GAIN SELECTION AND SIGNAL CONDITIONING VALUES

-Thomas Shaw

Overview

Purpose: Scale the sensed load voltage and load current into a range the ATmega328PB ADC can read accurately, with headroom for tolerances and noise.

• Input: 12.6–15.4 Vrms, 500 Hz ± 2% AC source, 2.5–7.5 VA load
• Output: two AC signals centred on 1.65 V, kept within 0.25–3.05 V
• ADC reference: 3.3 V AREF → 3.22 mV per LSB
• Op-amp: TLC082 (dual), single 5 V supply

Test Cases

Inductance fixed at L = 4 mH to test extreme values. R is chosen so the load reaches the VA limit. All cases at 500 Hz, so X_L = 2π·500·0.004 = 12.57 Ω.

|Z| = V² / S
R   = √(|Z|² − X_L²)
PF  = R / |Z|
I_pk = √2 · V / |Z|

Case	V_rms	S	|Z|	R	PF	I_rms	I_pk	P
1	12.6 V	7.5 VA	21.17 Ω	17.03 Ω	0.805	595.2 mA	841.8 mA	6.04 W
2	15.4 V	7.5 VA	31.62 Ω	29.02 Ω	0.918	487.0 mA	688.7 mA	6.88 W
3	15.4 V	2.5 VA	94.86 Ω	94.03 Ω	0.991	162.3 mA	229.6 mA	2.48 W
4	12.6 V	2.5 VA	63.50 Ω	62.25 Ω	0.980	198.4 mA	280.6 mA	2.45 W

• Case 1 → highest current, lowest PF
• Case 3 → lowest current, highest PF
• With L fixed at 4 mH, PF cannot go below ~0.80 inside 2.5–7.5 VA. Hitting the 0.6 PF spec corner at 7.5 VA would need ~5.4–8.1 mH.

ADC Window: 0.25 V to 3.05 V

A 0.25 V safety net is left on either side of the 0–3.3 V ADC range.

Low side (0.25 V): limited by the op-amp

The TLC082 is not rail-to-rail on its output. From the datasheet at VDD = 5 V:

Parameter	Condition	Value
V_OL	I_OL = 1 mA, 25 °C, typ	0.18 V
V_OL	I_OL = 1 mA, 25 °C, max	0.25 V
V_OL	I_OL = 1 mA, full temp range, max	0.35 V

• The output is only guaranteed to pull down to 0.25 V, so anything lower would clip.
• The ADC input draws almost no current, so the real V_OL sits closer to 0.18 V and gives some margin.

High side (3.05 V): limited by AREF

Parameter	Condition	Value
V_OH	I_OH = −1 mA, 25 °C	~4.1–4.3 V
V_OH	I_OH = −1 mA, full temp range, min	3.9 V

• The op-amp can swing well past 3.3 V, so the ADC saturates first (code 1023).
• The 0.25 V margin below AREF absorbs:
  • AREF tolerance
  • resistor tolerance
  • op-amp offset × gain (V_IO ≤ 1.9 mV)
  • noise and harmonics

Resolution

Centre    = 3.3 / 2 = 1.65 V
Swing     = ±1.40 V
Span      = 3.05 − 0.25 = 2.8 V
Codes     = 2.8 / 3.22 mV ≈ 869 of 1024 (85%)
Margin    = 0.25 V ≈ 78 LSB each side

Current Sensing

Shunt: 2 × 1 Ω in parallel (R6 ∥ R7)

R_shunt   = 1 ∥ 1 = 0.5 Ω
P_total   = I_rms² × 0.5
P_each    = (I_rms / 2)² × 1

Case	I_rms	V_is,pk	P_total	P per resistor	Peak inst. per resistor
1	595.2 mA	0.421 V	177.1 mW	88.6 mW	177 mW
2	487.0 mA	0.344 V	118.6 mW	59.3 mW	119 mW
3	162.3 mA	0.115 V	13.2 mW	6.6 mW	13 mW
4	198.4 mA	0.140 V	19.7 mW	9.8 mW	20 mW

• Worst case is 88.6 mW per resistor, which is about 35% of a 0.25 W rating. This keeps within the usual 50% derating.
• A single 0.5 Ω resistor would dissipate 177 mW, about 71% of rating, which is why the shunt is split.
• The shunt adds < 3% to load impedance, so its effect on load current is negligible.

Current amplifier (U4B)

Difference amplifier with matched ratios:

Ref	Value	Role
R9	10 kΩ	Inverting input (to GND)
R11	33 kΩ	Feedback
R10	10 kΩ	Non-inverting input (from V_is)
R12	33 kΩ	To 1.65 V reference

R11 / R9 = R12 / R10 = 3.3
V_im = 3.3 · V_is + 1.65 V

Gain choice:

• Worst case V_is,pk = 0.421 V (case 1) → 3.3 × 0.421 = 1.389 V, just under the ±1.40 V budget.
• Required transimpedance ≈ 1.40 / 0.8418 ≈ 1.66 V/A. Design gives 0.5 Ω × 3.3 = 1.65 V/A.

Case	V_im range	Amplitude
1	0.29 – 3.01 V	1.363 V
2	0.53 – 2.77 V	1.120 V
3	1.27 – 2.03 V	0.377 V
4	1.19 – 2.11 V	0.459 V

• The low-current cases only use about 27% of the range: ~237 codes peak-to-peak, ~0.6 mA_pk per code.
• This is still well within the ±5% full-scale accuracy spec.

Voltage Sensing

Divider (R4, R8)

Ref	Value
R4	270 kΩ
R8	15 kΩ

k = R8 / (R4 + R8) = 15 / 285 = 0.05263

• High resistance keeps divider current low (~54 µA at 15.4 V), so it has negligible effect on the load.

Thevenin equivalent of divider

Seen from V_vs, the divider acts as a source with internal resistance:

R_th = R4 ∥ R8 = 270k ∥ 15k = 14.21 kΩ
V_th = V_is + k · (V_ac − V_is)

R_th sits in series with R22, so the non-inverting input sees:

R22 + R_th = 68k + 14.21k = 82.21 kΩ ≈ R20 (82 kΩ)

This is why R22 = 68 kΩ rather than 82 kΩ. Using 82 kΩ would give ~96 kΩ effective and unbalance the amplifier.

Voltage amplifier (U5B)

Ref	Value	Role
R20	82 kΩ	Inverting input (from V_is)
R19	100 kΩ	Feedback
R22	68 kΩ (+14.21 kΩ R_th)	Non-inverting input (from V_vs)
R23	100 kΩ	To 1.65 V reference

G    = R19 / R20 = 100 / 82 = 1.2195
V_vm ≈ G · k · (V_ac − V_is) + 1.65
     ≈ 0.0641 · V_load + 1.65 V

Gain choice:

• Worst case V_pk = 15.4 × √2 = 21.78 V.
• Required gain = 1.40 / 21.78 = 0.0643. Design gives 0.0641.
• The differential input (V_vs − V_is) cancels the shunt voltage, so V_vm tracks the voltage across the load only.

Case	V_vm range	Amplitude
1	0.53 – 2.77 V	1.121 V
2	0.28 – 3.03 V	1.376 V
3	0.26 – 3.04 V	1.389 V
4	0.52 – 2.79 V	1.133 V

Difference Amplifier Mismatch

Nominal mismatch

Inverting ratio      R20 / R19          = 0.8200
Non-inverting ratio  (R22 + R_th) / R23 = 0.8221
ε = +0.26%

Effect	Value
Common-mode gain A_cm ≈ G·ε / (1 + G)	−0.0014 V/V
Differential gain A_d	1.219 V/V
CMRR	≈ 59 dB
Error from V_is (0.421 V_pk)	≈ 0.6 mV_pk (~0.05%)
DC offset	1.652 V (+2.3 mV, ~0.7 LSB)

The nominal mismatch is negligible.

Resistor tolerance

Ratio mismatch ε	Cause	A_cm	V_is error (pk)	Offset shift
0.26%	Nominal values	0.0014	0.6 mV	2.3 mV
1%	0.25–0.5% resistors	0.0055	2.3 mV	9 mV
2%	1% resistors, typical	0.011	4.6 mV	18 mV
4%	1% resistors, worst case	0.022	9.3 mV	36 mV

• Even at 4%, the V_is leakage is ~0.7% of full scale, well within ±5% spec.
• The leaked term is in phase with current, not voltage, so it barely affects V_rms.
• **Firmware must subtract the measured