Crystal oscillator circuit in PCB Design - Cell-by-Cell Formula Trace

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.

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 MHz

This 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

What is the total capacitance used in the frequency calculation?

A 11 pF

B 44 pF

C 22 pF

D 25 pF

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.

Practice

1. What is the main purpose of a crystal oscillator circuit in electronics?

easy

A. To amplify weak signals

B. To store electrical energy

C. To convert AC to DC power

D. To provide a precise and stable clock signal

Solution

1. Step 1: Understand the role of oscillators

   Oscillators generate repetitive signals used as timing references in circuits.

2. Step 2: Identify crystal oscillator function

   Crystal oscillators use a quartz crystal to produce a very stable and precise frequency.

3. Final Answer:

   To provide a precise and stable clock signal -> Option D

4. Quick Check:

   Crystal oscillator = precise clock signal [OK]

Hint: Crystal oscillators give stable timing signals [OK]

Common Mistakes:

• Confusing oscillator with amplifier
• Thinking it stores energy
• Mixing power conversion with timing

2. Which of the following is the correct way to connect the crystal in a typical Pierce oscillator circuit?

easy

A. Between the power supply and ground without capacitors

B. Directly in series with the power supply

C. Between the input and output of the inverter with capacitors to ground

D. In parallel with the load resistor only

Solution

1. Step 1: Recall Pierce oscillator configuration

   In a Pierce oscillator, the crystal is connected between the inverter input and output pins.

2. Step 2: Confirm capacitor placement

   Capacitors connect from these points to ground to form the feedback network.

3. Final Answer:

   Between the input and output of the inverter with capacitors to ground -> Option C

4. Quick Check:

   Pierce oscillator crystal connection = input-output with capacitors [OK]

Hint: Crystal goes between inverter input and output with caps to ground [OK]

Common Mistakes:

• Connecting crystal directly to power
• Omitting capacitors to ground
• Placing crystal only in parallel with load

3. Given a crystal oscillator circuit with a 10 MHz crystal and capacitors of 20 pF each, what is the expected output frequency?

medium

A. 5 MHz

B. 10 MHz

C. 20 MHz

D. Depends on capacitor values only

Solution

1. Step 1: Understand crystal frequency role

   The crystal sets the fundamental frequency of oscillation, here 10 MHz.

2. Step 2: Recognize capacitor effect

   Capacitors fine-tune frequency slightly but do not double or halve it.

3. Final Answer:

   10 MHz -> Option B

4. Quick Check:

   Crystal frequency = output frequency [OK]

Hint: Output frequency equals crystal frequency, capacitors adjust tuning [OK]

Common Mistakes:

• Assuming capacitors double frequency
• Confusing frequency division
• Ignoring crystal's dominant frequency

4. A crystal oscillator circuit is not oscillating. Which of the following is the most likely cause?

medium

A. Incorrect capacitor values causing no feedback

B. Crystal frequency too high for the circuit

C. Power supply voltage is too high

D. Using a resistor instead of a crystal

Solution

1. Step 1: Check feedback network

   Oscillation requires proper feedback; wrong capacitor values can break this loop.

2. Step 2: Evaluate other options

   Crystal frequency mismatch or power issues usually cause distortion, not no oscillation; resistor cannot replace crystal.

3. Final Answer:

   Incorrect capacitor values causing no feedback -> Option A

4. Quick Check:

   No oscillation often means feedback failure [OK]

Hint: No oscillation? Check capacitor feedback network first [OK]

Common Mistakes:

• Blaming crystal frequency without checking feedback
• Ignoring capacitor role
• Replacing crystal with resistor

5. You want to design a crystal oscillator circuit for a microcontroller that requires a 16 MHz clock. Which combination is best to ensure stable oscillation?

hard

A. Use a 16 MHz crystal with two 22 pF capacitors connected to ground

B. Use a 16 MHz crystal with no capacitors

C. Use a 8 MHz crystal with two 22 pF capacitors connected to ground

D. Use a 16 MHz crystal with one 100 pF capacitor in series

Solution

1. Step 1: Match crystal frequency to microcontroller clock

   The crystal frequency must be 16 MHz to match the required clock.

2. Step 2: Choose proper capacitor values and placement

   Two capacitors (~22 pF) connected from crystal terminals to ground stabilize oscillation.

3. Step 3: Evaluate other options

   No capacitors or wrong crystal frequency cause instability; one large capacitor in series is incorrect.

4. Final Answer:

   Use a 16 MHz crystal with two 22 pF capacitors connected to ground -> Option A

5. Quick Check:

   Correct crystal + proper capacitors = stable oscillation [OK]

Hint: Match crystal freq and use two small caps to ground [OK]

Common Mistakes:

• Skipping capacitors
• Using wrong crystal frequency
• Incorrect capacitor placement