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**Question:** Prove that in active mode BJT can be modeled as a dependent current controlled current source?

or

Show that for a BJT {\beta=\frac\alpha{1-\alpha}}

**Solution**

For the npn transistor, the currents are specified as in Fig 1

Figure 1:currents of a npn transistors

Apply KCL to figure 1 we get

{I_E=I_B+I_c.....................(i)}

Where I_{E}, I_{C}, and I_{B} are emitter, collector, and base currents respectively.

When transistors operate in the active mode, typically V_{BE}≈0.7

{I_C=\alpha I_E.............(ii)}

where α is called the common-base current gain. In Eq. (ii),α denotes the fraction of electrons injected by the emitter that are collected by the collector. Also

{I_C=\beta I_B.............(iii)}

where β is known as the common-emitter current gain. Typically,α takes values in the range of 0.98 to 0.999, while β takes values in the range of 50 to 1000.

From equation (i) we get

{\begin{array}{l}I_E=I_B+I_C\\I_E=I_B+\beta I_B\\I_E=(1+\beta)I_B...........(iv)\\\end{array}}

Again from equation (i) we get

{\begin{array}{l}I_E=I_B+I_C\\\\I_E=I_B+\alpha I_E\;\;\;\lbrack from(i)\rbrack\\\\I_E(1-\alpha)=I_B\\\\I_E=\frac{I_B}{1-\alpha}\\\\(1+\beta)I_B=\frac{I_B}{1-\alpha}\;\;\;\;\;\;\lbrack from\;(iv)\rbrack\\\\\beta=\frac1{1-\alpha}-1\\\\\beta=\frac{1-1+\alpha}{1-\alpha}\\\\\beta=\frac\alpha{1-\alpha}..............(v)\\\end{array}}

**These equations show that, in the active mode, the BJT can be modeled as a dependent current-controlled current source**. Thus, in circuit analysis, the dc equivalent model in Fig. 2(b) may be used to replace the npn transistor in Fig.2(a).

Figure2: (a) An npn transistor, (b) its dc equivalent model

**Since in Eq. (v) is large, a small base current controls large currents in the output circuit.**

prove that in active mode bjt can be modeled as a dependent current controlled current source