circuits
Current & conduction, Kirchhoff & Thévenin, emf & internal resistance, capacitors & RC, AC impedance & phasors, RLC resonance, AC power.
current & conduction · series, parallel & Kirchhoff · emf & internal resistance · capacitors & RC · AC & impedance · RLC & resonance · AC power
Current, conduction & resistance
Current & current density. \(I=\dd q/\dd t\) (amperes); conventional current flows from positive to negative, electron flow opposite. Current density \(\vv J\) = current per cross-sectional area; per carrier type \(\vv J=nq\vv v_d\) (drift model \(I=nqAv_d\)); species contributions add.
Charge conservation. Continuity: \(\nabla\cdot\vv J=-\partial\rho/\partial t\); steady currents have \(\nabla\cdot\vv J=0\) — the same \(I\) through every cross-section, and the junction rule.
Potential difference. \(V=W/q\), work per unit charge between two points.
Conductivity & resistance. \(\vv J=\sigma_c\vv E\); resistivity \(\rho_r=1/\sigma_c\) spans \(\sim10^{-8}\,\Omega\,\mathrm m\) (metals) to \(10^{16}\) (insulators). \(R=V/I\); \(R=\rho_rL/A=L/\sigma_cA\); ohmic if \(R\) constant at fixed \(T\).
Microscopic picture. A tiny drift \(v_d=qE\tau/m\) rides on large random thermal speeds (\(\tau\) = mean time between collisions); \(\sigma_c=nq^2\tau/m\). Ohm's law holds because \(\tau\) is field-independent. Metals: \(n\) fixed, \(\tau\) shrinks with \(T\) so \(\rho_r\) rises; drift speeds are mm/s scale.
Semiconductors. Carriers: conduction-band electrons and valence-band holes, thermally activated — \(\sigma_c\) rises steeply with \(T\), opposite to metals. Doping: donor impurities give n-type (electron) conduction, acceptors give p-type (hole) conduction; parts-per-million doping shifts \(\sigma_c\) by orders of magnitude.
Power & energy. \(P=IV=I^2R=V^2/R\) (collisions turn field work into heat); \(E=VIt=qV\).
Series, parallel & Kirchhoff
Series. \(R_s=\sum R_i\); same \(I\), voltages add.
Parallel. \(R_p^{-1}=\sum R_i^{-1}\); same \(V\), currents add.
Kirchhoff. Node: \(\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}\) (charge conservation). Loop: \(\sum\mathcal E=\sum IR\) (single-valued potential).
Thévenin. Any linear two-terminal network of sources and resistors ≡ one emf \(\mathcal E_{\mathrm{eq}}\) in series with one \(R_{\mathrm{eq}}\): \(\mathcal E_{\mathrm{eq}}\) = open-circuit voltage; \(R_{\mathrm{eq}}=\mathcal E_{\mathrm{eq}}/I_{\mathrm{short}}\) = resistance seen with all emfs shorted out.
Emf, internal resistance & dividers
Emf physically. \(\mathcal E=W_{\mathrm{source}}/q\), work per unit charge by non-electrostatic forces. Around a full circuit \(\oint\vv E\cdot\dd\vv s=0\) for electrostatic fields, so somewhere a source must push charge against \(\vv E\) — chemical forces in a voltaic cell, \(\vv v\times\vv B\) in a generator.
Internal resistance. Terminal \(V=\mathcal E-Ir\); short-circuit \(I=\mathcal E/r\).
Load power. \(P=I^2R=\mathcal E^2R/(R+r)^2\), maximum at \(R=r\).
Potential divider. \(V_{\mathrm{out}}=V_{\mathrm{in}}\,R_2/(R_1+R_2)\).
Combining cells. Series: \(\mathcal E_{\mathrm{tot}}=\sum\mathcal E\), \(r_{\mathrm{tot}}=\sum r\). \(N\) identical cells in parallel: \(\mathcal E\) unchanged, \(r_{\mathrm{eq}}=r/N\).
Capacitors & RC transients
Capacitance. \(C=Q/V\); parallel plate \(C=\varepsilon_0A/d\), \(\times\kappa\) with a dielectric (dielectrics).
Combinations. Parallel: \(C_p=\sum C_i\). Series: \(C_s^{-1}=\sum C_i^{-1}\).
Stored energy. \(U_C=\tfrac12QV=\tfrac12CV^2=Q^2/(2C)\).
RC charging. \(Q=CV_0(1-e^{-t/RC})\), \(I=(V_0/R)\,e^{-t/RC}\), \(V_C=V_0(1-e^{-t/RC})\).
RC discharging. \(Q=Q_0e^{-t/RC}\), \(I=I_0e^{-t/RC}\), \(V=V_0e^{-t/RC}\); time constant \(\tau=RC\).
Leaky capacitor. Fill with slightly conducting dielectric (resistivity \(\rho_r\), constant \(\kappa\)): it discharges through itself with \(\tau=\varepsilon\rho_r=\kappa\varepsilon_0\rho_r\) — independent of size and shape.
AC circuits & impedance
Sinusoidal steady state. Drive a linear circuit at \(\omega\): after transients decay, every voltage and current oscillates at \(\omega\), differing only in amplitude and phase. Represent \(V(t)=\mathrm{Re}\,\tilde Ve^{i\omega t}\); solve algebraically in the complex amplitudes, take the real part at the end.
Impedance. \(\tilde V=Z\tilde I\); admittance \(Y=1/Z\), \(\tilde I=Y\tilde V\). \(|Z|\) sets the amplitude ratio, \(\arg Z\) the phase by which \(V\) leads \(I\).
| Element | \(Z\) | Reactance | Phase |
|---|---|---|---|
| resistor | \(R\) | — | \(V\), \(I\) in phase |
| inductor | \(i\omega L\) | \(X_L=\omega L\) | \(V\) leads \(I\) by \(90^\circ\) |
| capacitor | \(1/i\omega C\) | \(X_C=1/\omega C\) | \(V\) lags \(I\) by \(90^\circ\) |
Combination. Impedances add in series, admittances in parallel — resistor rules with complex numbers. Limits as sanity check: \(\omega\to0\): capacitor open, inductor short; \(\omega\to\infty\): reversed.
Phasors. Complex amplitudes as arrows rotating at \(\omega\); the instantaneous value is the real-axis projection. Series elements share the current phasor: \(V_R\) along \(I\), \(V_L\) at \(+90^\circ\), \(V_C\) at \(-90^\circ\); the source phasor is their vector sum.
RLC & resonance
Natural response (series RLC). \(L\ddot Q+R\dot Q+Q/C=0\): with \(\alpha=R/2L\), \(\omega_0=1/\sqrt{LC}\) — underdamped (\(\alpha\lt\omega_0\)): decaying oscillation \(e^{-\alpha t}\cos(\omega_dt+\varphi)\), \(\omega_d=\sqrt{\omega_0^2-\alpha^2}\); overdamped (\(\alpha\gt\omega_0\)): sum of two decaying exponentials; critical at \(\alpha=\omega_0\). Single-inductor transients (\(\tau=L/R\)): induction.
Quality factor. \(Q_f=\omega\times\dfrac{\text{energy stored}}{\text{average power dissipated}}=\dfrac{\omega_0L}{R}\) (series RLC); stored energy decays as \(e^{-\omega_0t/Q_f}\) — about \(Q_f\) radians of ringing.
Driven series RLC. \(\mathcal E_0\cos\omega t\) gives \(I(t)=I_0\cos(\omega t+\varphi)\): \[I_0=\frac{\mathcal E_0}{\sqrt{R^2+(\omega L-1/\omega C)^2}},\qquad \tan\varphi=\frac{1/\omega C-\omega L}{R}.\] Current leads below resonance (capacitive), lags above (inductive).
Resonance. At \(\omega_0=1/\sqrt{LC}\) the reactances cancel: \(Z=R\) purely resistive, current maximal, \(V_L\) and \(V_C\) equal, opposite, and \(Q_f\) times larger than \(\mathcal E_0\). Half-power bandwidth \(\Delta\omega\approx R/L=\omega_0/Q_f\) — sharper with higher \(Q_f\).
Parallel RLC. Admittances add: \(Y=1/R'+i(\omega C-1/\omega L)\); at resonance \(|Y|\) is minimal — the source current dips while current circulates internally between \(L\) and \(C\).
AC power
Rms. \(V_{\rm rms}=V_0/\sqrt2\), \(I_{\rm rms}=I_0/\sqrt2\) for sinusoids — the DC-equivalent heating values.
Average power. \(P=\tfrac12\mathcal E_0I_0\cos\varphi=V_{\rm rms}I_{\rm rms}\cos\varphi\); power factor \(\cos\varphi\). Resistor: \(P=V_{\rm rms}^2/R\). Ideal inductors and capacitors dissipate nothing on average — energy sloshes in and out each quarter cycle.
Transformers. Voltage/current ratios and \(I^2R\) transmission-loss argument: electromagnetism — induction.