# 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

• is the closed loop gain of the system.
• is the open loop gain (with no feedback)
• is the feedback fraction, that feeds back a portion of the output voltage back to the input

### Loop Gain

For oscillation, need unity gain, so angle therefore must be real, so also must be real.

### Frequency Potential Divider () ### Transfer function of CR Network

= Gain of CR network ### Transfer function of RC Network

= Gain of RC network ### Transfer function of Inverse Frequency potential divider () ### Transfer function of Inverse Frequency potential divider () ### Transfer function of Frequency potential divider (Inductor) () ## BJT Transitors ### Four Resistor Bias Circuit ## AC BJT Analysis

### Amplifier Topologies ### Transistor Input Impedance

Where = 25mV, = Collector current at Q point.

### Gain of Collector Follower (Common Emitter) AC ### Input Impedance of Collector Follower (Common Emitter)

Into the transistor

### Output Impedance of Collector Follower (Common Emitter)

As current source has infinite impedance.

### Emitter Follower (Common Collector) • High Input, low ouput impedence
• High current gain
• So acts as impedence trasnformer and buffer

### Voltage Gain of Emitter Follower (Common Collector)

as So low voltage gain, so instead current amplifier.

### Output Impedence of Emitter Follower (Common Collector)

Where = source input impedance

### Output Impedence of Emitter Follower (Common Collector) Simple

Where = source input impedance

## MOSFETs DC

No current through gate in MOSFET (as voltage controlled) (infinite input impedence)

### Stages • Cut off (no current flows,
• Linear
• Saturation

Where = Threshold Voltage

### Linear Region Drain Current

, where = transconductance constant

### Small Signal Model ### MOSFET Bias Network

Must check the two different values to see which ones are valid solutions. ### MOSFET input impedence

As no current flows into gate

## MOSFET Common Source

Similar to BJT common emmitter amplifier  ### Overall Input Impedence

As two gate bias resistors act as impedances to input signals. Therefore used over BJTs when high impedence required.

Is actually in parallel with source (input) impedence if it has it.

### Overall Output Impedance

As current source has infinite impedence, therefore is the only impedence seen.

Unless there is an which would be in parallel with .

### Bypassed Gain ## Differential Amplifier

Long tail pair: Modes Can operate in two modes.

• Differential (Amplfies Difference between two input signals)
• Common mode (Works similar to regular BJT amp)

Common Mode Same signal is connected to both input terminals.

• Ideal differential amp rejects common mode input, but not realistic
• Defined by CMRR

Better amps, have high ratio of differnetial to common gain, AKA Common Mode Rejection Ratio (CMRR).

### Quiescent Current of Long Tail Pair

Current through shared emitter resistor, .

### Biasing

and are grounded, therefore collector voltages are the same.

### Collector Voltage of Grounded Long Tail Pair

And for matched transistors, .

### Differential Gain without ground Not really used

### Differential Gain - Single Ended ### Generalised Differential Amplifier Output

Both common mode and differential mode input signals are factored in.

## Impedance Laplace ## Op-Amps

### Active Filter Gain ### Active Filter Gain, Z2 = R2 || C

• Low Pass filter
• Cutoff where = Hz

## Misc

### Source Regulation

Fraction of change in load and input voltage