# Line regulation

## What is line regulation?

In linear regulators, the output voltage is affected by the input voltage. Line regulation measures how much it is affected: $$\mbox{Line regulation} = \frac{\Delta V_{out}}{\Delta V_{in}}$$ We want this specification to be as low as possible. There are several ways of finding out this relation, but the most systematic one, that can be applied the same way to all circuits, is the small-signal analysis. Basically, a circuit is working at some operating point and we analyze it for small variations of some parameter, this case the input voltage $V_{in}$. Since we are analyzing around a specific point, we approximate the circuit to a linear version (remember Taylor series?), just with resistances, capacitors, inductors and sources. From here we can get a closed-form relation between any two parameters of the circuit. Constant voltage sources, such as the average value of the input voltage $V_{in}$, become short-circuits and constant current sources, such as the average value of the load current, become open-circuits.

## Examples

### Shunt linear regulator

Let's try with the shunt regulator. For a given operating point, the zener diode can be replaced by a resistance $r_z$ (that is the small-signal model of the diode). The input voltage is our changing parameter, therefore we connect a voltage source $v_{in}$ to it. Therefore, we have: Then: $$v_{out}= \frac{r_z||R_L}{r_z||R_L + R_D} v_{in}$$ Since $r_z \ll R_L$ (be careful, this is true for the small-signal model. In the large signal analysis, this does not apply), we can approximate $r_z || R_L \approx r_z$, which gives us: $$v_{out}= \frac{r_z}{r_z + R_D} v_{in}$$

### Series regulator with feedback and MOSFET pass transistor

For the regulator with feedback, it is important to account with a parallel resistance $r_{ds}$ that exists in the small signal model of the MOSFET transistor. Also, for this case, there is a voltage source in the input, representing our change in the input voltage. To calculate the transfer function between input and output voltages, we start with the Kirchhoff Current Law (KCL) in the output node: $$i_{out} \approx g_m v_{sg} + \frac{v_{in} - v_{out}}{r_{ds}}$$ $$\frac{v_{out}}{R_L} \approx g_m (v_{in} - A_o \frac{R_2}{R_1+R_2} v_{out}) + \frac{v_{in} - v_{out}}{r_{ds}}$$ Good, now our only variables are $v_{out}$ and $v_{in}$. Working out the above equation, we end up with: $$\frac{v_{out}}{v_{in}} = \frac{R_L + g_m r_{ds} R_L}{R_L + r_{ds} + g_m r_{ds} R_L A_o \frac{R_2}{R_2+R_1}}$$ Recognizing that $R_L + r_{ds} \ll g_m r_{ds} R_L A_o \frac{R_2}{R_2+R_1}$, we can approximate the above equation to: $$\frac{v_{out}}{v_{in}} = \frac{1 + g_m r_{ds}}{g_m r_{ds} A_o \frac{R_2}{R_2+R_1}}$$ Even more, $g_m r_{ds} \gg 1$, which approximates the above equation to: $$\frac{v_{out}}{v_{in}} = \frac{R_2+R_1}{A_o R_2}$$ And there you have it, the line regulation reduces with the gain of the opamp, just like the load regulation.

### Datasheet

This is the line regulation spec in the 78xx series datasheet, named Input voltage regulation and measured for two input voltage ranges and this is in the LM317 datasheet, measured in a range of input-voltage differences and indicated in %/V, since it is an adjustable regulator. 