rms

Examples of rms in the following topics:

• Root Mean Square Values

  • The root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity.
  • Now using the definition above, let's calculate the rms voltage and rms current.
  • If we are concerned with the time averaged result and the relevant variables are expressed as their rms values.

• RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram

  • By combining Ohm's law (Irms=Vrms/Z; Irms and Vrms are rms current and voltage) and the expression for impedance Z, from:
  • Therefore, the rms current will be Vrms/XL, and the current lags the voltage by almost 90∘.
  • Therefore, the rms current will be given as Vrms/XC, and the current leads the voltage by almost 90∘.

• Power

  • Power delivered to an RLC series AC circuit is dissipated by the resistance in the circuit, and is given as $P_{avg} = I_{rms} V_{rms} cos\phi$.

• Speed Distribution of Molecules

  • The distribution has a long tail because some molecules may go several times the rms speed.
  • The most probable speed vp (at the peak of the curve) is less than the rms speed vrms.
  • The most likely speed v_p is less than the rms speed v_rms.
  • Although very high speeds are possible, only a tiny fraction of the molecules have speeds that are an order of magnitude greater than v_rms.

• Avogador's Number

• Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams

  • The rms current Irms through an inductor L is given by a version of Ohm's law: $I_{rms} = \frac{V_{rms}}{X_L}$ where Vrms is the rms voltage across the inductor and $X_L = 2\pi \nu L$ with $\nu$ the frequency of the AC voltage source in hertz.

• Exercises

  • Answer: $mg = kx$ , so $x = mg/k = .1 {\rm kg}~ 9.8 {\rm \frac{m}{s^2}}/ 15.8 {\rm \frac{N}{m}} = .06 {\rm m} = 6 {\rm cm}$ .
  • Answer: $\gamma _ {\rm critical} = 2 \omega _0 = 2 \times 2 \pi \times 2 {\rm s^{-1}} \approx 25 {\rm s^{-1}}$ .
  • Answer: $\sqrt{\frac{1}{LC}} = \omega _0 = 2 \pi f_0 = \frac{2 \pi}{1 \rm{sec}}$$C = \frac{1}{L (2 \pi)^2} \approx 1 \rm Farad$ .
  • Therefore $C = \frac{1}{L (2 \pi)^2} \approx 1 \rm Farad$ .

• Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams

  • Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor.
  • This is considered to be an effective resistance of the capacitor to AC, and so the rms current Irms in the circuit containing only a capacitor C is given by another version of Ohm's law to be $I_{rms} = \frac{V_{rms}}{X_C}$, where Vrms is the rms voltage.

• Resonance in RLC Circuits

  • where Irms and Vrms are rms current and voltage, respectively.

• Eigenvectors and Orthogonal Projections

  • $\mathbf{x} = I \mathbf{x} = \left(V_r V_r ^T + V_0 V_0 ^T \right) \mathbf{x} = (\mathbf{x})_{\rm row} + (\mathbf{x})_{\rm null}