combine

Operational Amplifier =

amplifies voltage difference by A₀ (open-loop gain)

V_out = A₀(V₊ − V₋)

A₀ (open-loop gain) is very large: ~100k to 100M V/V

3 Ideal Op Amp assumptions =

1. No current into inputs: I_in = 0

2. Output can drive infinite current: I_out → ∞ (current comes from supply rails)

3. Open-loop gain is infinite: A₀ → ∞

Op amp V_out is always limited by =

Supply voltages

(Will get clipping if Vout exceeds Vs)

If Op-Amp has CLNF, what can be assumed =

V₊ = V₋

ONLY valid with negative feedback present (Vout connected to V-).

Noninverting amplifier gain + circuit =

A = 1 + [R_f / R_i] Gain is ALWAYS ≥ 1

Output same polarity as input.

Opamp Gain Formula =

V_out/V_in

Unity gain buffer/voltage follower gain + circuit =

A = 1

Inverting amplifier gain + circuit =

A = −[R_f / R_i]

Difference amplifier Vout and circuit =

V_out = (V_in+ − V_in−) × [R_f / R_i]

Amplifies the difference between two inputs.

Uses MATCHED resistor pairs, voltage divider configuration (R_i on both inputs, R_f on both feedback paths)

Inverting summing amplifier =

V_out = −R_f × ([V_in1 / R_i1] + [V_in2 / R_i2] + ... + [V_inN / R_iN])

Multiple inputs through separate R_i's to V-, single R_f in feedback.

Noninverting summing amplifier =

V_out = (1 + [R_f / R_i]) × (Req) × (V_in1 + V_in2 + ... + V_inN)

Where R_i is the feedback-to-ground resistor.

V₊ = average of inputs if ALL input resistors are EQUAL.

Which opamps have + top / - top?

+ top

noninverting amp

noninverting summing amp

voltage follower

- top

difference amp

inverting amp

inverting summing amp

Comparator =

Open-loop (no feedback)

Output hits the supply rails:

V_out = V_DD if V₊ > V₋

V_out = V_SS if V₊ < V₋

Compares two analog signals → digital output.

V_DD and V_SS =

V_DD = positive Voltage

V_SS = negative Voltage

How to tell which rail will saturate =

look at where Vin enters (V+ or V-)

V+ = non inverting = same sign as Vin

V- = inverting = opposite sign as Vin

What does || mean =

Parallel

R1 || R2 = R1R2/(R1 + R2)

Voltage divider formula =

Given resistor R1 on top, R2 on bottom, Vout is in the middle node

Vo = Vi[R2/(R1+R2)]

For multiple resistors, add all of them on denominator, only Rx is added to numerator where Rx is the node Vout measured from

Current divider formula =

Given resistor R1 on the left, R2 on the right, Io is on the right

Io = Iin[R1/(R2+R3)]

For multiple resistors, simplify with Req

Wheatstone bridge formula =

Given source Vs

V is the voltage between the two middle nodes of voltage dividers

Rx is the bottom right, unknown resistor

V = VsRx/(Rx + R2) - VsR3/(R1 + R3)

Two parallel voltage dividers, in diamond pattern

supernodes method =

create a boundary around a Voltage source, then apply KCL so you can ignore the Voltage source

mesh current method =

create loops for KVL and solve

components in between two loops will be a sum of currents

supermeshes method =

creating boundary around a Current source, then apply KVL so you can ignore the Current source

removing a voltage source, circuit becomes ???

short circuit

removing a current source, circuit becomes ???

open circuit

thevenin circuit =

voltage source + resistor in series, Vopen = Vth

norton circuit

current source + resistor in parallel, Ishort = In

what maximizes power in a thevenin circuit =

Rload = Rth

superposition method =

linear circuits, can consider independent sources one at a time and sum them all up

dependent sources MUST BE KEPT

how many KCL eqns in a circuit

N - 1

N = # of nodes

how many KVL eqns in a circuit

B - N + 1

B = # of branches / components

N = # of nodes

for mesh analysis, how to figure out polarity of V_drop =

look at loop current, the terminal it enters is the sign

how to know if component is absorbing/delivering power =

P > 0 = absorb

• current going INTO positive terminal

P < 0 = deliver

• current going OUT OF positive terminal

passive sign convention =

label current into positive terminal as positive

Define DC =

direct current

Current and voltage is constant

define AC =

alternating current

Current or voltage varies over time (not constant)

define Vpk =

how far AC component gets away from DC

define Vpk-pk =

max V - min V

How to convert from frequency to angular frequency

ω = 2πf

How to draw dependent vs independent source =

dependent = diamond

Independent = circle

Define conductance =

inverse of resistance (G = 1/R)

How well a resistor conducts electricity

Unit is Siemens (S)

Resistor has 2 ohms or 0.5 Siemens

what are E series resistors =

resistors split up into X equally spaced (logarithmic) values

ex.) E24 series

Values are from 10^(X/24)

how to combine parallel resistors to one =

For two parallel resistors

1/Req = 1/R1 + 1/R2

how to combine series resistors to one =

add them up

how to combine voltage sources =

in series ONLY

how to combine current sources =

in parallel ONLY

Kirchoff's current law =

current into a node = current out of a node

Kirchoff's voltage law =

sum of all voltage rises and drops in a closed loop circuit is 0

Current formula =

I = dQ/dt

Current in parallel is =

Current in series is =

Current adds up in parallel

(It = I1 + I2 + I3...)

Current is the same in series

(It = I1 = I2)

Voltage in parallel is =

Voltage in series is =

Voltage is the same in parallel

(Vt = V1 = V2 = V3)

Voltage is split in series

(Vt = V1 + V2 + V3)

how to know if components are in parallel/series/neither =

parallel = both common nodes

Series = 1 common node with NO additional branches