LM2907 Bias Resistor Change - Corrected Analysis¶

Problem Statement¶

LM2907 experiences 5-10 second RPM dropout after alternator field cutoff due to AC coupling time constant mismatch. Scope shows only ~300ms actual zero-crossing loss, but coupling capacitor takes much longer to recover proper DC bias level.

Root Cause¶

Time constant τ = R × C = 100kΩ × 20µF = 2.0 seconds
Recovery time ≈ 2.3τ for 90% recovery = 4.6 seconds (not 5τ = 10+ seconds)
AC coupling cap retains charge from large signal, creates DC offset on small returning signal

Current Circuit Configuration¶

Component	Value	Power Rating	Performance
Series Resistor	4.7kΩ	2W	Adequate
Bias Resistor	100kΩ	0.1W	Adequate
Coupling Capacitor	20µF	N/A	N/A
Shunt Capacitor	6.8nF	N/A	N/A
Time Constant	2.0 seconds	N/A	Problem: 4.6 second dropout
High-Pass Corner	0.08 Hz	N/A	Preserves low frequencies

Proposed Circuit Configurations¶

Component	Current	Option 1	Option 2
Series Resistor	4.7kΩ	4.7kΩ	4.7kΩ
Bias Resistor	100kΩ	1kΩ	10kΩ
Coupling Capacitor	20µF	20µF	20µF
Shunt Capacitor	6.8nF	6.8nF	6.8nF
Time Constant	2.0 s	0.02 s	0.2 s
High-Pass Corner	0.08 Hz	8 Hz	0.8 Hz
90% Recovery Time	4.6 s	0.046 s	0.46 s

Power Consumption Analysis (Corrected)¶

Complex Impedance Calculation¶

The circuit impedance must account for both capacitive reactances using proper complex voltage division:

Circuit Model:

Stator → 4.7kΩ series → 20µF coupling → node → (R_bias || 6.8nF) to ground

Load Impedance Calculation: The load impedance is the parallel combination of the bias resistor and shunt capacitor:

Z_load = R_bias || Z_shunt = (R_bias × Z_shunt) / (R_bias + Z_shunt)
where Z_shunt = 1/(j2πfC_shunt)

Total Impedance:

Z_total = R_series + Z_coupling + Z_load
where Z_coupling = 1/(j2πfC_coupling)

Node Voltage (Complex Division):

V_node = V_in × (Z_load / Z_total)

Power Calculations: - Series resistor: P_series = |I_line|² × R_series - Bias resistor: P_bias = |V_node|² / R_bias

TVS Conduction Analysis¶

SMBJ12CA TVS breakdown voltage: ~13V

Node Peak Voltage Calculations:

Input Voltage	1kΩ Node Peak	10kΩ Node Peak	100kΩ Node Peak	TVS Conduction
12V pp	1.05V	4.08V	5.73V	None
24V pp	2.10V	8.16V	11.46V	None
48V pp	4.21V	16.32V	22.91V	10kΩ, 100kΩ conduct
60V pp	5.26V	20.40V	28.64V	10kΩ, 100kΩ conduct

Power Analysis at Key Operating Points¶

At 12V Peak-to-Peak (4.24V RMS, 60Hz) - No TVS Conduction¶

Complex Impedance Calculations (Corrected): - X_c(6.8nF, 60Hz) = 1/(2π × 60 × 6.8nF) ≈ 389kΩ - X_c(20µF, 60Hz) = 1/(2π × 60 × 20µF) ≈ 133Ω

Current Design (100kΩ bias): - Z_parallel = 93.8k - j24.1kΩ (magnitude = 96.9kΩ) - |Z_total| = 101.5kΩ (complex: 98.5k - j24.2kΩ) - I_rms = 4.24V / 101.5kΩ = 41.8µA - P_series = (41.8µA)² × 4.7kΩ = 0.008mW - P_bias = 0.16mW

Option 1 (1kΩ bias): - Z_parallel ≈ 1kΩ (shunt cap negligible) - |Z_total| ≈ 4.83kΩ - I_rms = 4.24V / 4.83kΩ = 0.88mA - P_series = (0.88mA)² × 4.7kΩ = 3.6mW - P_bias = 0.6mW

Option 2 (10kΩ bias): - Z_parallel = 9.78k - j1.46kΩ (magnitude = 9.89kΩ) - |Z_total| ≈ 10.8kΩ - I_rms = 4.24V / 10.8kΩ = 0.39mA - P_series = (0.39mA)² × 4.7kΩ = 0.7mW - P_bias = 0.8mW

At 24V Peak-to-Peak (8.49V RMS, 60Hz) - No TVS Conduction¶

Current Design (100kΩ bias): - I_rms = 8.49V / 79.7kΩ = 0.106mA - P_series = (0.106mA)² × 4.7kΩ = 0.053mW - P_bias = (0.106mA × 79.6kΩ)² / 100kΩ = 0.71mW

Option 1 (1kΩ bias): - I_rms = 8.49V / 4.93kΩ = 1.72mA - P_series = (1.72mA)² × 4.7kΩ = 13.9mW - P_bias = (1.72mA × 997Ω)² / 1kΩ = 2.95mW

Option 2 (10kΩ bias): - I_rms = 8.49V / 10.88kΩ = 0.78mA - P_series = (0.78mA)² × 4.7kΩ = 2.86mW - P_bias = (0.78mA × 9.74kΩ)² / 10kΩ = 5.78mW

At 48V Peak-to-Peak (16.97V RMS, 60Hz) - TVS Conducts for 10kΩ and 100kΩ¶

Option 1 (1kΩ bias) - No TVS Conduction: - I_rms = 16.97V / 4.93kΩ = 3.44mA - P_series = (3.44mA)² × 4.7kΩ = 55.7mW - P_bias = (3.44mA × 997Ω)² / 1kΩ = 11.8mW

Option 2 (10kΩ bias) - TVS Conducts: - Node peak exceeds 13V (16.32V peak) - TVS clamps during peaks, reducing bias resistor power - Conservative estimate: ~15-20mW (reduced from linear prediction of 23.1mW)

Current Design (100kΩ bias) - TVS Conducts: - Node peak exceeds 13V (22.91V peak) - TVS clamps during peaks, reducing bias resistor power - Conservative estimate: ~2-3mW (reduced from linear prediction of 2.87mW)

At 60V Peak-to-Peak (21.21V RMS, 60Hz) - TVS Conducts for 10kΩ and 100kΩ¶

Option 1 (1kΩ bias) - No TVS Conduction: - I_rms = 21.21V / 4.93kΩ = 4.30mA - P_series = (4.30mA)² × 4.7kΩ = 87.0mW - P_bias = (4.30mA × 997Ω)² / 1kΩ = 18.5mW

Option 2 (10kΩ bias) - TVS Conducts: - Node peak exceeds 13V (20.40V peak) - TVS clamps during peaks, reducing bias resistor power - Conservative estimate: ~20-25mW (reduced from linear prediction)

Current Design (100kΩ bias) - TVS Conducts: - Node peak exceeds 13V (28.64V peak) - TVS clamps during peaks, reducing bias resistor power - Conservative estimate: ~3-4mW (reduced from linear prediction)

Power Rating Requirements and Safety Analysis¶

Complete Power Analysis with TVS Effects¶

Input Voltage	Current (100kΩ)	Option 1 (1kΩ)	Option 2 (10kΩ)	Analysis Method
12V pp	0.18mW	0.74mW	1.45mW	Small-signal (no TVS)
24V pp	0.71mW	2.95mW	5.78mW	Small-signal (no TVS)
48V pp	~2-3mW	11.8mW	~15-20mW	TVS clamped / Small-signal
60V pp	~3-4mW	18.5mW	~20-25mW	TVS clamped / Small-signal

Safety Factor Analysis¶

Safety Factor with 0.1W Rating: | Input Voltage | Current (100kΩ) | Option 1 (1kΩ) | Option 2 (10kΩ) | |---------------|------------------|----------------|----------------| | 12V pp | 556× (excellent) | 135× (excellent) | 69× (excellent) | | 24V pp | 141× (excellent) | 34× (excellent) | 17× (excellent) | | 48V pp | 40× (excellent) | 8.5× (good) | 5.5× (adequate) | | 60V pp | 30× (excellent) | 5.4× (adequate) | 4.5× (adequate) |

Engineering Practice: Minimum 3× safety factor recommended for reliable operation.

Component Rating Recommendations¶

Current design (100kΩ): 0.1W adequate for all operating conditions (TVS clamping reduces power at high voltages)
Option 1 (1kΩ): 0.1W adequate for all operating conditions (5.4× safety factor at 60V, no TVS conduction)
Option 2 (10kΩ): 0.1W adequate for all operating conditions (4.5× safety factor at 60V with TVS clamping)

Key Findings¶

TVS clamping begins at 48V pp for 10kΩ and 100kΩ configurations
1kΩ configuration avoids TVS conduction up to 60V pp, maintaining linear operation
All configurations meet minimum 3× safety factor with 0.1W rating across full operating range
TVS clamping reduces actual power below small-signal predictions for affected configurations

Engine Off Power Consumption¶

Stator at ~12V DC (through rectifier diodes)
Coupling capacitor blocks all DC current
Only capacitor leakage: ~2-3µA
Power consumption: Negligible (~microwatts) for all options

Transfer Function Analysis¶

The complete transfer function includes both the high-pass filter formed by the coupling capacitor and the voltage divider effect:

H(jω) = [R_bias / (R_bias + 4.7kΩ)] × [jωτ / (1 + jωτ)]
where τ = R_bias × 20µF (coupling time constant)

Frequency Response Calculations¶

Reactance Values¶

Frequency	X_c(20µF)	X_c(6.8nF)	Coupling Effect	Shunt Effect
5 Hz	1592Ω	4.68MΩ	Significant	Negligible
10 Hz	796Ω	2.34MΩ	Moderate	Negligible
50 Hz	159Ω	468kΩ	Minor	Minor
100 Hz	80Ω	234kΩ	Minimal	Minor
1000 Hz	8Ω	23.4kΩ	Negligible	Moderate
2000 Hz	4Ω	11.7kΩ	Negligible	Significant

Required Stator Amplitude Analysis¶

Required stator amplitude to produce 25mVpp at LM2907 pin:

Frequency	100kΩ Bias	10kΩ Bias	1kΩ Bias	Notes
5 Hz	36.8mVpp	52.8mVpp	694mVpp	High-pass attenuation dominates
10 Hz	29.5mVpp	42.1mVpp	278mVpp	Moderate high-pass attenuation
50 Hz	26.8mVpp	38.1mVpp	154mVpp	Minor high-pass effect
100 Hz	26.4mVpp	37.4mVpp	145mVpp	Minimal high-pass effect
1000 Hz	26.2mVpp	36.8mVpp	142mVpp	High-frequency baseline
2000 Hz	26.2mVpp	36.7mVpp	141mVpp	Shunt capacitor starts affecting 1kΩ

Detailed Calculations¶

100kΩ Bias Resistor¶

Frequency	Coupling Gain	Divider Gain	Total Gain	Required Input
5 Hz	0.628	0.955	0.600	41.7mVpp
10 Hz	0.783	0.955	0.748	33.4mVpp
50 Hz	0.950	0.955	0.907	27.6mVpp
100 Hz	0.975	0.955	0.931	26.8mVpp
1000 Hz	0.998	0.955	0.953	26.2mVpp
2000 Hz	0.999	0.955	0.954	26.2mVpp

10kΩ Bias Resistor¶

Frequency	Coupling Gain	Divider Gain	Total Gain	Required Input
5 Hz	0.628	0.680	0.427	58.5mVpp
10 Hz	0.783	0.680	0.532	47.0mVpp
50 Hz	0.950	0.680	0.646	38.7mVpp
100 Hz	0.975	0.680	0.663	37.7mVpp
1000 Hz	0.998	0.680	0.679	36.8mVpp
2000 Hz	0.999	0.680	0.679	36.8mVpp

1kΩ Bias Resistor¶

Frequency	Coupling Gain	Divider Gain	Total Gain	Required Input
5 Hz	0.628	0.176	0.111	225mVpp
10 Hz	0.783	0.176	0.138	181mVpp
50 Hz	0.950	0.176	0.167	150mVpp
100 Hz	0.975	0.176	0.172	145mVpp
1000 Hz	0.998	0.176	0.176	142mVpp
2000 Hz	0.999	0.175	0.175	143mVpp

Key Frequency Dependencies¶

High-Pass Corner Frequencies¶

100kΩ: f_c = 0.08 Hz (minimal impact above 5 Hz)
10kΩ: f_c = 0.8 Hz (minor impact at 5-10 Hz)
1kΩ: f_c = 8.0 Hz (significant impact below 50 Hz)

Signal Strength Requirements vs Frequency¶

Relative to 100kΩ baseline at 1000 Hz (26.2mVpp):

Frequency	100kΩ Factor	10kΩ Factor	1kΩ Factor
5 Hz	1.40×	2.01×	8.59×
10 Hz	1.13×	1.61×	6.91×
50 Hz	1.02×	1.45×	5.73×
100 Hz	1.00×	1.43×	5.54×
1000 Hz	1.00×	1.40×	5.42×
2000 Hz	1.00×	1.40×	5.42×

Engineering Implications¶

Low Frequency Performance (5-10 Hz)¶

100kΩ: Excellent performance, <40% penalty at 5 Hz
10kΩ: Good performance, <60% penalty at 5 Hz
1kΩ: Poor performance, 8-9× penalty at 5 Hz

Mid Frequency Performance (50-100 Hz)¶

100kΩ: Baseline reference performance
10kΩ: 40-45% signal strength penalty
1kΩ: 5.5× signal strength penalty

High Frequency Performance (1-2 kHz)¶

100kΩ: Baseline reference (26.2mVpp)
10kΩ: Consistent 40% penalty (36.8mVpp)
1kΩ: Consistent 5.4× penalty (142mVpp)

Recovery Time Analysis (Corrected)¶

Exponential Recovery Formula¶

When the coupling capacitor has stored charge, recovery follows:

V(t) = V_initial × e^(-t/τ)

For practical recovery to LM2907 threshold (~25mVpp):

t_recovery = -τ × ln(1 - α)
where α = required recovery fraction

Recovery Definition: Time until LM2907 pin voltage swing exceeds ~25mVpp detection threshold (not complete return to baseline).

Recovery Target	Time Factor	100kΩ (τ=2.0s)	10kΩ (τ=0.2s)	1kΩ (τ=0.02s)
90% recovery	2.3τ	4.6 seconds	0.46 seconds	0.046 seconds
95% recovery	3.0τ	6.0 seconds	0.60 seconds	0.060 seconds
99% recovery	4.6τ	9.2 seconds	0.92 seconds	0.092 seconds

Practical LM2907 Recovery: Based on ~25mVpp minimum input threshold for reliable frequency detection, 90% recovery provides adequate signal restoration for normal operation.

DC Drift Analysis¶

Mechanism 1: Field Collapse (Primary Issue)¶

When alternator field cuts off, the coupling capacitor retains charge creating a DC offset that decays with time constant τ = R_bias × C_coupling. This is the primary cause of the 5-10 second dropout.

Mechanism 2: TVS Clamping Effects (Secondary)¶

TVS clamping effects occur when node voltage exceeds ~13V:

TVS Conduction Thresholds: - 12V-24V pp operation: No TVS conduction for any configuration - 48V pp operation: TVS conducts for 10kΩ and 100kΩ (node peaks: 16.3V, 22.9V) - 60V pp operation: TVS conducts for 10kΩ and 100kΩ (node peaks: 20.4V, 28.6V) - 1kΩ configuration: No TVS conduction up to 60V pp (max node peak: 5.3V)

During field collapse: Signal levels return to small values (<24V pp), so TVS effects become negligible for the dropout recovery problem.

Technical Tradeoffs Summary¶

Parameter	Current (100kΩ)	Option 1 (1kΩ)	Option 2 (10kΩ)	Assessment
Recovery Time	4.6 seconds	0.046 seconds	0.46 seconds	All options solve dropout
High-Pass Corner	0.08 Hz	8 Hz	0.8 Hz	1kΩ attenuates low RPM
Signal Sensitivity	Reference	5.4× worse	40% penalty	1kΩ needs stronger signals
Power @ 48V pp	~2-3mW	11.8mW	~15-20mW	All acceptable with 0.1W
TVS Immunity @ 48V	Clamps	No clamp	Clamps	1kΩ avoids nonlinearity
TVS Immunity @ 60V	Clamps	No clamp	Clamps	1kΩ avoids nonlinearity
Component Rating	0.1W adequate	0.1W adequate	0.1W adequate	All meet 3× safety minimum
Implementation	N/A	1 resistor	1 resistor	Both simple changes

Recommendation¶

Option 2 (10kΩ bias resistor) provides the optimal balance:

Advantages: - Fast recovery: 0.46 seconds (90% recovery) eliminates dropout problem - Acceptable sensitivity: 40% signal strength penalty vs current design - Minimal frequency impact: 0.8Hz corner barely affects engine frequencies (>5Hz) - Simple implementation: Single component change - Adequate power rating: 0.1W provides 4.5× safety margin at 60V pp with TVS clamping - Better EMI immunity: Lower impedance node vs 100kΩ

Limitations: - TVS conduction at 48V+: Node voltage exceeds 13V at 48V pp and above - Signal strength requirement: Verify stator provides >37mVpp for reliable operation

Requirements: - Component change: Replace 100kΩ bias resistor with 10kΩ, 0.1W rated - Voltage consideration: TVS clamping occurs at 48V pp and above - Recovery verification: Confirm 0.5-second recovery meets system requirements

Alternative: If sub-0.1 second recovery is required and linear operation up to 60V pp is needed, Option 1 (1kΩ) provides 0.046-second recovery with no TVS conduction, but with 5.4× sensitivity penalty and potential low-RPM detection issues due to 8Hz high-pass corner.

Implementation¶

Replace 100kΩ bias resistor with 10kΩ, 0.1W rated component

Benefits: - Eliminates 5-10 second LM2907 dropout - Maintains reasonable signal sensitivity - Preserves low-frequency response - Adequate power rating with 0.1W resistor - Improved EMI immunity

Verification Steps: 1. Confirm 0.5-second recovery meets system response requirements 2. Test signal detection capability with 40% sensitivity reduction 3. For >48V pp operation, verify TVS clamping behavior 4. Validate power dissipation under actual stator voltage conditions