# BJT Mismatch parameters # Mismatch parameters [top]

Bipolar mismatch can happen due to 3 parameters:
1. `\$I_S\$` Mismatch
2. `\$β\$` Mismatch
3. `\$R_E\$` Mismatch

There are 2 methods to extract the Bipolar Mismatch parameters. Let us analyze both methods to determine which method represents all 3 mismatch sources adequately.

# Measurement Method 1 [top] In this method the setup is made as shown above. Here the emitter currents are the same and the difference in the Vbe’s can be directly measured in the 2 side by side devices.
Wehave:
`\$\$ΔV_{BE}=V_{BE1}-V_{BE2}\$\$`

## `\$I_S\$` Mismatch only [top]

If the BJT has `\$I_S\$` mismatch only:
`\$\$ΔV_{BE}=V_{BE1}-V_{BE2}=nV_Tln{I_EI_{S2}}/{I_EI_{S1}}=nV_TlnI_{S2}/I_{S1}\$\$`

`\$\$ΔV_{BE}≈nV_T{ΔI_{S}}/I_{S}\$\$`

## Beta mismatch Only [top]

If BJT has `\$β\$` mismatch only:
`\$\$ΔV_{BE}=V_{BE1}-V_{BE2}=nV_TlnI_{C1}/I_{C2}\$\$`

`\$\$I_{C1}=β_1/{1+β_1}I_E\$\$`

`\$\$I_{C2}=β_2/{1+β_2}I_E\$\$`

`\$\$ΔV_{BE}=V_{BE1}-V_{BE2}=nV_Tln({β_1(1+β_2)}/{β_2(1+β_1)})\$\$`

Set:
`\$\$β_1=β+{Δβ}/2\$\$`

`\$\$β_2=β-{Δβ}/2\$\$`

`\$\$ΔV_{BE}=nV_Tln({(β+{Δβ}/2)(1+β-{Δβ}/2)}/{(β-{Δβ}/2)(1+β+{Δβ}/2)})\$\$`

`\$\$ΔV_{BE}=nV_Tln(1+{Δβ}/{β(1+β)-{Δβ}/2-({Δβ}/2)^2})\$\$`

`\$\$ΔV_{BE}=nV_T{Δβ}/{β(1+β)}\$\$`

## Re Mismatch only [top]

If BJT has `\$R_E\$` mismatch only:
`\$\$ΔV_{BE}=V_{BE1}-V_{BE2}=I_E(R_{E1}-R_{E2})\$\$`

# Measurement Method 2 [top] In this method the setup is made as shown above. Here the Vbe’s of both side by side transistors are forced equal and their collector currents can be measured.
`\$\$ΔV_{BE}=V_Tln{I_{C2}}/I_{C1}\$\$`

## `\$I_S\$` Mismatch only [top]

`\$\$I_{C1}=I_{S1}exp(V_{BE}/{nV_T})\$\$`
`\$\$I_{C2}=I_{S2}exp(V_{BE}/{nV_T})\$\$`

`\$\$I_{C2}/I_{C1} = I_{S2}/I_{S1}\$\$`

`\$\$ΔV_{BE}=V_Tln{I_{S2}}/I_{S1}\$\$`

## Beta mismatch Only [top]

Since `\$I_S\$` will be same and `\$V_{BE}\$` is the same the measured `\$I_C\$` will be the same so we do not detect any mismatch due to Beta mismatch.

## Re Mismatch only [top]

`\$\$ΔV_{BE}=V_Tln{I_{C2}}/I_{C1}\$\$`
`\$\$I_{C1}=αI_{E1}\$\$`

`\$\$I_{C2}=αI_{E2}\$\$`

`\$\$I_{C2}/I_{C1}=I_{E2}/I_{E1}\$\$`

`\$\$ΔV_{BE}=V_Tln{I_{E2}}/I_{E1}\$\$`

`\$\$I_{E1}R_{E1}+V_{BE1}=I_{E2}R_{E2}+V_{BE2}\$\$`

`\$\$I_{E1}=I_T+{ΔI}/2\$\$`

`\$\$I_{E2}=I_T-{ΔI}/2\$\$`

`\$\$R_{E1}=R_E+{ΔR_E}/2\$\$`

`\$\$R_{E2}=R_E-{ΔR_E}/2\$\$`

`\$\$I_{E1}R_{E1}-I_{E2}R_{E2}=I_TΔR_E+R_EΔI\$\$`

`\$\$⇒I_TΔR_E+R_EΔI=nV_Tln{I_{E2}}/I_{E1}\$\$`

`\$\$⇒I_TΔR_E+R_EΔI=nV_Tln(1-{ΔI}/I_{E1})\$\$`

`\$\$⇒I_TΔR_E+R_EΔI≈-nV_T{ΔI}/I_{E1}=-nV_T{I_{E1}-I_{E2}}/I_{E1}=-nV_T(1-I_{E2}/I_{E1})\$\$`

Setting `\$ΔI=I_{E1}{I_{E1}-I_{E2}}/I_{E1}≈I_{T}(1-I_{E2}/I_{E1})\$`
`\$\$⇒I_TΔR_E+R_EI_{T}(1-I_{E2}/I_{E1})=-nV_T(1-I_{E2}/I_{E1})\$\$`

Solving for `\$I_{E2}/I_{E1}\$`:
`\$\$I_{E2}/I_{E1}=1+{I_TΔR_E}/{nV_T+I_TR_E}\$\$`

`\$\$∴ΔV_{BE}=V_TlnI_{E2}/I_{E1}=V_Tln(1+{I_TΔR_E}/{nV_T+I_TR_E})≈V_T{I_TΔR_E}/{nV_T+I_TR_E}\$\$`

# Comparison of the Methods [top]

Mismatch Factor Method 1 Method 2Comment
`\$I_S\$``\$ΔV_{BE}=nV_TlnI_{S2}/I_{S1}\$``\$ΔV_{BE}=V_Tln{I_{S2}}/I_{S1}\$`Both methods represent this mismatch
`\$β\$``\$\$ΔV_{BE}=nV_T{Δβ}/{β(1+β)}\$\$``\$ΔV_{BE}=0\$`Method 2 cannot measure the mismatch due to Beta
`\$R_E\$``\$ΔV_{BE}=I_E(R_{E1}-R_{E2})\$``\$ΔV_{BE}≈V_T{I_TΔR_E}/{nV_T+I_TR_E}\$`Method 1 gives a better reading

This Method 1 should be the method used to measire the mismatch parameter for a BJT since it includes effects from all 3 mismatch parameters.
NOTE: Mismatch will increase with temperature due to `\$V_T\$` dependance