#### Theory of Static Characteristics III and Transfer Characteristic

Inverter Design:

Figure: Bipolar Transistor Inverter Circuit

Usually select IC mid range for a high value of βF.

Let consider a supply of  = 5V.

A usual mid-range value of IC would be 3 to 5 mA.

For IC = 5mA,

Rc = VCC/ICMAX =1 KΩ

A usual βF = 50 for an integrated transistor.

When we employ a base overdrive factor of σu = 5. Then,

σu = βFRC/R => 5 = (50 x 1 KΩ)/RB => RB = (50 KΩ)/5 = 10 KΩ

The idea is just to ensure that RB gives sufficient current to overdrive the transistor. Rewriting the relationship, the selection is really:

RB = (βFu) Rc

Operation and Output Characteristic:

Figure: Collector Current as the function of Base-Emitter Bias

The collector current which flows via the transistor depends on the bias applied to the base emitter junction according to normal exponential law for the p-n junction.

Generally,

VBE cut - in = 0.6V

VBE on = 0.7V

VBE sat = 0.8V

Referring to the output characteristic of figure shown below that shows an operating load line, the below can be seen. At very low input voltages, VBE is small and no current flows in the collector of transistor, apart from certain small leakage current. Therefore, the output voltage is at Vo = VCE = VCC. If the input voltage to inverter is slowly raised, ultimately the base-emitter junction reaches the point of cut-in if Vi = VBE CUT-IN at point A on the characteristic as shown in figure below. Here, IC ≈ 0 and hence Vo = VCE = VCC still. Since the input voltage is further raised, base current begins to flow and the transistor enters the forward active mode where IC = βF IB and the operating point travels upward all along the load line. Ultimately, the point B is reached where IB = ICMAX/ βF and the transistor arrives at the edge of saturation. In this condition, the collector current reaches its maximum value and the output voltage drops to its minimum of Vo = VCESAT ≈ 0.1 – 0.2 V. Further raise in the input voltage overdrives the transistor deeper into saturation however has little effect on collector current or output voltage.

Figure: Output IC vs. VCE Characteristic for Single Transistor Inverter

Transfer Characteristic:

The transfer characteristic of a logic gate is just a plot of steady-state output voltage vs. input voltage over its range of operation such as that shown in figure below. Initially if Vi = 0, then T is OFF and hence IC = 0 and VO = VCC. As Vi is slowly raised, a point is ultimately reached at A, where the transistor starts to turn ON. This is termed to as the cut-in point for the transistor or the “edge of conduction”. Since Vi is further raised, the transistor becomes fully conducting and enters the forward active mode among the points A and B on the characteristic. As the collector current rises, the output voltage drops from VCC towards ground till the transistor reaches the edge of saturation at point B. The slope of characteristic among points A and B is basically determined by the gain of circuit. Finally, as the transistor is driven fine into saturation, the output voltage levels off at Vo = VCESAT. Critical points on the characteristic are points A and B as these are points that define the transition region in the output between the high and low logic level. They as well define the boundaries of operation between, Cut-off, Forward Active and Saturation modes of operation of the transistor.

Logic Voltages:

Output Levels: From the transfer characteristic for single transistor inverter, it can be seen that the output HI and LO logic voltages are fine defined.

VOL = VCESAT; VOH = VCC

Input Levels: The input logic voltages are not as evidently defined and can occupy ranges where the output remains at the accurate logic level. The critical points, A and B, on the characteristic can be employed as critical points to define the limiting values for such ranges.

Point A defines the input voltage that just starts to turn on the transistor and is, therefore, the maximum input LO voltage.

Therefore:

ViLMAX = VBE CUT-IN; Usually 0.6 V

Point B defines the minimum input voltage that will just keep the transistor at the edge of saturation region and therefore as well at the edge of linear forward active region. At this point:

Vi = ViHMIN

ICMAX = βF IB

(VCC – VCESAT)/RC = βF (ViHMIN - VBESAT)/RB

[(VCC – VCESAT) RB]/ βF RC = ViHMIN - VBESAT

ViHMIN = VBESAT + (RB/ βF RC) (VCC – VCESAT) = VBESAT + (1/σu) (VCC – VCESAT)

For component values which are established above:

ViHMIN = 0.8 + (1/5) (5 – 0.2) = 0.8 + 0.2 x 4.8 = 0.8 + 0.96 = 1.76 V

Figure: Transfer Characteristic of Bipolar Transistor Inverter

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