Merge branch 'main' of https://github.com/ryanrocket/caravel_ft8_receiver into main
diff --git a/docs/documentation.md b/docs/documentation.md
index 205a7ac..a1dfa38 100644
--- a/docs/documentation.md
+++ b/docs/documentation.md
@@ -29,7 +29,7 @@
blah
## 2. PDK Characterization
-Proper characterization of the PDK devices is paramount for accurate circuit design in future steps. Once values such as $g_m$ and $V_{TH}$ are obtained, processes like gm/Id design can be utilized to derive circuit topologies and values.
+Proper characterization of the PDK devices is paramount for accurate circuit design in future steps. Once values such as $g_{m}$ and $V_{TH}$ are obtained, processes like gm/Id design can be utilized to derive circuit topologies and values.
### Characterization of `nfet_01v8`
@@ -43,9 +43,8 @@
.saveall
```
Once the simulation has finished, run `plot -i(v2)` to view the drain current vs. $V_{GS}$ graph. This graph helps to give us the transconductance $g_m$ of the MOSFET, which indicates how efficiently the device can convert a voltage to a current. To derive this value from the simulation, you can either run the command `print @m.xm1.msky130_fd_pr__nfet_01v8[gm]` or use the typical analytical expression:
-```math
-$$g_{m} = \frac{\partial{I_D}}{\partial{}V_{GS}} = \mu_{n}C_{OX}\frac{W}{L}(V_{GS}-V_{TH}) = \frac{2I_D}{V_{GS}-V_{TH}} \tag*{(1)}$$
-```
+$$g_{m} \ = \ \frac{\partial{I_D}}{\partial{}V_{GS}} \ = \ \mu_{n}C_{OX}\frac{W}{L}(V_{GS}-V_{TH}) \ = \ \frac{2I_D}{V_{GS}-V_{TH}}$$
+
To find the threshold voltage $V_{TH}$ of the device, you can simply run the same command as above for the parameter: `print @m.xm1.msky130_fd_pr__nfet_01v8[vth]`
#### 2. Sweep of $V_{DS}$
@@ -65,8 +64,10 @@
.saveall
```
For a given DC sweep, one can obtain the $V_{DSAT}$ value by running `print @m.xm1.msky130_fd_pr__nfet_01v8[vdsat]`. Or, use the expression $V_{DSAT}=V_{GS}-V_{TH}$. Now that the key values of the device have been extracted, one can now determine some other Figures of Merit, such as on resistance:
-$$R_{on}=[\mu_{n}C_{OX}\frac{W}{L}(V_{GS}-V_{TH})]^{-1} \tag*{(2)}$$
+$$R_{on} \ = \ [\mu_{n}C_{OX}\frac{W}{L}(V_{GS}-V_{TH})]^{-1}$$
And to determine the behavior of drain current past saturation:
-$$\int_0^LI_D\mathrm dx=\mu_{n}C_{OX}\int_0^{V_{GS}-V_{TH}}[V_{GS}-V_{TH}-V(x)]\mathrm dV\tag*{(3)}$$
+$$\int_0^LI_D\mathrm dx \ = \ \mu_{n}C_{OX}\int_0^{V_{GS}-V_{TH}}[V_{GS}-V_{TH}-V(x)]\mathrm dV$$
-$$\therefore I_D=\frac{1}{2}\mu_nC_{OX}\frac{W}{L}(V_{GS}-V_{TH})^2(1+\lambda V_{DS}) \ \ \ \ \mathrm{for} \ V_{DS}>V_{DSAT} \tag*{(4)}$$
+$$
+\therefore \ I_D \ = \ \frac{1}{2}\mu_nC_{OX}\frac{W}{L}(V_{GS}-V_{TH})^2(1+\lambda V_{DS}) \ \ \ \ \mathrm{for} \ V_{DS}>V_{DSAT}
+$$
