deck

The Schematic

Get id from gain, bandwidth, and slew rate

f_u \approx f_{3dB} \times Av = \frac{1}{2\pi R_{out}C_{out}} \times g_{m1,2}R_{out} \\ = \frac{g_{m1,2}}{2\pi C_{out}} \\ = \frac{I_{d1,2}}{\pi C_{out}V_{ov1,2}}

f_u \approx f_{3dB} \times Av = \frac{1}{2\pi R_{out}C_{out}} \times g_{m1,2}R_{out} \\ = \frac{g_{m1,2}}{2\pi C_{out}} \\ = \frac{I_{d1,2}}{\pi C_{out}V_{ov1,2}}

Bandwidth & Gain

I_{d1,2} > f_u \pi C_{out} V_{ov1,2}

I_{d1,2} > f_u \pi C_{out} V_{ov1,2}

For target bandwidth,

Slew Rate

I_2 = i_{d5-10}

I_2 = i_{d5-10}

\frac{I_{ss}}{2} = i_{d1,2}

\frac{I_{ss}}{2} = i_{d1,2}

I_1 = \frac{I_{ss}}{2} + I_2

I_1 = \frac{I_{ss}}{2} + I_2

SR_- = (I_2 - max\{(I_1 - I_{ss}), 0\}) / C_{out}

SR_- = (I_2 - max\{(I_1 - I_{ss}), 0\}) / C_{out}

SR_+ = (I_1 - I_2) / C_{out}

SR_+ = (I_1 - I_2) / C_{out}

Let I1 - Iss <= 0, so I2 can be minimize.
Also SR- = I2 / Cout

Slew Rate (Continue)

I_1 \le I_{ss} \\ \rArr I_2 + \frac{I_{ss}}{2} \le I_{ss} \\ \rArr \frac{I_{ss}}{2} \ge I_2 = SR_- * C_{out}

I_1 \le I_{ss} \\ \rArr I_2 + \frac{I_{ss}}{2} \le I_{ss} \\ \rArr \frac{I_{ss}}{2} \ge I_2 = SR_- * C_{out}

\rArr I_{d1,2} \ge I_{d5-10} = SR_- * C_{out}

\rArr I_{d1,2} \ge I_{d5-10} = SR_- * C_{out}

Let's recall previous result,

I_{d1,2} > f_u \pi C_{out} V_{ov1,2}

I_{d1,2} > f_u \pi C_{out} V_{ov1,2}

I_{d1,2} \ge I_{d5-10} = SR_- * C_{out}

I_{d1,2} \ge I_{d5-10} = SR_- * C_{out}

Then, the minimal requirement will be,

I_{d1,2} = max\{ f_u \pi C_{out} V_{ov1,2}, SR_- * C_{out} \}

I_{d1,2} = max\{ f_u \pi C_{out} V_{ov1,2}, SR_- * C_{out} \}

I_{d5-10} = SR_- * C_{out}

I_{d5-10} = SR_- * C_{out}

More about gain

Av = g_{m1}R_{out}

Av = g_{m1}R_{out}

Now, gm1 is known, gain is given, Rout can be calculated.

R_{out} = r_{op} \parallel r_{on}

R_{out} = r_{op} \parallel r_{on}

r_{op} = g_{m7}r_{o7}r_{o9}

r_{op} = g_{m7}r_{o7}r_{o9}

r_{on} = g_{m5}(r_{o1} \parallel r_{o3})r_{o5}

r_{on} = g_{m5}(r_{o1} \parallel r_{o3})r_{o5}

Let

More about gain (Continue)

r_{op} = k * r_{on}

r_{op} = k * r_{on}

R_{out} = k*r_{on} \parallel r_{on} = \frac{k*r_{on}^2}{(k+1)r_{on}} \\ \rArr r_{on} = \frac{k+1}{k}R_{out} \\ \rArr r_{op} = (k+1)R_{out}

R_{out} = k*r_{on} \parallel r_{on} = \frac{k*r_{on}^2}{(k+1)r_{on}} \\ \rArr r_{on} = \frac{k+1}{k}R_{out} \\ \rArr r_{op} = (k+1)R_{out}

Get\ r_{op}\ and\ r_{on}\ from\ R_{out}

Get\ r_{op}\ and\ r_{on}\ from\ R_{out}

More about gain (Continue)

Get\ r_{o7-10}\ from\ r_{op} = g_{m7}r_{o7}r_{o9}

Get\ r_{o7-10}\ from\ r_{op} = g_{m7}r_{o7}r_{o9}

Let

r_{o7} = r_{o9}

r_{o7} = r_{o9}

\rArr r_{o7} = r_{o9} = \sqrt{\frac{r_{op}}{g_{m7}}}

\rArr r_{o7} = r_{o9} = \sqrt{\frac{r_{op}}{g_{m7}}}

More about gain (Continue)

Get\ r_{o1,3,5}\ from\ r_{on} = g_{m5}(r_{o1} \parallel r_{o3})r_{o5}

Get\ r_{o1,3,5}\ from\ r_{on} = g_{m5}(r_{o1} \parallel r_{o3})r_{o5}

Let\ r_{o5} = m*(r_{o1} \parallel r_{o3}) \\ r_{o1} = n*(r_{o3})

Let\ r_{o5} = m*(r_{o1} \parallel r_{o3}) \\ r_{o1} = n*(r_{o3})

Skip the calculation...

r_{o1} \parallel r_{o3} = \sqrt{\frac{r_{on}}{m*g_{m5}}} \\ r_{o5} = m*(r_{o1} \parallel r_{o3})

r_{o1} \parallel r_{o3} = \sqrt{\frac{r_{on}}{m*g_{m5}}} \\ r_{o5} = m*(r_{o1} \parallel r_{o3})

Text

More about gain (Continue)

All ro can be modeled as functions of Rout

when k, m, n are set

In my case, k = 6, m = 1.9, n = 4.

From ro to gate length

r_o = \frac{1+\lambda V_{ds}}{\lambda I_d}

r_o = \frac{1+\lambda V_{ds}}{\lambda I_d}

Vds can be set and Id is known, we can calculate lambda.

The\ relationship\ of\ \lambda\ vs\ gate\ length\ can\ be\ get\ from\ simulation.

The\ relationship\ of\ \lambda\ vs\ gate\ length\ can\ be\ get\ from\ simulation.

From gate length to width

I_d = 0.5 * \mu n,p * C_{ox} * \frac{W}{L} * V_{ov}^2 * (1 + \lambda V_{ds})

I_d = 0.5 * \mu n,p * C_{ox} * \frac{W}{L} * V_{ov}^2 * (1 + \lambda V_{ds})

\mu_n * C_{ox} = 258 \mu A/V^2 \\ \mu_p * C_{ox} = 94 \mu A/V^2

\mu_n * C_{ox} = 258 \mu A/V^2 \\ \mu_p * C_{ox} = 94 \mu A/V^2

From model parameters,

The Initial Design

I write an auto-generation tool, so complicated hand calculation can be avoided.

The tool is open-source:

https://github.com/dragonman225/opamp-generator

The Final Design after fine-tuning

Mb1,b2: w=165u, l=0.22u

M1,2: w=360u, l=0.5u

M3,4: w=130u, l=0.9u

M5,6: w=85u, l=0.8u

M7,8: w=300u, l=0.56u

M9,10: w=55u, l=0.56u

Id1,2 = 700uA

Id5-10 = 515uA

Project

Get id from gain, bandwidth, and slew rate

More about gain

More about gain (Continue)

More about gain (Continue)

More about gain (Continue)

More about gain (Continue)

All ro can be modeled as functions of Rout

From ro to gate length

From gate length to width

The Initial Design

The Final Design after fine-tuning

Simulation Results

Layout

Layout (Bias Circuit)

The End

deck

deck

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