# The Bandgap Reference

#### Saturday, 2014 March 01

Consider the bandgap reference circuit shown above. The op-amp is known to be ideal, such that no current flows into the inputs and

\[ \begin{align} V_a &= V_b \end{align} \]

\(Q_1\) is a single diode-connected BJT, and \(Q_2\) is composed of \(N\) diode-connected BJTs in parallel. This means that the reverse saturation current for \(Q_1\) is \(I_S\), while for \(Q_2\), it is \(N I_S\). All in all, the collector currents of those transistors are

\[ \begin{align} I_1 &= I_S e^\frac{V_{BE1}}{V_T} \\ I_2 &= N I_S e^\frac{V_{BE2}}{V_T} \end{align} \]

It also follows that \[ \begin{align} V_{BE1} &= V_T \ln \frac{I_1}{I_S} \\ V_{BE2} &= V_T \ln \frac{I_2}{N I_S} \end{align} \]

Since \(V_a = V_b\), the difference between \(V_{BE1}\) and \(V_{BE2}\) falls on \(R_3\) as \(\Delta V_{BE}\):

\[ \begin{align} \Delta V_{BE} &= V_{BE1} - V_{BE2} \\ \Delta V_{BE} &= V_T \ln \frac{N I_1}{I_2} \end{align} \]

The currents \(I_1\) and \(I_2\) can also be expressed as

\[ \begin{align} I_1 &= \frac{V_{REF} - V_a}{R_1} \\ I_2 &= \frac{V_{REF} - V_b}{R_2} \end{align} \]

Therefore, we have

\[ \boxed{ \begin{align} \Delta V_{BE} &= V_T \ln \frac{N R_2}{R_1} \end{align} } \]

Now, determining \(V_{REF}\) is fairly straightforward. KVL from \(V_{REF}\) to ground along \(R_1\) and \(Q_1\) gives us

\[ \begin{align} V_{REF} &= I_1 R_1 + V_{BE1} \end{align} \]

Since \(V_a = V_b\), \(I_1 R_1 = I_2 R_2\)

\[ \begin{align} V_{REF} &= I_2 R_2 + V_{BE1} \end{align} \]

Since the voltage drop on \(R_3\) is \(\Delta V_{BE}\)

\[ \begin{align} \Delta V_{BE} &= I_2 R_3 \end{align} \]

we can express \(I_2 R_2\) in terms of \(R_3\) by multiplying with \(\frac{R_3}{R_3}\)

\[ \begin{align} I_2 R_2 &= I_2 R_2 \frac{R_3}{R_3} \\ I_2 R_2 &= \frac{R_2}{R_3} \Delta V_{BE} \end{align} \]

Finally, we have

\[ \boxed{ \begin{align} V_{REF} &= \frac{R_2}{R_3} \Delta V_{BE} + V_{BE1} \end{align} } \]

## References

- Kuijk, K.E., “A precision reference voltage source,"
*Solid-State Circuits, IEEE Journal of*, vol. 8, no. 3, pp. 222–226, June 1973. - Gray, et al.,
*Analysis and Design of Analog Integrated Circuits*, 5th ed. (2010)