Concept Flow
Crystal
Capacitor C1 (22 pF)
Capacitor C2 (22 pF)
Resistor R1 (10 kΩ)
This flow shows the main components of the crystal oscillator circuit and how they relate to the frequency calculation formula.
0
0
Crystal oscillator circuit in PCB Design - Cell-by-Cell Formula Trace
Formula
Oscillation Frequency = 1 / (2 * π * sqrt(L * C))
Where:
L = Equivalent inductance of crystal (assumed 9 uH)
C = Total load capacitance = C1 * C2 / (C1 + C2) = 22 pF * 22 pF / (22 pF + 22 pF) = 11 pF
Convert units:
L = 9 uH = 9e-6 H
C = 11 pF = 11e-12 F
Calculate:
sqrt(L * C) = sqrt(9e-6 * 11e-12) = 9.95e-9
Denominator = 2 * π * 9.95e-9 = 6.26e-8
Frequency = 1 / 6.26e-8 = 15,974,000 Hz ≈ 16 MHzThis code snippet shows the step-by-step calculation of the oscillation frequency using the crystal's inductance and the capacitors' capacitance.
Step-by-Step Trace
| Step | Expression | Evaluates To | Explanation |
|---|---|---|---|
| 1 | L = Equivalent inductance of crystal | 9 uH | Crystal behaves like an inductor with 9 microhenries inductance. |
| 2 | C = C1 * C2 / (C1 + C2) = 22 pF * 22 pF / 44 pF | 11 pF | Load capacitors C1 and C2 are in series across the crystal. |
| 3 | Convert units: L = 9 uH, C = 11 pF | L=9e-6 H, C=11e-12 F | Units converted to standard SI units. |
| 4 | Calculate sqrt(L * C) | 9.95e-9 | Square root of product of inductance and capacitance. |
| 5 | Calculate denominator: 2 * π * sqrt(L * C) | 6.26e-8 | Multiply by 2π to complete denominator. |
| 6 | Calculate frequency: 1 / denominator | 15,974,000 Hz ≈ 16 MHz | Final oscillation frequency calculated. |
The oscillation frequency is approximately 16 MHz, which matches the expected output for this 16 MHz crystal oscillator setup.
Variable Tracker
| Variable | Value | Unit | Description |
|---|---|---|---|
| L | 9e-6 | H | Equivalent inductance of crystal after conversion |
| C1 | 22 | pF | Capacitor 1 value |
| C2 | 22 | pF | Capacitor 2 value |
| C | 11 | pF | Total load capacitance (C1 and C2 in series) |
| C | 11e-12 | F | Total capacitance converted to Farads |
| sqrt(L*C) | 9.95e-9 | s | Square root of inductance times capacitance |
| Denominator | 6.26e-8 | s | 2 * π * sqrt(L*C) |
| Frequency | 15974000 | Hz | Calculated oscillation frequency |
Key Moments
Why do we calculate C = C1 * C2 / (C1 + C2)?
Why convert uH and pF to H and F before calculation?
What does the final frequency represent?
Sheet Trace Quiz - 3 Questions
Test your understanding
What is the total capacitance used in the frequency calculation?
Key Result
The crystal oscillator frequency is calculated using the formula 1 / (2π√(L*C)), where L is the crystal's equivalent series inductance and C is the total load capacitance from C1 and C2 in series combination. Unit conversions to standard SI units are essential before calculation.
Transcript
We start with the crystal's equivalent inductance, assumed as 9 microhenries. Next, we calculate the series equivalent of capacitors C1 and C2, each 22 picofarads, to get a total load capacitance of 11 picofarads. We convert these values to henries and farads respectively for the formula. Then, we calculate the square root of the product of L and C. Multiplying this by 2π gives the denominator. Finally, taking the reciprocal of this denominator yields the oscillation frequency, approximately 16 megahertz.