To test the understanding of the lesson

Question

Give the instantaneous power supplied by a generator of emf \(e(t)\) supplying a current \( i(t)\).

Solution

The instantaneous supplied power is :

\(p(t)=e(t)i(t)\)

Question

What is the average power (or active) dissipated in sinusoidal regime in a complex impedance \(\underline Z\) (we note \(\varphi=arg(\underline Z)\)) ?

Solution

Note \(U_{eff}\) and \(I_{eff}\) the effective (RMS) voltage across the impedance and the effective (RMS) intensity of the current through the impedance.

The average power dissipated in the impedance is :

\(P=U_{eff}I_{eff}cos \varphi\)

If \(R\) is the real part of the complex impedance \(\underline Z\) :

\(P=RI^2_{eff}\)

If \(G\) is the real part of the complex admittance \(\underline Y =1/ \underline Z\) :

\(P=GU^2_{eff}\)

Question

Give the definition of gain in decibel (\(dB\)) associated with a transfer function \(\underline H (j\omega)=\frac{\underline v_S}{\underline v_E}\).

Solution

Let \(G(\omega)\) the transfer function \(\underline H (j\omega)=\frac{\underline v_S}{\underline v_E}\) module.

The gain in \(dB\) is given by :

\(G_{dB}=20\;log\;G(\omega)\)

Question

Give the definition of a cut-off frequency.

Solution

The cut-off frequency \(\omega_c\) is such that :

\(G(\omega_c)=\frac{G_{max}}{\sqrt{2}}\)

Where, equivalently :

\(G_{dB}(\omega_c)=G_{dB,max}-3\;dB\)

Question

Give the relationship between the quality factor \(Q\) of a damped oscillator and pass-band (pass-band filter).

Solution

\(Q=\frac{\omega_0}{\Delta \omega}\)

Thus, the smaller the pass-band, the better the filter quality (or \(Q\) large).

Question

State the rules of the voltage divider and current divider.

Solution

  • Voltage divider :

    Two resistors in series :

    \(u_{R_1}=\frac{R_1}{R_1+R_2}u\;\;\;and\;\;\;u_{R_2}=\frac{R_2}{R_1+R_2}u\)

  • Current divider :

    For two resistances in parallel :

    \(i_{R_1}=\frac{R_2}{R_1+R_2}i\;\;\;and\;\;\;i_{R_2}=\frac{R_1}{R_1+R_2}i\)

Question

Define the characteristic of a dipole.

Solution

The characteristic of a dipole is the graph of the current \(i\) in the dipole as a function of the voltage \(u\) at its terminals (or otherwise, \(u\) in terms of \(i\)).

Question

  • What is the time constant of an RC series circuit ?

  • What is the time constant of a series RL circuit ?

Solution

  • For the RC circuit : \(\tau = RC\)

  • For the RL circuit : \(\tau = L/R\)

Question

  • What are the impedances of a capacitor and a coil ?

  • What are the admittances of a capacitor and a coil ?

Solution

  • For capacitor : \(\underline z_C=1/jC\omega\)

    For the coil : \(\underline z_L=r+jL\omega\), where \(r\) is the coil resistance.

  • For capacitor : \(\underline y_C=jC\omega\)

    For the coil : \(\underline y_L=\frac{1}{r+jL\omega}\)