Current

• $I=\frac{dq}{dt}$ is the electric current (in $\mathrm{A}$) of a circuit in which a charge $q$ (in $\mathrm{C}$) passes through some surface in the circuit in an interval of time $t$ (in $\mathrm{s}$)

  • The SI unit of current is the ampere (abbreviated $\mathrm{amp}$ or $\mathrm{A}$), defined as $1\mathrm{A}=1C/s$

• $\vec{\mathbf{J}}=\frac{I}{A}\hat{n}$ is the current density (in $A/{\mathrm{m}}^2$)

  • $A$ is the cross-sectional area (in ${\mathrm{m}}^2$)
  • $\hat{n}$ is a unit vector normal to the cross-sectional area

• A current is positive if it flows from the positive terminal of a voltage source to the negative terminal

Conventional current

• The conventional direction of (possitve) current, also known as conventional current, is arbitrarily defined to be the direction in which positive charges would flow

Alternating current (AC)

$v(t)=V_m\cos (\omega t+\varphi ),i(t)=I_m\cos (\omega t)$

• $v(t)$ and $i(t)$ are the instantaneous voltage and current at time $t$
• $V_m$ and $I_m$ are the peak voltage and current
• $\omega$ is the angular frequency (in radians per second)
• $\varphi$ is the phase angle between the voltage and current
• $I_{\text{rms}}=\frac{I_m}{\sqrt{2}}$ and $V_{\text{rms}}=\frac{V_m}{\sqrt{2}}$ are the rms values of the current and voltage
• The phasor representation of the voltage and current are given by:
  • $\tilde{V}=V_m\angle {\varphi }_v$
  • $\tilde{I}=I_m\angle {\varphi }_i$
• $I_{\text{rav}}=\frac{2}{\pi }{I}_m$ is the rectified average current
• $V_m=I_{m}Z$

RLC series circuit

• (inductive circuit) $X_L>X_C$, then $\varphi >0$ and the circuit is inductive
  • (pure inductor) $R=0$, $X_C=0$, then $Z=X_L$ and $\varphi =\frac{\pi }{2}$
• (capacitive circuit) $X_L<X_C$, then $\varphi <0$ and the circuit is capacitive
  • (pure capacitor) $R=0$, $X_L=0$, then $Z=X_C$ and $\varphi =-\frac{\pi }{2}$
• (resistive circuit) $X_L=X_C$, then $\varphi =0$ and the circuit is resistive
  • (pure resistor) $X_L=0$, $X_C=0$, then $Z=R$ and $\varphi =0$

Resonance

• When $X_L\approx X_C$, which is also $\omega \approx {\omega }_0$, the circuit is said to be in resonance
  • ${\omega }_0=\frac{1}{\sqrt{LC}}$ is the resonant angular frequency
  • $f_0=\frac{{\omega }_0}{2\pi }$ is the resonant frequency
  • At resonance, $Z$ is minimum and equal to $R$, and the current is maximum and in phase with the voltage

Power

• $p(t)=v(t)i(t)$
• $P_{avg}=\overline{p(t)}=\frac{1}{2}{V}_m{I}_m\cos \varphi =V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$ is the average power delivered to the circuit
  • $\cos \varphi$ is the power factor
  • If $\varphi =0$, then $P_{avg}=V_{\text{rms}}{I}_{\text{rms}}$ (maximum power transfer)
  • If $\varphi =\pm \frac{\pi }{2}$, then $P_{avg}=0$ (no power transfer)

Transformers

• $\frac{V_2}{V_1}=\frac{N_2}{N_1}$
  • $V_1$ is the primary voltage (the supply voltage)
  • $V_2$ is the secondary voltage (the output voltage)
  • $N_1$ and $N_2$ are the number of turns in the primary and secondary coils
• $V_1{I}_1=V_2{I}_2$
  • $I_1$ is the primary current
  • $I_2$ is the secondary current