7071 * Vmax) is calculated from the peak value (Vmax).
Form Factor for Triangular Waves:
The form factor concept applies not just to sine waves but also to other repetitive, non-sinusoidal waveforms. One example is the triangular waveform.
True to its name, a triangular wave is shaped like a triangle. It has positive and negative linear ramps with equal slopes.
These ramps rise to a positive peak value and then fall to a negative peak value in one complete cycle.
The triangular wave’s period, measured from peak to peak, is shown in the image.
Making Triangular Waves:
- We can create triangular waveforms by combining a standard RC oscillator with an integrator circuit.
Triangular Wave Properties:
- The resulting waveform is rectangular, swinging between +Vmax and -Vmax.
- The peak-to-peak value is simply the difference between the positive and negative peak values.
- Since the triangle is symmetrical, the RMS value (VRMS) of a triangular voltage or current is Vmax / √3.
- Because a full cycle forms an isosceles or equilateral triangle, the average DC value (VDC) of each half-cycle is Vmax / 2.
Triangular Waveform Form Factor:
The formula to calculate the form factor for a triangular waveform is:
- FF = VRMS / VDC
- = (VMAX / √3) / (VMAX / 2)
- = 2 / √3
- = 1.155
For a symmetrical triangular wave, this form factor is calculated to be 1.155. This means it is peakier than the smoother sine wave we discussed earlier.
Sawtooth Waveform:
The sawtooth waveform is a close relative of the triangular wave, and its name reflects its resemblance to a saw blade’s teeth.
A sawtooth wave typically rises steadily from zero to its peak value (Vmax) at time T. Then it drops rapidly back to zero (or sometimes a negative value) over its period (T). This rise is also known as a linear ramp.
Sawtooth Waveform Properties:
The sawtooth waveform’s rise is a straight line similar to a series of right-angled triangles. This means:
- The average DC value (positive, equivalent to DC) is simply Vmax / 2, just like a triangular wave.
- Because the sawtooth is symmetrical and repetitive, its effective RMS value (VRMS) is Vmax / √3.
Sawtooth Waveform Form Factor:
The following formula can be used to calculate the form factor for a sawtooth waveform.
- FF = VRMS / VDC
- = (VMAX / √3) / (VMAX / 2)
- = 2 / √3
- = 1.155
Interestingly, the calculated form factor for a sawtooth wave is 1.155. This looks exactly same as a triangular wave. This makes sense because both waveforms can be visualized as a series of connected triangles. The slopes of these triangles are defined by Vmax / T.
Note: Sawtooth waveforms can have ramps that increase positively or negatively.
Square and Pulse Waveforms:
Now we will discuss the form factor of square waves and pulses, another type of periodic waveform.
Square and Pulse Properties:
- In square waves and rectangular pulses, frequency and period have the same meaning.
- So do the peak values (maximum or minimum) since these waves switch between two constant voltage levels, as shown in the image.
Square Waveform
- For any periodic waveform, the average DC value (Vavg) equals its average value over just the positive half-cycle (between 0 and T/2, or π).
- In a square wave, this average value (Vavg) is the same as its maximum value (Vmax). This is because, the full-cycle average (over T or 2π) is zero (positive and negative parts cancel out). So, Vavg = Vmax.
- A square wave’s instantaneous value is always either its positive maximum (+Vmax) or negative maximum (-Vmax).
- This means the squared maximum voltage value is V2max, and its square root (RMS value, VRMS) is simply Vmax. Therefore, VRMS = Vmax for a square wave.
Square Wave Form Factor:
Since the maximum and RMS amplitudes are identical, the form factor for a symmetrical square wave is calculated using the formula:
- Form factor = VRMS Value / VAVE Value
- = VMAX / VMAX
- = 1
Looking at the calculations, we see the form factor for a symmetrical square wave is 1.0 (unity form factor). This makes sense because the square wave has a flat peak with no “peakiness.”
Pulse Waveform Form Factor:
The form factor for a series of rectangular pulses depends on its duty cycle (also called mark-space ratio).
A typical pulse wave with amplitude Vmax and period T has a 50% duty cycle. This means its mark-space ratio (TON/TOFF) is 50:50 or 1:1.
Different pulse widths (TON) and intervals (TOFF) will result in a non-unity mark-space ratio.
A pulse train is a series of pulses where each pulse width (TON) is defined, along with the interval (TOFF) between pulses. The period (T) of a pulse train is the sum of TON and TOFF, as shown in the image.
Waveform of a Positive Pulse
Pulse Waveform Properties:
If a pulse train has a 1:1 duty cycle (TON = TOFF), the average voltage (Vavg) is half the pulse’s peak value (Vmax).
So, Vavg = Vmax / 2.
For other duty cycles, the output voltage will be higher or lower than Vmax / 2.
This will depend on the specific TON and TOFF values. We can define Vavg as Vavg = Vmax * (TON / T).
Here TON / T is the duty cycle (DC) representing the on-time as a proportion of the period.
The RMS value (VRMS) of a rectangular pulse (always positive) is based on the square root of its duty cycle’s average.
In other words VRMS is Vmax multiplied by the square root of its duty cycle: VRMS = Vmax * √(TON / T).
Mark-Space Ratio and Form Factor:
With a constant amplitude (Vmax), the pulse train’s mark-space ratio affects its form factor a lot.
This can be shown using mark-space ratios of 10% (1:10), 50% (5:10), and 90% (9:10) with a Vmax of 1 volt, as shown below:
- When duty cycle is 10%
- FF = RMS Value / Average Value
- = Vmax (√TON/T)
- = 1(√1/10) / 1(1/10)
- = 0.316 / 0.1
- = 3.16
- When duty cycle is 50%
- FF = RMS Value / Average Value
- = Vmax (√TON/T)
- = 1(√5/10) / 1(5/10)
- = 0.7071 / 0.5
- = 1.414
- When duty cycle is 90%
- FF = RMS Value / Average Value
- = Vmax (√TON/T)
- = 1(√9/10) / 1(9/10)
- = 0.95 / 0.9
- = 1.06
Duty Cycle vs. Form Factor:
- Look at the form factor as the duty cycle of the pulse train changes. As the duty cycle increases from 10% to 90%, the form factor goes down from over 3 to almost 1.0 (unity).
Why this happens:
- At a 10% duty cycle, the pulse looks like a sharp spike, resulting in a high form factor.
- At a 90% duty cycle, the pulse is close to a flat line, approaching a unity form factor.
Form Factor Summary:
This section reviewed form factor. It’s the ratio between a waveform’s RMS value (effective value) and its average DC value.
What Does Form Factor Tell Us?
Form factor describes a waveform’s shape by indicating how close it is to a perfect sine wave. Different waveforms have different form factors based on their shapes.
Form Factor Values:
Form factor can range from 1 (unity) for smooth waveforms to 3 or 4 for peaky waveforms like pulses. A value of 1 means the RMS value is the same as the peak value.
Form Factor Reference Table:
The following table summarizes the form factor formulas and values for the different waveform types discussed earlier.
| Waveform Type | Form Factor Formulas | Values |
| Sine Wave | π/2√2 | 1.11 |
| Half-wave Rectified Sine Wave | π/2 | 1.57 |
| Full-wave Rectified Sine Wave | π/2√2 | 1.11 |
| Triangle Waveform | 2/√3 | 1.155 |
| Sawtooth Waveform | 2/√3 | 1.155 |
| Symmetrical Square Wave | 1 | 1.0 |
| 50% Pulse Waveform | Vmax√Duty Cycle Vmax × Duty Cycle | 1.414 |
References: Form Factor