BJT Transistor Nodal Analysis

Basic BJT Equations:

i_C = \alpha i_E \hspace{20 mm} i_B = (1- \alpha ) i_E = \frac{i_E}{\beta + 1} i_C = \beta i_B \hspace{20 mm} i_E = ( \beta + 1) i_B \beta = \frac{ \alpha }{ 1 - \alpha } \hspace{21mm} \alpha = \frac{ \beta }{ \beta + 1 }

It is also important to know that |V_{EB}| = |V_{BE}| can be modeled as .7V .

These equations are not very informative by themselves so a few examples are demonstrated below. In both examples we will assume \beta is very large. What this means for our calculations is i_B \approx 0 . Since i_B \approx 0 we also assume that i_C \approx i_E .

Finding missing voltages in a BJT circuit

Example 1. Solve for V3:

bjt-voltagesThere are several ways to find V_3 . The more “difficult” way is to first find the emitter current, i_E , and then use Ohm’s Law. Since we know i_C \approx i_E , we can find the collector current, i_C , and then solve for V_3 .

\frac{-4-(-10)}{2.4k } = i_C = 2.5mA 12 - (i_E)(5.6k) = 12 - (i_C)(5.6k) = V_3 V_3 = 12 - (2.5mA)(5.6k) = -2 V

The easier way to find V_3 is to recall that |V_{EB}| behaves like a diode. For this pnp BJT: V_{EB} = V_E - V_B = .7 .

We know that V_B = -2.7V so V_E = V_3 = -2 V

\beta may not always be a very large number. Had that been the case here, we would have started by finding the collector current (since it’s voltage drop and resistance are given) and since i_B \neq 0 anymore, we would use the formulas above to the find the base and collector current.

Finding BJT Bias Voltages and Currents

Example 2 Solve for V2 and I1:


Here we will want to start by finding I_1 . I_1 also equals I_E which approximately equals I_C and this collector current will allow us to find V_2 .

I_1 = \frac{10.7-.7}{10k} = 1mA V_2 = (1mA)(10k) - 10.7 = -.7 V

Notice that V_E was given as .7 V .  If this had not been given, we would have been able to find it because V_{EB} = V_E - V_B = .7 V and V_B = 0 V .

Similar to the previous example, if \beta was not a very large number. We would first find the emitter current and then use the equations in the table to find the other branch currents.

Note that both of these examples used pnp BJTs. The difference in an npn BJT is the base-emitter voltage is reveresed. You would use V_{BE} = V_B - V_E = .7 V.

General Rule of Thumb

Most of these problems are very simple to solve. Typically \beta is given and you will need to use Ohm’s Law to identify one of the currents. After one of the currents is found you will be able to solve for the other currents using the basic equations listed above. If one of the currents is not immediately obvious, the base-emitter voltage is likely needed. Most problems have you deduce the emitter voltage from the base, but it is easily possible to find the base voltage from the emitter voltage and then use that to find the base current.

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