Characteristic Curves and Transistor Equations

A transistor has three terminals, and its behavior depends on two voltages and two currents. To present these dependencies clearly, they are resolved into characteristic curve fields: one curve shows the relationship between two quantities while a third is held fixed as a parameter. For the bipolar transistor these are the input characteristic I_B(U_BE), the transfer characteristic I_C(U_BE) or I_C(I_B), and the output characteristic field I_C(U_CE) with I_B as parameter. For the MOSFET the input characteristic disappears, since the gate current is practically zero in static operation; what remains are the transfer characteristic I_D(U_GS) and the output characteristic field I_D(U_DS) with U_GS as parameter.

Characteristic curves describe the device alone. Only the external circuit determines where on the curve the transistor actually operates. For a load resistor R_L between supply voltage U_DD and drain, U_DS = U_DD − I_D·R_L holds – a straight line in the output characteristic field, the load line. Its intersection with the curve for the currently applied gate voltage is the operating point. If U_GS changes, the operating point moves along the load line; this movement is precisely the amplification mechanism of a transistor. The location of the operating point determines whether a transistor acts as a linear amplifier, a switch, or a current source.

Transistor equations are mathematical approximations that describe these curves with a few parameters. They allow circuits to be calculated by hand, but they are models with a limited range of validity. The models in circuit simulators such as SPICE are considerably more complex, yet build on these same underlying equations.

The basic structure of a bipolar transistor is covered in the chapter Bipolar Transistors; the following describes its electrical behavior in operation.

The input characteristic of an npn transistor is the diode characteristic of the forward-biased base-emitter junction: base current rises exponentially with U_BE, becoming significant only above about 0.6 V for silicon. Because collector current is coupled to base current through current gain B, the transfer characteristic I_C(U_BE) is likewise exponential. A change in U_BE of only 60 mV multiplies the collector current tenfold at room temperature – the bipolar transistor is an extremely steep, but also temperature-sensitive, converter of voltage into current.

The output characteristic field I_C(U_CE) shows, for each base current, a curve that rises steeply at small U_CE and then transitions into a nearly horizontal branch. The steep region below about 0.2 V is the saturation region: both pn junctions are forward-biased, and the transistor behaves like a closed switch with a small residual voltage U_CEsat. In the horizontal branch, the active region, the base-collector junction is reverse-biased, and collector current is determined almost entirely by base current – here the transistor operates as an amplifier. The slight upward slope of the curves in the active region is the Early effect: as U_CE increases, the space-charge region of the collector junction widens into the base, the effective base width shrinks, and collector current rises. Extending the curves to the left, they intersect the voltage axis approximately at a common point, the Early voltage −U_A (typically 50 to 150 V).

With no base current, only a negligible residual current flows – the transistor is off. At sufficiently high collector-emitter voltage, avalanche breakdown of the collector junction eventually sets in; the breakdown voltage U_CEO marks the upper limit of the permissible operating range.