DC & AC Circuit Analysis
Fundamental laws, solving techniques, and alternating current properties.
1. DC Analysis & Ohm's Law
Click V, I, or R to see formula
Voltage: The potential difference pushing the current.
Current: The rate of flow of electric charge.
Resistance: The opposition to current flow.
2. AC Waveforms & Phasors
AC signals are represented as rotating vectors (phasors) in the complex plane.
AC Circuit Analysis
3. Average & RMS Values
Fundamental$$ V_{avg} = \frac{1}{T}\int_0^T v(t)\,dt \quad ; \quad V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2(t)\,dt} $$
$$ \text{Peak Factor} = \frac{V_{peak}}{V_{rms}} \quad ; \quad \text{Form Factor} = \frac{V_{rms}}{V_{avg}} $$
4. Signal Waveform Reference
8 Waveforms5. Phasor & Power Fundamentals
Vector Domain$$ \mathbf{V} = V_{rms}\angle\phi \quad ; \quad Z = R + jX $$
$$ P = VI\cos\phi \quad ; \quad Q = VI\sin\phi \quad ; \quad S = VI $$
6. Series RL Circuit
Lagging$$ V = \sqrt{V_R^2 + V_L^2} \quad ; \quad \phi = \tan^{-1}(X_L/R) $$
Current LAGS voltage by $\phi$. $\cos\phi$ is lagging.
7. Series RC Circuit
Leading$$ V = \sqrt{V_R^2 + V_C^2} \quad ; \quad \phi = \tan^{-1}(X_C/R) $$
Current LEADS voltage by $\phi$. $\cos\phi$ is leading.
8. Series RLC & Resonance
Resonance$$ Z = \sqrt{R^2 + (X_L - X_C)^2} \quad ; \quad f_0 = \frac{1}{2\pi\sqrt{LC}} $$
9. Parallel RL Circuit
Admittance$$ I = \sqrt{I_R^2 + I_L^2} \quad ; \quad Y = \sqrt{G^2 + B_L^2} $$
Voltage is common. Current LAGS by $\phi$.
10. Parallel RC Circuit
Admittance$$ I = \sqrt{I_R^2 + I_C^2} \quad ; \quad Y = \sqrt{G^2 + B_C^2} $$
Voltage is common. Current LEADS by $\phi$.
11. Parallel Resonance (Practical)
Tank Circuit$$ \omega_r = \sqrt{\frac{1}{LC} - \frac{R^2}{L^2}} \quad ; \quad Z_{max} = \frac{L}{CR} $$
Acts as a BAND-STOP filter. $I$ is minimum at $f_r$.
12. Series Resonance
Acceptor$$ f_0 = \frac{1}{2\pi\sqrt{LC}} \quad ; \quad Q = \frac{\omega_0 L}{R} $$
Voltage Magnifier: $V_L = V_C = QV$. $Z$ is minimum ($R$).
13. Series vs Parallel Comparison
Switch to Parallel| Feature | Series |
|---|---|
| Impedance | Minimum (R) |
| Current | Maximum |
| Filter | Band-Pass |
14. Power in AC (Complete)
Power Triangle$$ S = P + jQ \quad ; \quad pf = \cos\phi = P/S $$
$P$ (Watts), $Q$ (VAR), $S$ (VA).
15. Electric vs Magnetic Analogy
Magnetism$$ EMF(V) \leftrightarrow MMF(NI) \quad ; \quad R \leftrightarrow \mathcal{S} $$
Current $I \leftrightarrow \Phi$ (Flux). Conductivity $\sigma \leftrightarrow \mu$.
16. Three Phase Systems
STAR (Y)$$ V_L = \sqrt{3}V_{ph} \quad ; \quad I_L = I_{ph} $$
Balanced Load: $P = \sqrt{3}V_L I_L \cos\phi$. Phase diff: 120°.
17. Graph Theory: Topology
Network Science$$ \text{Rank} = N - 1 \quad ; \quad \sum \text{deg}(k) = 2B $$
$N$ = Nodes, $B$ = Branches. Euler: $N - B + l = 1$.
18. Trees, Twigs & Links
Terminology| Term | Notation | Formula |
|---|---|---|
| Twig | Tree Branch | $N-1$ |
| Link | Co-tree | $B-N+1$ |
| Possible Trees | det[A][A'] | $N^{N-2}$ (for complete) |
19. Incidence Matrix [A]
KCL Basis$$ [A][I_b] = [0] \quad ; \quad \text{Sum of column rows} = 0 $$
Reduced matrix $[\tilde{A}]$ by deleting reference row.
20. Tie-set (Loop) Matrix
KVL Basis$$ [B_f][V_b] = [0] \quad ; \quad \text{Rank} = B - N + 1 $$
Each row = 1 Fundamental Loop using 1 Link.
21. Cut-set Matrix [Q]
Twig Dependent$$ [Q_f][I_b] = [0] \quad ; \quad \text{Rank} = N - 1 $$
Each row corresponds to one fundamental Cut-set.
22. Principle of Duality
Morphing Duals$$ R \leftrightarrow G \quad ; \quad L \leftrightarrow C \quad ; \quad V \leftrightarrow I $$
Node analysis is dual to Mesh analysis.