Differences in Early Voltage \$V_A\$ in \$V_{CE}\$ vs. \$I_C\$ Graphs for Constant \$I_B\$ and Constant \$V_{BE}\$

Most of the techniques for measuring \$V_{_\text{A}}\$ become relatively inaccurate as \$V_{_\text{A}}\to\infty\$. But this isn't a problem because the effect of basewidth modulation (\$V_{_\text{A}}\$) becomes far less important as \$V_{_\text{A}}\to\infty\$.

For practical values of \$V_{_\text{A}}\$ a better method is to hold \$V_{_\text{BE}}\$ constant and set \$V_{_\text{BC}}\$ to two different values and measure the collector current for each. One of the measurement points should be made with \$V_{_\text{BC}}\approx 0\:\text{V}\$ (near). The other can be made with \$\vert V_{_\text{BC}}\vert \approx 1\:\text{V}\$, for example.

Also, \$V_{_\text{BE}}\$ should be kept small enough that the collector currents are as low as practical in order to mitigate the effects of the bulk resistances in the emitter, base, and collector.

In such a case, only two measurement points are needed and the resulting value is readily computed as:

$$V_{_\text{A}}=\frac{V_{_{\text{BC}_2}}\cdot I_{_{\text{C}_1}}-V_{_{\text{BC}_1}}\cdot I_{_{\text{C}_2}}}{I_{_{\text{C}_2}}-I_{_{\text{C}_1}}}$$

For example:

One might then compute:

$$V_{_\text{A}}=\frac{1\:\text{V}\cdot 118.53245\:\mu\text{A}-0\:\text{V}\cdot 119.71668\:\mu\text{A}}{119.71668\:\mu\text{A}-118.53245\:\mu\text{A}}\approx 100.09$$

And that is correct for the LTspice simulation model of the 2N2222.

The above technique assumes that basewidth modulation, that part due to variation in the width of the emitter-base space charge layer, is negligible (the so-called Late Effect.) This is a valid approach. But if curious to test the idea, then just repeat the above but at two different \$V_{_\text{BE}}\$ constant values, instead. If it changes much then there is a significant Late Effect and the Gummel-Poon model should be used.

Hopefully, this suggests a simple procedure you can try out. Best wishes.

(see the update at the end.)

The slope of the Ic=f(Vce) curves at a certain fixed operational point (Ic,o, Vce,o) within the active region can be expressed using small-signal parameters (h21=hfe, h11=hie):

• hybrid parameters: For Ib=const the slope is dI/dV=h22
• y-parameters: For Vbe=const the slope is dI/dV=y22.

The relationship between both parameters can be found elsewhere:

y22=h22-h12 * h21/h11=h22-h12 * gm=h22-h12 * Ic,o/Vt

Hence, the slopes of both curves (Ib=const resp. Vbe=const) are not identical, unless the last term can be neglected. Is it allowed to neglect it?

Example (typical values at Ic,o=1.3mA)):

h22=25µS, h12=260E-6; and

y22=25µS-260E-6 *1.3mA/26mV=25µS-13µS

This example shows that - from the beginning - both slopes cannot be considered as equal (or nearly equal). Hence, the Early voltages derived from both slopes cannot be identical. Therefore, the question still is: Which set of curves must be used for finding the „correct“ Early voltage?

My present understanding is as folllows: We can define two DIFFERENT Early voltages VEA - depending on the used set of output characteristics (IB=const or VBE=const.). Both must be different because the conductance parameters h22 and y22, respectively, are different.

In this context, it is woth mentioning that papers from Berkeley Univ. are defining VEA for IB=const. and papers from Stanford Univ. are defining VEA for VBE=const.

Surprisingly, both are using the same approximation for the transistors dynamic output resistance ro~VEA/IC. (Although they have different definitions for VEA).

Lecture notes from the Columbia Univ. define VEA also for VBE=const. Even more confusing is a contribution from the Georgia Inst. of Tech.. Here, the Early voltage VEA is definde for IB=const. . However, in the same paper this parameter is used in the equation y22=IC/(VEA+VCE).

As we know: y22 is the output conductance for VBE=const.

Can somebody explain these contradictions?

EDIT: Discovered just now: In another lecture note from Berkeley Univ. VBE is held constant. Now we have two different definitions from the same source.

EDIT/UPDATE

The Early voltage VEA appears as a correcting factor (1+Vce/VEA) in Shockleys equation for the B-E junction Ic=f(Vbe). Therefore, it seems to be logical to define this parameter VEA for the alternative with Vbe=const. only.