Circuit Analysis

name: circuit-analysis

description: > Expert circuit analysis assistant for electrical engineers and students. Use this skill whenever the user needs: help with circuit analysis using Kirchhoff's laws, nodal and mesh analysis, Thevenin/Norton equivalents, op-amp circuit design, filter design, frequency response analysis, or simulation of analog and digital circuits. Includes both theoretical foundations and practical design guidance. trigger: Any electrical engineering problem involving circuits - from simple DC analysis to complex AC networks and filter design. license: MIT compatibility: opencode metadata: audience: engineers category: electrical-engineering

Circuit Analysis — Theory and Design

Covers: DC Analysis · AC Analysis · Network Theorems · Circuit Simulation · Filters · Operational Amplifiers · Transient Response

Fundamental Laws

Kirchhoff's Current Law (KCL)

At any node (junction): the algebraic sum of currents entering equals sum leaving.

Σ I_in = Σ I_out

Kirchhoff's Voltage Law (KVL)

Around any closed loop: the algebraic sum of voltage drops equals zero.

Σ V = 0

Ohm's Law

V = IR

In generalized form (for any two-terminal element):

V = ZI

Where Z is impedance (complex in AC).

Power

Instantaneous power:

p(t) = v(t)i(t)

Average power (periodic):

P_avg = (1/T) ∫ p(t) dt = VI cos(θ_v - θ_i) = VI cos(φ)

For purely resistive: P = VI = I²R = V²/R

Energy

W = ∫ p(t) dt

Circuit Elements

Passive Elements

Energy Storage Comparison

Active Elements

• Voltage source: Fixed voltage, current determined by circuit
• Current source: Fixed current, voltage determined by circuit
• Dependent sources: Output depends on another voltage/current

Equivalent Circuits

Series combination:

• Resistors: R_eq = R₁ + R₂ + ...
• Inductors: L_eq = L₁ + L₂ + ...
• Capacitors: 1/C_eq = 1/C₁ + 1/C₂ + ...

Parallel combination:

• Resistors: 1/R_eq = 1/R₁ + 1/R₂ + ...
• Inductors: 1/L_eq = 1/L₁ + 1/L₂ + ...
• Capacitors: C_eq = C₁ + C₂ + ...

Analysis Methods

Nodal Analysis

1. Select a reference node (ground)
2. Write KCL at each non-reference node
3. Express voltages in terms of node voltages
4. Solve resulting equations

Procedure:

At node V₁: (V₁ - V₂)/R₁ + V₁/R₂ = I₁
At node V₂: (V₂ - V₁)/R₁ + (V₂ - V₃)/R₃ = 0

Mesh Analysis

1. Define mesh currents for each independent loop
2. Write KVL around each mesh
3. Solve for mesh currents
4. Find branch currents from mesh currents

Procedure:

Mesh I₁: -V₁ + I₁R₁ + (I₁ - I₂)R₂ = 0
Mesh I₂: (I₂ - I₁)R₂ + I₂R₃ + V₂ = 0

Choosing Method

Systematic Solution

For linear circuits with n nodes or m meshes:

• Write equations in matrix form
• Use Cramer's rule or matrix inversion
• Consider symbolic vs. numerical solutions

Network Theorems

Superposition

For linear circuits: total response = sum of responses from each source acting alone.

Steps:

1. Deactivate all independent sources except one
2. Find circuit response
3. Repeat for each source
4. Sum all responses

Deactivation rules:

• Voltage source → short circuit (V = 0)
• Current source → open circuit (I = 0)

Thevenin's Theorem

Any linear two-terminal network can be replaced by:

• Thevenin voltage V_th: open-circuit voltage
• Thevenin resistance R_th: resistance with sources deactivated

Finding R_th:

• Deactivate sources → find equivalent resistance
• Or: R_th = V_th / I_sc (open-circuit / short-circuit current)

Norton's Theorem

Complement to Thevenin:

• Norton current I_N: short-circuit current
• Norton resistance R_N = R_th

Equivalence: V_th = I_N R_th

Maximum Power Transfer

For a given Thevenin equivalent, maximum power to load occurs when:

R_L = R_th

Maximum power:

P_max = V_th²/(4R_th)

Source Transformation

Voltage source V with series R ↔ Current source I = V/R with parallel R

Delta-Wye (Δ-Y) Transformation

Δ to Y:

R₁ = (R_ab R_ac)/(R_ab + R_ac + R_bc)

Y to Δ:

R_ab = (R₁ R₂ + R₂ R₃ + R₃ R₁)/R₃

Millman's Theorem

For n parallel branches with series voltages:

V_out = (Σ V_i/R_i) / (Σ 1/R_i)

AC Circuit Analysis

Phasor Representation

v(t) = V_m cos(ωt + θ) → V = V_m ∠θ

Euler's relation: e^(jθ) = cos θ + j sin θ

Impedance

Complex Power

S = VI* = P + jQ

• Real power: P = VI cos φ [W]
• Reactive power: Q = VI sin φ [VAR]
• Apparent power: |S| = VI [VA]

Power factor: pf = cos φ = P/|S|

AC Circuit Analysis Procedure

1. Convert all sources to phasors
2. Replace L and C with jωL and 1/jωC
3. Solve using KCL/KVL (complex algebra)
4. Convert results back to time domain

Resonance

At resonance, imaginary part of impedance = 0.

Series resonance:

ω₀ = 1/√(LC)

• Minimum impedance (resistive)
• Current maximum
• Q = ω₀L/R = 1/(ω₀CR)

Parallel resonance:

ω₀ = 1/√(LC) × √(1 - 1/Q²)

• Maximum impedance
• Current minimum

Quality Factor Q

Q = energy stored / energy dissipated per cycle

For series RLC: Q = ω₀L/R = 1/(ω₀CR)

Operational Amplifiers

Ideal Op-Amp Characteristics

Golden Rules

1. Inputs draw no current (Z_in = ∞)
2. Inputs at same voltage (V+ = V- for negative feedback)

Basic Configurations

Op-Amp Specifications (Real)

Common Op-Amp Circuits

Instrumentation amplifier: High CMRR, high input impedance

Integrator: V_out = -(1/RC) ∫ V_in dt

Differentiator: V_out = -RC dV_in/dt

Log/Anti-log: Using diode/transistor characteristics

Active filters: Sallen-Key, Multiple Feedback

Frequency Response

Transfer Function

H(jω) = Y(jω)/X(jω) = |H(jω)| e^{j∠H(jω)}

• Magnitude: |H(jω)|
• Phase: ∠H(jω)

Bode Plots

Magnitude (log-log):

• 20 log₁₀|H(jω)| in dB
• Slope changes at poles/zeros

Phase (log-linear):

• ∠H(jω) in degrees
• -45°/decade per pole, +45°/decade per zero

Bode Plot Rules

Filter Types

First-Order Filters

RC Low-pass:

H(s) = 1/(1 + sRC)
f_c = 1/(2πRC)

RC High-pass:

H(s) = sRC/(1 + sRC)
f_c = 1/(2πRC)

Second-Order Filters

Sallen-Key low-pass:

H(s) = 1/(1 + s(3-k)/ω₀ + s²/ω₀²)

Where k = R_f/R₁ (gain).

Butterworth: Maximally flat magnitude Chebyshev: Ripple in passband Elliptic: Ripple in both bands

Bandwidth and Q

BW = f_c / Q

For second-order: Q determines peaking at resonance.

Transient Analysis

Time Domain Solution

For RC circuits:

v(t) = V_final + (V_initial - V_final)e^{-t/τ}

Where τ = RC (time constant)

For RL circuits: τ = L/R

First-Order Step Response

Second-Order Systems

For series RLC:

s² + (R/L)s + 1/(LC) = 0

Damping ratio:

ζ = R/(2)√(C/L) = R/(2)√(C/L)

Underdamped Response

v(t) = A e^{-αt} cos(ω_d t + φ)

Where:

• α = ζω_n
• ω_d = ω_n √(1 - ζ²)
• ω_n = 1/√(LC) (natural frequency)

Laplace Transform Solution

V(s) = H(s)V_in(s)

Transform pairs:

• 1/s → u(t)
• 1/(s+a) → e^{-at}u(t)
• ω/(s² + ω²) → sin(ωt)u(t)

Circuit Simulation

SPICE Basics

.Title: Circuit Name
* Comments
R1 1 2 1k
C1 2 0 1uF
V1 1 0 DC 5 AC 1
.AC DEC 10 1 10k
.PROBE
.END

Analysis Types

Model Parameters

• Resistor: R = value, TC1 = temp coefficient
• Capacitor: C = value, V = voltage coefficient
• MOSFET: LEVEL, L, W, VTO, KP...

Simulation Best Practices

• Check convergence options
• Use appropriate time steps
• Include initial conditions (.IC)
• Verify with hand calculations
• Model parasitics for accuracy

Common Errors to Avoid

• Forgetting to deactivate sources when finding Thevenin resistance
• Confusing series and parallel combinations
• Using wrong reference for nodal analysis
• Applying superposition to power (not linear)
• Forgetting to use complex numbers in AC analysis
• Confusing ω and f in reactance calculations
• Not considering op-amp saturation
• Using wrong filter cutoff formula
• Ignoring load effects in source transformation
• Confusing transient and steady-state responses

Key References

• Fundamentals of Electric Circuits by Sadiku — Standard textbook
• Microelectronic Circuits by Sedra/Smith — Op-amps and more
• SPICE for Circuits and Electronics by Rashid — Simulation
• The Art of Electronics by Horowitz & Hill — Practical design