Final Exam Preparation for Circuits and Mech

Electric charge (Q)

Property of matter that can be positive or negative and creates electric forces. Unit: coulomb (C).

Electric current (I)

Rate at which charge flows. I = ΔQ / Δt. Unit: ampere (A) = coulomb per second.

Conventional current direction

Current is drawn as moving from positive terminal to negative terminal, even though electrons move the opposite way.

Voltage (V)

Energy per unit charge between two points. V = W / Q. Unit: volt (V).

Reference node (ground)

Special node defined as 0 V. All other node voltages are measured relative to this node.

Power (P)

Rate of energy transfer in a circuit. P = V × I. Unit: watt (W).

Energy (W or E)

Total work done or energy transferred. For constant power, E = P × t. Unit: joule (J).

Passive sign convention

If current enters the terminal marked “+”, then P = V × I is power absorbed. Negative P means the element delivers power.

Resistor

Component that opposes current flow and obeys V = I × R. Unit: ohm (Ω).

Resistance (R)

Measure of how strongly a component resists current. R = V / I. Unit: ohm (Ω).

Conductance (G)

Reciprocal of resistance. G = 1 / R. Unit: siemens (S).

Ohm’s Law

Relationship in a resistor: V = I × R, or I = V / R, or R = V / I.

Independent voltage source

Source that maintains a specified voltage regardless of current drawn (within ratings).

Independent current source

Source that maintains a specified current regardless of voltage across it (within ratings).

Dependent source

Source whose value depends on another voltage or current somewhere else in the circuit.

Open circuit

Connection with essentially infinite resistance. Current is zero, but voltage can be nonzero.

Short circuit

Connection with essentially zero resistance. Voltage between the two points is nearly zero.

Node

Point where two or more elements are connected. All points on a perfect conductor form one node.

Branch

Single element, or series group treated as one element, connecting two nodes.

Loop

Any closed path in a circuit that begins and ends at the same node.

Mesh

A loop that does not contain another loop inside it. Used for mesh analysis.

Series connection

Elements share the same current and are connected end to end with no branching node between them.

Parallel connection

Elements share the same two nodes, so they have the same voltage across them.

Kirchhoff’s Current Law (KCL)

Sum of currents entering a node equals sum of currents leaving the node.

Kirchhoff’s Voltage Law (KVL)

Sum of voltage rises and drops around any closed loop is zero.

Series resistors

Equivalent resistance adds: R_eq = R1 + R2 + … + Rn.

Parallel resistors

Equivalent conductance adds: 1 / R_eq = 1 / R1 + 1 / R2 + … + 1 / Rn.

Two-resistor parallel shortcut

For R1 and R2 in parallel: R_eq = (R1 × R2) / (R1 + R2).

Voltage divider rule

In a series chain, voltage across resistor Rk is Vk = Vtotal × (Rk / sum of all series resistors).

Current divider rule

In two parallel resistors, current in R1 is I1 = I_total × (R2 / (R1 + R2)), and similarly for R2.

Single-loop circuit

One closed path with one current. Use KVL and Ohm’s Law to find that loop current.

Single-node (parallel) circuit

Several elements share two nodes. Use KCL and Ohm’s Law to find branch currents and node voltage.

Current reference direction

Arrow you choose for analysis. If final current is negative, actual direction is opposite to the arrow.

Voltage polarity marking

You choose which terminal is “+” when writing equations. Negative answer means opposite polarity.

Equivalent resistance idea

Replace multiple resistors by a single resistor that draws the same current for a given applied voltage.

DC source with internal resistance

Modelled as ideal source in series with internal resistor R_int.

Linear element

Element for which voltage and current are related by a straight line (constant R).

Nonlinear element

Element for which the V–I relationship is not a straight line (e.g., diode).

Basic nodal analysis step 1

Choose a reference node and label it 0 V.

Basic nodal analysis step 2

Label unknown node voltages V1, V2, … relative to reference.

Basic nodal analysis step 3

Apply KCL at each non-reference node, writing currents as (Vnode − Vother) / R.

Basic nodal analysis step 4

Solve the resulting simultaneous equations for the node voltages.

Supernode

Formed when a voltage source connects two unknown nodes. Treat them as one larger node region.

Supernode equation

Apply KCL to the supernode, plus one extra equation for the voltage source constraint.

Basic mesh analysis step 1

Define a mesh current around each mesh, usually clockwise.

Basic mesh analysis step 2

Write KVL for each mesh, using element voltages written in terms of mesh currents and resistances.

Basic mesh analysis step 3

For shared elements, use the difference between the two mesh currents.

Basic mesh analysis step 4

Solve the simultaneous equations to find the mesh currents.

Supermesh

Created when a current source lies between two meshes. Combine those meshes into one loop for KVL.

Supermesh procedure

Skip the current source when writing KVL around the combined loop, then use a separate equation relating the two mesh currents to the source.

Power check in DC circuits

Total power supplied by sources should equal total power absorbed by resistors (with sign convention).

Linearity property

In a linear circuit, doubling all sources doubles all currents and voltages.

Superposition theorem

Total response equals sum of responses from each independent source acting alone, others turned off.

Turning off a voltage source

Replace ideal independent voltage source with a short circuit when using superposition.

Turning off a current source

Replace ideal independent current source with an open circuit when using superposition.

Source transformation (V to I)

Voltage source Vs in series with R is equivalent to current source Is = V_s / R in parallel with R.

Source transformation (I to V)

Current source Is in parallel with R is equivalent to voltage source Vs = I_s × R in series with R.

Thevenin equivalent

Replace a linear network seen from two terminals by one voltage source VTh in series with resistor RTh.

Norton equivalent

Replace a linear network seen from two terminals by one current source IN in parallel with resistor RTh.

Finding V_Th

Remove the load and find the open-circuit voltage across the two terminals.

Finding I_N

Short the two output terminals and find the resulting short-circuit current.

Finding R_Th with only independent sources

Turn off all independent sources and find equivalent resistance seen from the output terminals.

Thevenin–Norton relationship

VTh = IN × RTh, and RTh is the same in both forms.

Maximum power transfer condition

Max power is delivered to the load when Rload equals the Thevenin resistance RTh.

Maximum power transfer tradeoff

Rload = RTh gives maximum power, but not maximum efficiency.

Sinusoidal source

AC voltage described by v(t) = V_peak sin(ωt + φ) or similar.

Amplitude of sinusoid

The maximum (peak) value of the sinusoidal voltage or current.

RMS value of sinusoid

Effective value equal to peak divided by √2 for a sine wave.

Frequency (f)

Number of cycles per second of a sinusoid. Measured in hertz (Hz).

Period (T)

Time for one full cycle. T = 1 / f.

Angular frequency (ω)

Angle rate of sinusoid in radians per second. ω = 2πf.

Phase angle (φ)

Shift of sinusoid along time axis. Positive phase means leading, negative means lagging.

Complex number (rectangular form)

Number of the form x + j y, where j² = −1.

Complex number (polar form)

Number expressed as magnitude ∠ angle, like |z|∠θ.

Magnitude of complex number

|z| = √(x² + y²) for z = x + j y.

Angle of complex number

θ = arctangent(y / x), adjusted for the correct quadrant.

Phasor

Complex number that represents the magnitude and phase of a sinusoidal voltage or current at one frequency.

Phasor representation idea

Replace time-varying sinusoids with constant complex numbers, do circuit math, then convert back at the end.

Impedance (Z)

AC version of resistance. Z = Vphasor / Iphasor. Often complex.

Impedance of resistor

Z_R = R, a real number. Voltage and current are in phase.

Impedance of inductor

Z_L = j ω L. Voltage leads current by 90 degrees.

Impedance of capacitor

Z_C = 1 / (j ω C). Current leads voltage by 90 degrees.

Series impedances

Total impedance is the sum: Z_total = Z1 + Z2 + … + Zn.

Parallel impedances

Total admittance adds: 1 / Z_total = 1 / Z1 + 1 / Z2 + … + 1 / Zn.

AC nodal analysis

Same steps as DC nodal analysis, but use complex impedances instead of resistances.

AC mesh analysis

Same steps as DC mesh analysis, but write voltages and currents using complex impedances.

Instantaneous power in AC

p(t) = v(t) × i(t). Average over one cycle is the real power.

Real (average) power P

Average power in AC. P = Vrms × Irms × cos(φ).

Reactive power Q

Power that alternates between source and reactive elements. Q = Vrms × Irms × sin(φ).

Apparent power S

Product of RMS voltage and RMS current. S = Vrms × Irms.

Power factor

Power factor = cos(φ) = real power / apparent power.

Filter

Network that passes some frequency range and attenuates others.

Transfer function H(jω)

Ratio of output phasor to input phasor as a function of frequency: H(jω) = Vout / Vin.

Magnitude response

Plot of |H(jω)| versus frequency. Shows how much the circuit amplifies or attenuates each frequency.

Phase response

Plot of angle of H(jω) versus frequency. Shows phase shift for each frequency.

Decibel (dB) gain

Voltage gain in dB = 20 log10(|Vout / Vin|).

Cutoff (corner) frequency

Frequency where filter output has dropped to 1 / √2 of its low-frequency or high-frequency level (−3 dB).

First-order low-pass RC filter

Resistor in series, capacitor to ground, output across the capacitor.

Low-pass RC transfer function

H(jω) = 1 / (1 + j ω R C).

Low-pass RC cutoff frequency

f_c = 1 / (2 π R C).