ECE 65 reference sheet

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Op-amps

          v_o = A(v_+ - v_-)

          i_- = i_+ = 0

The op-amp has both a maximum output voltage and current.

          V_{s^-} < V_{sat^-} \leq v_o \leq V_{sat^+} < V_{s^+}

Negative feedback

          v_+ = v_-

Diodes

On

v_D = V_{D_0}

if i_D \geq 0.

Off

i_D = 0

if v_D < V_{D_0}.

Zener

On

v_D = V_{D_0}

if i_D \geq 0.

Off

i_D = 0

if -V_Z < v_D < V_{D_0}.

Zener

v_D = -V_Z

if i_D \leq 0.

Half-wave rectifier

When v_i \geq V_{D_0}, the diode is on, and v_o = v_i - V_{D_0}.

When v_i < V_{D_0}, the diode is off, and v_o = 0.

Circuit

Transfer

Output

Flipping the diode upside down flips the transfer function around.

Full-wave rectifier

Circuit

Transfer

Output

Clipper

When v_i \geq V_{D_0}, the diode is on, and v_o = V_{D_0}.

When v_i < V_{D_0}, the diode is off, and v_o = v_i.

              v_o \leq V_{D_0}

              v_o \leq V_{D_0} + V_{DC}

              v_o \leq V_{D_0} + V_Z

              v_o \geq -V_{D_0} - V_Z

              -V_{D_0} - V_{DC_2} \leq v_o \leq V_{D_0} + V_{DC_1}

              -V_{D_0} - V_{Z_2} \leq v_o \leq V_{D_0} + V_{Z_1}

Circuit

Transfer

Output

Peak detector

Circuit

A load resistor will make the capacitor leak voltage.

Output

The shape of the output signal depends on \frac{\tau}{T}. As \frac{\tau}{T} decreases, the circuit departs from a peak detector.

An ideal \frac{\tau}{T} \to \infty.
A good \frac{\tau}{T} >> 1.
For \frac{\tau}{T} << 1, the capacitor discharges very fast, and the output resembles a rectifier circuit.

Clamp

Identical to the peak detector circuit except the output voltage is taken from the diode instead.

              v_o = v_i - (V_p - V_{D_0})

              v_o = v_i - (V_p - V_{D_0} - V_{DC})

              v_o = v_i - (V_p - V_{D_0} - V_Z)

              v_o = v_i + (V_p - V_{D_0})

Circuit

Output

BJTs

The arrow is on the emitter node and points in the direction of the emitter current.

Off

i_B = 0, i_C = 0
if v_{BE} < V_{D_0} (NPN) or v_{EB} < V_{D_0} (PNP).

On

v_{BE} = V_{D_0} (NPN) or v_{EB} = V_{D_0} (PNP) if i_B \geq 0.

Active

i_C = \beta i_B
if v_{CE} \geq V_{D_0} (NPN) or v_{EC} \geq V_{D_0} (PNP).

Saturation

v_{CE} = V_{sat} (NPN) or v_{EC} = V_{sat} (PNP)
if i_C < \beta i_B.

MOSFETs

The arrow is on the source node and points in the direction of the drain current.

NMOS

V_{OV} = v_{GS} - V_{tn}

PMOS

V_{OV} = v_{SG} - |V_{tp}|

Off

i_D = 0

if V_{OV} < 0.

Triode

              i_D = 0.5 \mu_n \, C_{ox} \, \dfrac{W}{L} \left(2 V_{OV} v_{DS} - v_{DS}^2\right)

              i_D = 0.5 \mu_p \, C_{ox} \, \dfrac{W}{L} \left(2 V_{OV} v_{SD} - v_{SD}^2\right)

if V_{OV} \geq 0 and
v_{DS} \leq V_{OV} (NMOS) or v_{SD} \leq V_{OV} (PMOS).

If i_D = 0 but the MOSFET is on, then it's in the triode region, and v_{DS} = 0.

Saturation (active)

              i_D = 0.5 \mu_n \, C_{ox} \, \dfrac{W}{L} \cdot V_{OV}^2 \left(1 + \lambda v_{DS}\right)

              i_D = 0.5 \mu_p \, C_{ox} \, \dfrac{W}{L} \cdot V_{OV}^2 \left(1 + \lambda v_{SD}\right)

if V_{OV} \geq 0 and
v_{DS} \geq V_{OV} (NMOS) or v_{SD} \geq V_{OV} (PMOS).

When analyzing logic gates, if you don't know what state one of the MOSFETs in series is in, trying assuming that it's off.

Small signal model

BJTs

g_m = \frac{I_C}{V_T}

r_o = \frac{V_A}{I_C}

r_\pi = \frac{V_T}{I_B} = \frac{\beta}{g_m}

MOSFETs

g_m = \frac{2I_D}{V_{OV}}

r_o = \frac{1}{\lambda I_D}

For transistor amplifier configurations, "common" refers to the grounded pin. Input is connected to base/gate, and output is taken from the non-grounded pin.

Common collector

Also known as an emitter follower.

              A_{vo} = \frac{g_m (R_E \parallel r_o)}{1 + g_m (R_E \parallel r_o)}

              R_o = \frac{1}{g_m} \parallel r_\pi \parallel R_E \parallel r_o

              R_i = R_B \parallel (r_\pi + (\beta + 1)(r_o \parallel R_E \parallel R_L))

Common drain

Also known as a source follower.

              A_{vo} = \frac{g_m (R_S \parallel r_o)}{1 + g_m (R_S \parallel r_o)}

              R_o = \frac{1}{g_m} \parallel R_S \parallel r_o

R_i = R_G

Common collector/drain circuits have a gain of slightly less than 1 V/V.

Common emitter

A_{vo} = -g_m (R_C \parallel r_o)

R_o = R_C \parallel r_o

R_i = R_B \parallel r_\pi

With an emitter resistor:

              A_{vo} = \frac{-R_C}{R_E + \left(1 + \dfrac{R_C}{r_o}\right) \dfrac{R_E + r_\pi}{\beta}}

              R_o = \frac
                {(r_\pi \parallel R_E) + \frac{1}{g_m} \parallel r_o}
                {\dfrac{\dfrac{1}{g_m} \parallel r_o}{R_C \parallel r_o} + \dfrac{r_\pi \parallel R_E}{R_C}}

              R_i = R_B \parallel \left(R_E + r_\pi + \frac{\beta R_E}{1 + \frac{R_E + R_C \parallel R_L}{r_o}}\right)

Common source

A_{vo} = -g_m (R_D \parallel r_o)

R_o = R_D \parallel r_o

R_i = R_G

With a source resistor:

              A_{vo} = \frac{-g_m R_D}{1 + g_m R_S + \dfrac{R_D}{r_o}}

R_o = R_D \parallel (r_o \, (1 + g_m R_S))

R_i = R_G

Voltage amplifier model

Gain

Circuit voltage gain:

              A = \frac{v_o}{v_{sig}}

Amplifier voltage gain:

              A_v = \frac{v_o}{v_i}

Open-loop gain:

              
              A_{vo} = \left.\frac{v_o}{v_i}\right|_{R_L \to \infty}

Resistance

Input resistance:

              R_i = \frac{v_i}{i_i}

Output resistance:

              R_o = \left.-\frac{v_o}{i_o}\right|_{v_i = 0}

Method

Draw the bias circuit: open capacitors and set signal sources to zero. Find the bias point parameters.
Find the small signal parameters g_m, r_o, and r_\pi using the bias point parameters.
Draw the signal equivalent circuit: short capacitors, set DC voltage and current sources to zero, and replace transistors with their small signal models.
Find the amplifier parameters R_i, R_o, and A_{vo}.
Use the voltage amplifier model and calculated parameters to find the amplifier circuit gain A.