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The transistors have finite β and an infinite early voltage. Derive the expression for \$ I_0 \$ in terms of \$ I_{REF} \$and β.

In this question Io has to be represented in terms of \$I_{REF} \$.... The solution matches to that given in text book. But they have perhaps done it without considering the doubled area of transistor Q3. I have done the same yet. My question is that what would be the effect on analysis ie whether there will be any change in the analysis(current equations) due to the doubling of area of transistor Q3 as shown?

What is the effect of doubling of the emitter area of transistor Q3 for this current mirror circuit as shown?

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Attempted Soln(Without considering the doubled area of transistor Q3):

$$I_{REF}=I_{C1}+I_{B3} \tag 1$$ Since Q1 and Q2 are matched $$I_{C1}=\beta I_{B1}=\beta I_{B2}=I_{C2} \tag 2$$

$$I_{E3}=I_{C2}+I_{B1}+I_{B2}$$

From eq (2)

$$I_{E3}=I_{C1}+2I_{B1}$$

$$I_{E3}=I_{C1}+2\frac{I_{C1}}{\beta}$$ $$I_{E3}=I_{C1}(1+\frac{2}{\beta})$$ $$I_{C1}=\frac{I_{E3}}{(1+\frac{2}{\beta})} \tag 3$$

$$I_{E3}=\frac{\beta}{\beta +1}I_0 \tag 4$$

From (3) and (4) $$I_{C1}=\frac{\beta +1}{\beta +2} I_0$$

Eq (1) Becomes:

$$I_{REF}=\frac{I_0}{\beta}+\frac{\beta +1}{\beta +2} I_0$$ $$I_{REF}=I_0(\frac{\beta +2+{\beta}^2 +\beta}{\beta(\beta+2)})$$ $$I_0=\frac{1}{1+\frac{2}{\beta(\beta+2)}}I_{REF}$$ $$$$

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    \$\begingroup\$ Doubling the emitter area doubles \$I_{SAT}\$ and therefore reduces \$V_{BE}\$ by \$26\:\textrm{mV}\cdot\operatorname{ln}\left(2\right)\approx 18\:\textrm{mV}\$. It also reduces resistance. I don't think it affects \$\beta\$ as the terms there don't include emitter area. If the question is asking you to estimate a change in stiffness of the current sink, I don't think that's much affected either (lessened only slightly.) \$\endgroup\$ – jonk Jun 30 '17 at 15:53
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Doubling the emitter area of Q3 will drop that Vbe by 18 millivolts, reducing the Vce of Q1 by 18 millivolts.

Without knowing the Early Voltage, and the plot of how beta varies with Ic, I think this (doubled emitter area) is a process-specific and Imirror-specific tweak.

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  • \$\begingroup\$ The question is: Consider the Wilson current source in the figure. The transistors have finite \$ \beta \$ and an infinite early voltage. Derive the expression for \$ I_0 \$ in terms of \$I_{REF} \$ and \$ \beta \$. Sir can you provide the mathematical/intuitive reasoning to understand why this phenomenon happens. Will the emitter current of Q3 double because of doubling of area of E in Q3? \$\endgroup\$ – Soumee Jun 28 '17 at 7:48

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