# 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

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