If adding emitter resistors can reduce the temperature variation in \$V_{BE}\$, why do we still need to find a matched pair of transistors to construct a current mirror?
Electronic – Matched biasing transistor vs. emitter resistor in current mirror
bjtcurrent-mirror
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Best Answer
You are right about the emitter resistor reducing the temperature variation of the current mirror. However, not the collector current is mirrored but the base current / voltage is mirrored, meaning that you still have some dependency on the gain of the second pair of your current mirror.
In the following circuit, the current through \$Q_1\$ will be approximately given by:
$$I_{C,Q_1} \propto I_B\cdot \beta_{Q_1}$$
$$I_{C,Q_1} \propto I_{C, Q_2}\dfrac{\beta_{Q_2}+1}{\beta_{Q_1}+1} \dfrac{\beta_{Q_1}}{\beta_{Q_2}}$$
simulate this circuit – Schematic created using CircuitLab
EDIT #1
Consider the following circuit:
The base voltage which is common for both transistor is given approximately by (for the sake of simplicity the internal emitter resistance is neglected ):
$$V_B = V_{BE,1}+ R_{E,1} (I_{C,1}(1+\dfrac{1}{\beta_1})) = V_{BE,2}+ R_{E,2} (I_{C,2}(1+\dfrac{1}{\beta_2}))$$
For a matched transistor and resistor, the above formula yields:
$$I_{C,1}=I_{C,2}$$
Assumming now that, \$\beta\$ is not matched, the collector current of \$Q_2\$ is given by:
$$I_{C,2}=I_{C,1}\dfrac{\beta_1 +1}{\beta_2 +1}\dfrac{\beta_2}{\beta_1}$$
As already explained in this question, \$\beta\$, which is defined by collector to base current ratio, has some temperature dependence given by:
$$\beta = \dfrac{I_C}{I_B}=\dfrac{I_C}{I_E - I_C}$$ where \$I_E\$ can be written in terms of thermal voltage \$V_T\$, which is dependent on the temperature according to:
$$V_T=\dfrac{k_BT}{q}$$
The following simulation, can show that although not very significant, the beta of the transistor still play a role on the mirrored current despite the introduction of matched emitter degeneration resistors.
This simulation, modifies the beta of the second transistor, by adding a \$\pm 50%\$ tolerance to its nominal beta value. All other parameters are left untouched. Furthermore, the simulation is run for 3 different temperatures in order to accout their variations on the final collector curent.
As you can see in the above plot, the output current (\$I_{Q,2}\$) has a dependence on the temperature, and consequently on beta.