Single-Phase Full-Wave Diode Rectifiers

Single-Phase Full-Wave Diode Rectifier:

• Bridge Rectifier:

  • Lower peak diode voltage.

  • Suitable for high-voltage applications.

• Center-Tapped Transformer Rectifier:

  • Provides electrical isolation.

  • Only one diode voltage drop between source and load.

  • Desirable for low-voltage, high-current applications.

• Full-Wave Bridge Rectifier:

  

  • Diode Conducting Pairs:

    • D1 & D2 conduct together.

    • D3 & D4 conduct together.

  • Kirchhoff’s Voltage Law (KVL):

    • D1 & D3 cannot be on simultaneously.

    • D2 & D4 cannot conduct simultaneously.

  • Load Current:

    • Positive or zero.

    • Never negative.

  • Load Voltage:

    • +$v_s$ when D1 & D2 are on.

    • +$v_s$ when D3 & D4 are on.

• Reverse-Biased Diode Voltage:

  • Maximum is the peak value of the source.

  • With D1 on, the voltage across D3 is -$v_s$.

• Source Current:

  • Current entering the bridge: $i_{D1} - i_{D4}$.

  • Symmetric about zero.

  • Average source current: zero.

  • RMS source current = RMS load current.

  • Source current = load current for half the period.

  • Source current = -load current for the other half.

  • Squares of load and source currents are equal, so RMS currents are equal.

• Output Voltage Frequency:

  • Fundamental frequency: $2\omega$ (twice the AC input frequency).

• Output Voltage Fourier Series:

  • Consists of a DC term and even harmonics of the source frequency.

• The Center-Tapped Transformer Rectifier

  

  • Kirchhoff’s Voltage Law (KVL):

    • Only one diode can conduct at a time.

  • Load Current:

    • Positive or zero.

    • Never negative.

  • Output Voltage:

    • +$v_{s1}$ when D1 conducts.

    • -$v_{s2}$ when D2 conducts.

  • Transformer Secondary Voltages:

    • Related to the source voltage by $v_{s1} = v_{s2} = v_s \left(\frac{N_2}{2N_1}\right)$.

• Reverse-Biased Diode Voltage:

  • Maximum is twice the peak value of the load voltage.

  • Shown by KVL around the transformer secondary windings, D1, and D2.

• Source Current:

  • Current in each half of the transformer secondary is reflected to the primary.

  • Average source current: zero.

• Transformer Function:

  • Provides electrical isolation between the source and the load.

• Output Voltage Frequency:

  • Fundamental frequency: $2\omega$ (twice the AC input frequency).

• Resistive Load

  • $$\nu_o(\omega t)=\begin{cases}V_m\sin\omega t&\quad\mathrm{for~}0\leq\omega t\leq\pi\\-V_m\sin\omega t&\quad\mathrm{for~}\pi\leq\omega t\leq2\pi\end{cases}$$
    The voltage across a resistive load
  • $$\begin{aligned} V_o&=\frac{1}{\pi}\int\limits_0^\pi V_m\sin\omega td(\omega t)=\frac{2V_m}{\pi}\\ I_o & =\frac{V_o}{R}=\frac{2V_m}{\pi R} \end{aligned}$$
    The dc component of the output voltage is the average value, and load current is simply the resistor voltage divided by resistance

  • Power absorbed by the load resistor $= I_{\mathrm{rms}}^2 \cdot R$, where $I_{\mathrm{rms}} = \dfrac{I_m}{\sqrt{2}}$

  • Power factor $= 1$

• RL-Load

  

  • Load Current ($i_o$):

    • Reaches a periodic steady-state condition after a start-up transient.

  • Full-Wave Rectified Sinusoidal Voltage Across the Load:

    • Can be expressed as a Fourier series.

    • Consists of a DC term and the even harmonics.

  $$\begin{aligned} \nu_o(t) & =V_o+\sum_{n=2,4\ldots}^\infty V_n\cos{(n\omega_0t+\pi)}\\ V_o & =\frac{2V_m}{\pi}\quad\text{and}\quad V_n=\frac{2V_m}{\pi}\left(\frac{1}{n-1}-\frac{1}{n+1}\right) \end{aligned}$$

• Current in the RL Load:

  • Computed using superposition.

  • Each frequency is considered separately and results are combined.

• DC Current and Current Amplitude:

  • $$\begin{aligned} I_0 & =\frac{V_0}{R} \quad ~\text{and}~\quad I_n & =\frac{V_n}{Z_n}=\frac{V_n}{|R+jn\omega L|} \end{aligned}$$
    Computed for each frequency.
  • $$i(\omega t)\approx I_o=\frac{V_o}{R}=\frac{2V_m}{\pi R}\quad ~\text{and}~ \quad~ I_{\mathrm{rms}}\approx I_o$$
    (load inductance may be relatively large or made large by adding external inductance) If

• Harmonic ($n$) Effects:

  • $n~\uparrow~\Rightarrow~V_n~\downarrow$.

  • For an RL load, as $n~\uparrow ~\Rightarrow ~Z_n~\uparrow$.

  • Combination of $V_n~\downarrow ~ + ~Z_n~\uparrow~\Rightarrow ~I_n~\downarrow$ rapidly with $n~\uparrow$.

• Significance in RL Load:

  • DC term and only a few AC terms are usually necessary to describe the current in an RL load.

• RLE-Load

  

• Applications:

  • DC motor drive circuit.

  • Battery charger.

• Modes of Operation:

  • Continuous-current mode.

  • Discontinuous-current mode.

• $$\boxed{I_o=\frac{V_o-V_{\mathrm{dc}}}{R}=\frac{\frac{2V_m}{\pi}-V_{\mathrm{dc}}}{R}}$$
  The dc (average) component of current in this circuit:

• Continuous-Current Mode:

  • Load current is always positive for steady-state operation.

  • One pair of diodes is always conducting.

  • Voltage across the load is a full-wave rectified sine wave.

  • Only modification to the RL load analysis is in the DC term of the Fourier series.

• Discontinuous-Current Mode:

  • Load current returns to zero during every period.

• Capacitance Output Filter

  

  • Produces an output voltage that is essentially DC.

  • Analysis similar to that of the half-wave rectifier with a capacitance filter.

  • Capacitor discharge time is smaller than in the half-wave circuit due to the rectified sine wave in the second half of each period.

  • Output voltage ripple for the full-wave rectifier is approximately half that of the half-wave rectifier.

  • Peak output voltage is less in the full-wave circuit due to two diode voltage drops rather than one.

• Output voltage is a positive sine function when one of the diode pairs is conducting.

• Output voltage is a decaying exponential otherwise.

• $$\nu_o(\omega t)=\begin{cases}\left|V_m\sin\omega t\right|&\text{one diode pair on}\\(V_m\sin\theta)e^{-(\omega t-0)/\omega RC}&\text{diodes off}\end{cases}$$
  Assuming ideal diodes,

• The diodes become reverse biased at $\theta$

  $$\theta=\tan^{-1}(-\omega RC)=-\tan^{-1}(\omega RC)+\pi$$

• The maximum output voltage ($V_m$) , and the minimum output voltage is determined by evaluating $\nu_o$ at the angle at which the second pair of diodes turns on, which is at $\omega t = \pi + \alpha$. At that boundary point

  $$\begin{aligned} & (V_m\sin\theta)e^{-(\pi+\alpha-\theta)/\omega RC}=-V_m\sin\left(\pi+\alpha\right) \\ \Rightarrow &~(\sin\theta)e^{-(\pi+\alpha-\theta)/\omega RC}-\sin\alpha=0\\ & \text{Solve numerically for}~\alpha \end{aligned}$$

• $$\boxed{\Delta V_o=V_m-\begin{vmatrix}V_m\sin{(\pi+\alpha)}\end{vmatrix}=V_m(1-\sin{\alpha})}$$
  The peak-to-peak voltage variation, or ripple, is the difference between maximum and minimum voltages

• $\alpha$ is larger for the full-wave rectifier and the ripple is smaller for a given load

• The peak-to-peak ripple

  $$\boxed{\Delta V_o\approx\frac{V_m\pi}{\omega RC}=\frac{V_m}{2fRC}}$$

• Note that the approximate peak-to-peak ripple voltage for the full-wave rectifier is one-half that of the half-wave rectifier