SLYA042 July 2024 FDC1004 , FDC1004-Q1

11 Derivation of V_{OS, EXT} for a Single Stage Current Sense Amplifier (CSA)

This derivation assumes a single-stage, linear CSA. Derivation first begins by determining V_B (bias voltage across bias resistor) when load (I_L) is 0A as this is essentially V_{OS, EXT RTI}. Then this is divided by gain error factor (GEF) to get the referred-to-shunt offset error (V_{OS, EXT RTS}). Note that this derivation also requires a more specific equation for GEF which assumes that the R_EXT value at each pin can be different.

Equation 24.

L e t \frac{I_{O S}}{2} = I_{O S 2} L e t R_{F B} + R_{I N T} = R_{F I} L e t I_{L} = 0 - A (1) I_{S H} - I_{N} - I_{L} = 0 (2) I_{P} - I_{X} - I_{B} - I_{C M, O N} = 0 (3) I_{N} - I_{Y} + I_{B} - I_{C M, O N} = 0 (4) I_{Y} + I_{O S 2} - I_{F B} = 0 (5) I_{X} - I_{O S 2} - I_{R E F} = 0 (6) V_{C M} - I_{S H} R_{S H} - I_{N} R_{F 2} - I_{F B} R_{F I} - V_{O U T} = 0 (7) V_{C M} - I_{P} R_{E X T 1} - I_{R E F} R_{F I} - I_{F B} R_{F I} - V_{R E F} = 0 (8) V_{C M} - I_{R E F} R_{I N T} + + I_{F B} R_{I N T} = 0 (9) V_{O U T} = V_{R E F} + V_{B} \frac{R_{F B}}{R_{I N T}} (10) V_{S H} - I_{P} R_{E X T 1} - V_{B} + R_{E X T 2} I_{N} = 0

(4) into (3) and (5) into (2)

Equation 25.

I_{P} = I_{C M, O N} + I_{B} + I_{O S 2} + I_{R E F} I_{N} = I_{C M, O N} - I_{B} - I_{O S 2} + I_{F B}

(2) into (7)

Equation 26.

I_{R E F} = \frac{V_{C M} - V_{R E F} - R_{E X T 1} (I_{C M, O N} + I_{B} + I_{O S 2})}{R_{F I} + R_{E X T 1}}

(7) into (8)

Equation 27.

I_{F B} = \frac{V_{C M} - V_{R E F} - R_{E X T 1} (I_{C M, O N} + I_{B} + I_{O S})}{R_{F I} + R_{E X T 1}} - \frac{V_{B}}{R_{I N T}}

(1), (3), and (9) into (6).

Equation 28.

V_{B U S} - V_{R E F} - V_{B} (\frac{R_{F B}}{R_{I N T}}) + V_{B} (\frac{R_{S H} + R_{E X T 2})}{R_{B}} - (I_{C M} - I_{O S 2}) (R_{S H} + R_{E X T 2}) - I_{F B} (R_{F I} + R_{S H} + R_{E X T 2}) = 0

(8) into (6). Solve for V_B

Equation 29.

L e t C_{E V} = \frac{R_{F I} + R_{S H} + R_{E X T 2}}{R_{F I} + R_{E X T 1}} = \frac{1 + \frac{R_{S H} + R_{E X T 2}}{R_{F I}}}{1 + \frac{R_{E X T 1}}{R_{F I}}} V_{B} = \frac{(V_{R E F} - V_{B U S}) (1 - C_{E V}) + I_{B, C M O N} (R_{S H} + R_{E X T 2} - R_{E X T 1} C_{E V}) - \frac{I_{O S}}{2} (R_{S H} + R_{E X T 2} + R_{E X T 1} C_{E V})}{1 + (R_{S H} + R_{E X T 2}) (\frac{1}{R_{I N T}} + \frac{1}{R_{B I A S}}) + \frac{R_{E X T 1} C_{E V}}{R_{B I A S}}} V_{B} = V_{O S, E X T R T I} w h e n I_{L} = 0

Determine Gain Error Factor when R_EXT1 does not equal R_EXT2. You can simply remove I_{CM, ON}, I_OS sources, and set V_CM and V_REF sources to 0V as these only affect the offset. Gain is constant throughout operational V_CM.

(5) into (2)

Equation 30.

L e t I_{C M, O N} a n d I_{O S} = 0 L e t V_{C M} = V_{R E F} = 0 I_{P} = I_{B} + I_{R E F}

(2) into (7)

Equation 31.

I_{R E F} = \frac{0 - 0 - R_{E X T 1} (I_{B} + 0 + 0)}{R_{F I} + R_{E X T 1}}

(2), (8) into (10).

Equation 32.

V_{S H} - V_{B} - I_{B} (R_{E X T} + R_{E X T 2}) - I_{R} R_{E X T 1} + I_{R} R_{E X T 2} - \frac{V_{B} R_{E X T 2}}{R_{I N T}} = 0

(7) into (10). Solve for V_B/V_SH = GEF

Equation 33.

L e t C_{E G} = \frac{R_{E X T 1}}{R_{F I} + R_{E X T 1}} \frac{V_{B}}{V_{S H}} = G E F = \frac{1}{(1 + \frac{R_{E X T 2}}{R_{I N T}}) + \frac{R_{E X T 1} + R_{E X T 2}}{R_{B}} - \frac{C_{E G} (R_{E X T 1} - R_{E X T 2})}{R_{B}}} V_{O S, E X T R T S} = \frac{V_{O S, E X T R T I}}{G E F}

11 Derivation of VOS, EXT for a Single Stage Current Sense Amplifier (CSA)

11 Derivation of V_{OS, EXT} for a Single Stage Current Sense Amplifier (CSA)