Bandgap Reference

The need for temperature-independent references are essential for modern applications and rapidly changing environments. This article discusses the negative and positive temperature coefficients (TC) of a bipolar device and how to cancel their effects to create a stable reference.

CTAT: Complementary to Absolute Temperature

For a bipolar device, the forward voltage of a pn-junction diode exhibits a negative TC.

\[\begin{split} \begin{align*} I_C &= I_s\ e^{(V_{BE}/V_T)} \text{ where } V_T = \frac{kT}{q} \\ I_S &= bT^{4+m}\ e^{(-\frac{Eg}{kT})} \\[0.5em] V_{BE} &= V_T\ ln{\left(\frac{I_C}{I_S} \right)} \\[0.5em] \dfrac{\partial V_{BE}}{\partial T} &= \dfrac{\partial V_T}{\partial T}\ ln{\dfrac{I_C}{I_S}} - \dfrac{V_T}{I_S} \dfrac{\partial I_S}{\partial T} \\[0.5em] \dfrac{V_T}{I_S} \dfrac{\partial I_S}{\partial T} &= (4+m)\dfrac{V_T}{T} + \dfrac{E_g}{kT^2}V_T \\[0.5em] \dfrac{\partial V_{BE}}{\partial T} &= \dfrac{V_T}{T}\ ln{\dfrac{I_C}{I_S}} - (4+m)\dfrac{V_T}{T} + \dfrac{E_g}{kT^2}V_T \\[0.5em] &= \dfrac{V_{BE}-(4+m)V_T - E_g/q}{T} \end{align*} \end{split}\]

Thus, at \(T=300K\) and \(V_{BE} \approx 750\)mV, the change in TC voltage with respect to temperature is \(\partial V_{BE}/ \partial T \approx -1.5\)mV.

PTAT: Proportional to Absolute Temperature

Fig. 2 PTAT Circuit

Figure 1 shows two bipolar transistors operating with ideal current sources. The difference between their base-emitter voltages is directly proportional to absolute temperature. The circuit emphasizes design choices by scaling bias current n and number of devices m.

\[\begin{split} \begin{align*} \Delta V_{BE} &= V_{BE1} - V_{BE2} \\[0.5em] &= V_T\ ln{(\dfrac{n I_0}{I_{S}})} - V_T\ ln{(\dfrac{I_0}{mI_{S}})} \\[0.5em] &= V_T\ ln{(nm)} \\[0.5em] &= \dfrac{kT}{q}\ ln{(nm)} \\[0.5em] \dfrac{\partial}{\partial T} \Delta V_{BE} &= \dfrac{\partial}{\partial T} \dfrac{kT}{q}\ ln{(nm)}\\[0.5em] &= \dfrac{k}{q}\ ln{(nm)} \end{align*} \end{split}\]

The positive temperature coefficient is proportional to \(\frac{k}{q}\) such that

\[\dfrac{\partial }{\partial T}\Delta V_{BE} = \alpha\ 0.087 \text{mV/C.}\]

Bandgap Reference

Now a temperature independent reference can be obtained by combining the negative and positive coefficients mentioned previously. The reference is defined as

\[ \begin{align*} V_{REF} &= \alpha_1 V_{BE} + \alpha_2 V_T ln(nm) \end{align*} \]

For simplicity, \(\alpha_1\) is chosen to be 1. Then \(V_{REF}\) is

\[ \begin{align*} V_{REF} &= V_{BE} + \alpha_2 V_T ln(m) \end{align*} \]

where \(m\) is the number of devices.