Equations

Below are some of the main equations that I have found useful to have on hand.

Use ./generateTables.sh ../src/es2c0/equations.md in the scripts folder.

Oscillators
Closed Loop Gain	$A_{c l} (jω) = \frac{v _{o}}{v _{i}} = \frac{A}{1 - A β ( jω )}$
Loop Gain	$L (j w) = A β (j w)$
Frequency Potential Divider ( $s$ )	$\frac{V _{o}}{V _{i}} (s) = \frac{Z _{2}}{Z _{1} + Z _{2}} = \frac{s CR}{1 + 3 s CR + s ^{2} C ^{2} R ^{2}}$
Frequency Potential Divider ( $j w$ )	$\frac{V _{o}}{V _{i}} (j w) = \frac{jω CR}{1 - ω ^{2} C ^{2} R ^{2} + 3 jω CR}$
Frequency of Unity Gain (0 phase shift)	$ω_{0} = \frac{1}{RC} \Rightarrow f_{0} = \frac{1}{2 π RC}$
60 Degrees of phase shift in CR network	$ω_{60°} = \frac{1}{s q r t 3 \times RC}$
Transfer function of CR Network	$\frac{v _{o}}{v _{i}} = \frac{jω RC}{1 + jω RC}$
Transfer function of RC Network	$\frac{v _{o}}{v _{i}} = \frac{1}{1 + s CR} = \frac{1}{1 + j wCR}$
Transfer function of Inverse Frequency potential divider ( $s$ )	$\frac{V _{o}}{V _{i}} (s) = \frac{Z _{2}}{Z _{1} + Z _{2}} = \frac{( 1 + s CR ) ^{2}}{1 + 3 s CR + s ^{2} C ^{2} R ^{2}}$
Transfer function of Inverse Frequency potential divider ( $j w$ )	$\frac{V _{o}}{V _{i}} (j w) = \frac{( 1 + j wCR ) ^{2}}{( 1 - w ^{2} C ^{2} R ^{2} ) + 3 j wCR}$
Transfer function of Frequency potential divider (Inductor) ( $s$ )	$\frac{V _{o}}{V _{i}} (s) = \frac{s L R}{( R ^{2} + s ^{2} L ^{2} ) + 3 s L R}$
Transfer function of Frequency potential divider (Inductor) ( $j w$ )	$\frac{V _{o}}{V _{i}} (j w) = \frac{j w L R}{( R ^{2} - w ^{2} L ^{2} ) + 3 j w L R}$
Frequency of Unity Gain (0 phase shift) (Inductor)	$ω_{0} = \frac{R}{L} \Rightarrow f_{0} = \frac{R}{2 π L}$

BJT Transitors
Common Emitter Forward Gain, $β$	$β = \frac{I _{C}}{I _{B}} = \frac{α}{1 - α}$
Common Base Forward current gain, $α$	$α = \frac{β}{β + 1}$
NPN Emitter Current	$I_{E} = I_{B} + I_{C} = \frac{I _{C}}{α} = I_{C} + \frac{I _{C}}{β} = (1 + β) I_{B}$
Emmitter Voltage Rule of Thumb	$V_{E} \approx \frac{V _{cc}}{10}$
Thevin Resistance Rule of Thumb	$R_{T H} = \frac{β R _{E}}{10}$
Four Resistor Bias Circuit $R_{T H}$	$R_{T H} = R_{1} // R_{2} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$
Four Resistor Bias Circuit $V_{T H}$	$V_{T H} = V_{CC} \frac{R _{2}}{R _{1} + R _{2}}$
Transconductance	$g_{m} = \frac{I _{CQ}}{V _{T}} ≊ 40 I_{CQ}$

AC BJT Analysis
Amplifier Topologies	$x$
Transistor Input Impedance	$r_{π} = \frac{β V _{T}}{I _{CQ}}$
Gain of Collector Follower (Common Emitter) AC	$A_{v} = \frac{V _{O}}{V _{in}} = \frac{- β R _{C}}{r _{π} + ( 1 + β ) R _{E}} \approx - \frac{R _{C}}{R _{E}}$
Input Impedance of Collector Follower (Common Emitter)	$R_{in} = r_{π} + (1 + β) R_{E}$
Output Impedance of Collector Follower (Common Emitter)	$R_{o u t} = R_{C}$
Emitter Follower (Common Collector)	$N / A$
Voltage Gain of Emitter Follower (Common Collector)	$A_{v} = \frac{R _{E} ( 1 + β )}{r _{π} + R _{E} ( 1 + β )}$
Current Gain of Emitter Follower (Common Collector)	$A_{i} = \frac{( 1 + β ) i _{b}}{i _{b}}$
Input Impedance of Emitter Follower (Common Collector)	$R_{in} = r_{π} + (1 + β) R_{E}$
Output Impedence of Emitter Follower (Common Collector)	$R_{o u t} = \frac{R _{E} ( r _{π} + R _{S} )}{( r _{π} + R _{S} ) + R _{E} ( 1 + β )}$
Output Impedence of Emitter Follower (Common Collector) Simple	$R_{o u t} = \frac{r _{π}}{( 1 + β )} = \frac{1}{g _{m}}$

MOSFETs DC
Stages	$V_{D S (s a t)} = V_{GS} - V_{TN}$
Linear Region Drain Current	$i_{D} = K_{n} [2 (V_{GS} - V_{TN}) - V_{D S}] V_{D S}$
Saturation Drain Current	$i_{D} = K_{n} (V_{GS} - V_{TN})^{2}$
Saturation Drain Current -> VGS	$V_{GS} = (\frac{i _{d}}{K _{n}})^{1/2} + V_{TN}$
Small Signal Model	$i_{D} = g_{m} \times V_{GS}$
Transconductance $g_{m}$	$g_{m} = \frac{2 I _{D Q}}{V _{GS} - V _{TN}}$
MOSFET Bias Network	$sqrt i_{D} = \frac{- K _{n}^{- 1/2} \pm ( K _{n}^{- 1} - 4 R _{s} ( V _{TN} - V _{T H} ) ) ^{\frac{1}{2}}}{2 R _{s}}$
MOSFET input impedence	$R_{in} = \infty$

MOSFET Common Source
Overall Input Impedence	$R_{in} = R_{T H}$
Overall Output Impedance	$R_{o u t} = R_{D}$
Bypassed Gain	$A_{v} = \frac{- g _{m} R _{D} V _{GS}}{V _{GS}} = - g_{m} R_{D}$
Common Drain (Source Follower)	$z$
Output Impedance	$R_{o u t} = \frac{1}{g m}$

Differential Amplifier
Quiescent Current of Long Tail Pair	$I_{Q} = I_{E 1} + I_{E 2}$
Biasing	$I_{Q} = \frac{( V _{EE} ) + V _{E}}{R _{EE}}$
Collector Voltage of Grounded Long Tail Pair	$V_{C_{1}} = V_{C_{2}} = V_{CC} - I_{C} R_{C}$
Differential Gain without ground	$A_{i d} = \frac{V _{o d}}{V _{d}} = g_{m} R_{C}$
Differential Gain - Single Ended	$\frac{V _{o}}{V _{d}} = \frac{g _{m} R _{C}}{2}$
Differential Input Resistance	$R_{i d} = 2 r_{π}$
Differential Output Resistance	$R_{o d} = 2 R_{E}$
Common Mode Gain	$A_{c m} = \frac{V _{o 1}}{V _{c m}} = \frac{- β R _{C}}{r _{π} + 2 ( 1 + β ) R _{EE}} \approx \frac{- R _{C}}{2 R _{E} E}$
Common Mode Input Resistance	$R_{i c} = \frac{r _{π}}{2} + R_{EE} (1 + β)$
CMRR - Common Mode Rejection Ratio	$CMRR = \frac{A _{i d}}{A _{c m}}$
Generalised Differential Amplifier Output	$V_{o u t} = A_{i d} V_{i d} + A_{c m} V_{c m}$

Impedance Laplace
Resistor	$Z = R$
Capacitor	$Z = \frac{1}{s C} = \frac{1}{j wC} = - \frac{1}{wC} j$
Inductor	$Z = s L$
Resistor Capacitor Series	$Z = R + \frac{1}{s C}$
Resistor Capacitor Parallel	$Z = \frac{R}{1 + s CR}$
Resistor Inductor Series	$Z = R + s L$
Resistor Inductor Parallel	$Z = \frac{s L R}{R + s L}$

Op-Amps
Non-inverting Gain	$A = 1 + \frac{R 2}{R 1}$
Inverting Gain	$A = - \frac{R 2}{R 1}$

Misc
Source Regulation	$\frac{Δ V _{L}}{Δ V _{i}}$
Load Regulation	$\frac{Δ V _{L}}{V _{L e x p ec t e d}}$

Oscillators

Closed Loop Gain

$A_{c l} (jω) = \frac{v _{o}}{v _{i}} = \frac{A}{1 - A β ( jω )}$

$A_{c l}$ is the closed loop gain of the system.
$A$ is the open loop gain (with no feedback)
$β$ is the feedback fraction, that feeds back a portion of the output voltage back to the input

Loop Gain

$L (j w) = A β (j w)$ For oscillation, need unity gain, so angle $= 0°$ therefore must be real, so $β$ also must be real.

Frequency Potential Divider ( $s$ )

$\frac{V _{o}}{V _{i}} (s) = \frac{Z _{2}}{Z _{1} + Z _{2}} = \frac{s CR}{1 + 3 s CR + s ^{2} C ^{2} R ^{2}}$

Frequency Potential Divider ( $j w$ )

$\frac{V _{o}}{V _{i}} (j w) = \frac{jω CR}{1 - ω ^{2} C ^{2} R ^{2} + 3 jω CR}$

Frequency of Unity Gain (0 phase shift)

$ω_{0} = \frac{1}{RC} \Rightarrow f_{0} = \frac{1}{2 π RC}$

60 Degrees of phase shift in CR network

$ω_{60°} = \frac{1}{s q r t 3 \times RC}$

Transfer function of CR Network

$\frac{v _{o}}{v _{i}} = \frac{jω RC}{1 + jω RC}$ $\frac{v _{o}}{v _{i}}$ = Gain of CR network

Transfer function of RC Network

$\frac{v _{o}}{v _{i}} = \frac{1}{1 + s CR} = \frac{1}{1 + j wCR}$ $\frac{v _{o}}{v _{i}}$ = Gain of RC network

Transfer function of Inverse Frequency potential divider ( $s$ )

$\frac{V _{o}}{V _{i}} (s) = \frac{Z _{2}}{Z _{1} + Z _{2}} = \frac{( 1 + s CR ) ^{2}}{1 + 3 s CR + s ^{2} C ^{2} R ^{2}}$

Transfer function of Inverse Frequency potential divider ( $j w$ )

$\frac{V _{o}}{V _{i}} (j w) = \frac{( 1 + j wCR ) ^{2}}{( 1 - w ^{2} C ^{2} R ^{2} ) + 3 j wCR}$

Transfer function of Frequency potential divider (Inductor) ( $s$ )

$\frac{V _{o}}{V _{i}} (s) = \frac{s L R}{( R ^{2} + s ^{2} L ^{2} ) + 3 s L R}$

Transfer function of Frequency potential divider (Inductor) ( $j w$ )

$\frac{V _{o}}{V _{i}} (j w) = \frac{j w L R}{( R ^{2} - w ^{2} L ^{2} ) + 3 j w L R}$

Frequency of Unity Gain (0 phase shift) (Inductor)

$ω_{0} = \frac{R}{L} \Rightarrow f_{0} = \frac{R}{2 π L}$

BJT Transitors

Common Emitter Forward Gain, $β$

$β = \frac{I _{C}}{I _{B}} = \frac{α}{1 - α}$

Common Base Forward current gain, $α$

$α = \frac{β}{β + 1}$

NPN Emitter Current

$I_{E} = I_{B} + I_{C} = \frac{I _{C}}{α} = I_{C} + \frac{I _{C}}{β} = (1 + β) I_{B}$

Emmitter Voltage Rule of Thumb

$V_{E} \approx \frac{V _{cc}}{10}$

Thevin Resistance Rule of Thumb

$R_{T H} = \frac{β R _{E}}{10}$

Four Resistor Bias Circuit $R_{T H}$

$R_{T H} = R_{1} // R_{2} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$

Four Resistor Bias Circuit $V_{T H}$

$V_{T H} = V_{CC} \frac{R _{2}}{R _{1} + R _{2}}$

Transconductance

$g_{m} = \frac{I _{CQ}}{V _{T}} ≊ 40 I_{CQ}$

AC BJT Analysis

Amplifier Topologies

$x$

Transistor Input Impedance

$r_{π} = \frac{β V _{T}}{I _{CQ}}$ Where $V_{T}$ = 25mV, $I_{C} Q$ = Collector current at Q point.

Gain of Collector Follower (Common Emitter) AC

$A_{v} = \frac{V _{O}}{V _{in}} = \frac{- β R _{C}}{r _{π} + ( 1 + β ) R _{E}} \approx - \frac{R _{C}}{R _{E}}$

Input Impedance of Collector Follower (Common Emitter)

$R_{in} = r_{π} + (1 + β) R_{E}$ Into the transistor

Output Impedance of Collector Follower (Common Emitter)

$R_{o u t} = R_{C}$ As current source has infinite impedance.

Emitter Follower (Common Collector)

$N / A$

High Input, low ouput impedence
High current gain
So acts as impedence trasnformer and buffer

Voltage Gain of Emitter Follower (Common Collector)

$A_{v} = \frac{R _{E} ( 1 + β )}{r _{π} + R _{E} ( 1 + β )}$ $A_{v} \approx 1$ as $r_{π} >> RE (1 + β)$ So low voltage gain, so instead current amplifier.

Current Gain of Emitter Follower (Common Collector)

$A_{i} = \frac{( 1 + β ) i _{b}}{i _{b}}$

Input Impedance of Emitter Follower (Common Collector)

$R_{in} = r_{π} + (1 + β) R_{E}$

Output Impedence of Emitter Follower (Common Collector)

$R_{o u t} = \frac{R _{E} ( r _{π} + R _{S} )}{( r _{π} + R _{S} ) + R _{E} ( 1 + β )}$ Where $R_{S}$ = source input impedance

Output Impedence of Emitter Follower (Common Collector) Simple

$R_{o u t} = \frac{r _{π}}{( 1 + β )} = \frac{1}{g _{m}}$ Where $R_{S}$ = source input impedance

MOSFETs DC

No current through gate in MOSFET (as voltage controlled) (infinite input impedence) $i_{G} = 0 \to i_{D} = i_{S}$

Stages

$V_{D S (s a t)} = V_{GS} - V_{TN}$

Cut off (no current flows, $V_{GS} < V_{TN})$
Linear $V_{D S} > V_{D S (s a t)}$
Saturation $V_{D S} < V_{D S (s a t)}$

Where $V_{TN}$ = Threshold Voltage

Linear Region Drain Current

$i_{D} = K_{n} [2 (V_{GS} - V_{TN}) - V_{D S}] V_{D S}$

$for 0 < V_{D S} < (V_{GS} - V_{TN})$ , where $K_{n}$ = transconductance constant

Saturation Drain Current

$i_{D} = K_{n} (V_{GS} - V_{TN})^{2}$

Saturation Drain Current -> VGS

$V_{GS} = (\frac{i _{d}}{K _{n}})^{1/2} + V_{TN}$

Small Signal Model

$i_{D} = g_{m} \times V_{GS}$

Transconductance $g_{m}$

$g_{m} = \frac{2 I _{D Q}}{V _{GS} - V _{TN}}$

MOSFET Bias Network

$sqrt i_{D} = \frac{- K _{n}^{- 1/2} \pm ( K _{n}^{- 1} - 4 R _{s} ( V _{TN} - V _{T H} ) ) ^{\frac{1}{2}}}{2 R _{s}}$ Must check the two different values to see which ones are valid solutions.