CHAPTER 6

Single-Phase AC to DC Conversion

Single-phase AC supplies most homes and offices due to relatively low power demand and simple wiring. Since more and more devices are DC-based, the AC/DC conversion is required. The AC/DC conversion and converter are commonly called as the rectification and rectifier, respectively. Diodes automatically pass forward current and block reverse current. The feature of one-direction conduction supports circuit constructions for the AC/DC conversion.

6.1  Half-Wave Rectification

Figure 6.1 shows a simple rectifier using only one diode. The topology is named as the half-wave rectifier since only the positive cycle of v_ac can appear at the output terminal and supply the load. The AC source voltage is expressed as v_ac = V_m sin(ωt). The input current passes through the diode and becomes the same as the output, i_d = i_o, as shown in Fig. 6.2. The average voltage of the output can be determined by (6.1), the RMS value is derived by (6.2), and the averaged power is computed by (6.3). The output voltage and current are discontinuous when a pure resistive load is applied at the output terminal. The power quality is considerably low at both the input and output port. The power factor for the AC side can be computed as 0.707. Following the DC quality discussion in Sec. 4.6, the form factor of v_o is 0.637, which is also low.

FIGURE 6.1 Half-wave rectifier circuit using one diode.

FIGURE 6.2 Waveforms of half-wave rectifier circuit.

6.1.1  Capacitor for Filtering

To improve the DC quality of v_o and i_o, capacitors can be applied in parallel with the load, as illustrated in Fig. 6.3. The capacitor, C_O, is expected to smoothen the DC voltage, v_o. The diode is conducting and connect the source only if the instantaneous value of v_ac is higher than v_o. The time period refers to as “D on,” as illustrated in Fig. 6.4. The voltage of v_o follows the same as the V_m sin(ωt) during the on-state of the diode. The diode is naturally turned off by the reverse-bias condition when v_ac = V_m sin(ωt) < v_o. During the off-state, the capacitor C_O is discharged by load power consumption, which leads v_o to decrease. The peak-to-peak voltage ripple is ΔV_O, as indicated in Fig. 6.4. The averaged value of v_o can be approximated by

FIGURE 6.3 Half-wave rectifier with C filtering.

FIGURE 6.4 Waveforms of half-wave rectifier circuit with C filter.

When ΔV_O is assigned to be low enough, the conduction time of D is much shorter than the off-state period. The diode is reverse-biased and off-conducting for most of the line cycle, as illustrated in Fig. 6.4. When the off-state of D is approximated as the whole line cycle, the following can be established:

According to (6.5), the capacitance can be sized for C_O when ΔV_O is specified, as expressed by

where f_b is the fundamental frequency of v_ac, ω = 2πf_b.

6.1.2  Case Study

A case study can demonstrate the design of a half-wave rectifier. The input voltage, v_ac, is rated as 230 V for the RMS value with a frequency of 50 Hz. The averaged value of the output voltage is specified as 320-V DC without any loss consideration. The load resistance is rated by R = 1024 Ω. Following (6.4), the peak-to-peak ripple of the output voltage, ΔV_O, can be approximated to be 10.54 V. The capacitance at the DC side can be rated as C_O = 593 μF by (6.6). Significant capacitance is generally required to maintain the output voltage with low ripples for high power ratings due to the half-cycle conduction.

6.2  Full-Wave Bridge Rectifier

Figure 6.5a illustrates the passive four-switch bridge used for single-phase AC to DC conversion. For the positive half-cycle of the input voltage (v_ac), the diagonal pair of D_AH and D_BL are forward-biased, as illustrated in Fig. 6.5b. The current flow is from D_AH to R and then D_BL for returning to the source. Meanwhile, another diagonal pair of D_BH and D_AL are reverse-biased. When the negative half-cycle of the input voltage (v_ac) appears, the diagonal pair of D_BH and D_AL are forward-biased, as illustrated in Fig. 6.5c. The current flow is from D_BH to R and then D_AL for return. The output voltage is always positive regardless of the polarity of v_ac, which gains the name of full-wave rectification.

FIGURE 6.5 Full-wave rectifier for single-phase AC to DC conversion: (a) circuit; (b) positive cycle; (c) negative cycle.

Considering ideal diodes, the waveforms of voltage and current are illustrated in Fig. 6.6. The DC voltage, v_o, is continuous and repeats the same every half of the line cycle. The double-line frequency appears at v_o, which is equal to 2ω, corresponding to the definition, v_ac = V_m sin(ωt). The average voltage of the output can be derived in (6.7). Furthermore, the RMS value can be computed by (6.8), the same RMS value of v_ac without consideration of any non-ideal factors. The input AC signals, v_ac and i_ac, are in phase and show unity power factor without distortion. Without filtering, the output power quality is low for DC loads since the peak-to-peak ripple is equal to V_m. The form factor is derived to be which clearly shows better DC quality than the half-wave rectifier.

FIGURE 6.6 Waveform of full-wave rectifier in operation with resistive loads.

6.2.1  Capacitor for Filtering

A capacitor can be implemented across the load to improve the quality of v_o, as shown in Fig. 6.7. The circuit operation is commonly called the peak detection since the diodes conduct only if |v_ac| > v_o. The operation principle is simple by following the diode I-V characteristics, and can be described by the following:

FIGURE 6.7 Circuit of full-wave rectifier in operation with C and R.

•   When v_ac > v_o, the diagonal diode pair, D_AH and D_BL, conducts.

•   When −v_ac > v_o, the diagonal diode pair, D_BH and D_AL, conducts.

•   When |v_ac| > v_o, one diagonal diode pair is forward-biased; C_O is charged and load is supplied by the AC source.

•   When |v_ac| ≤ v_o, all diodes are reverse-biased to isolate the load from the source; C_O is discharged to keep v_o steady.

When a significant volume of C_O is considered, the operation of the peak detection is as illustrated in Fig. 6.8. The waveform of v_o is close to a straight line and rides on the top of |v_ac|. When all diodes are reverse-biased, the AC source is disconnected from the load. The right side becomes a simple RC circuit that the capacitor, C_O, is discharged to slow down the voltage drop of v_o, as shown by the zoom-in plot in Fig. 6.8. The condition can be expressed by

FIGURE 6.8 Voltage waveform of full-wave rectifier in operation with C filtering.

where T_OFF refers to the period of the off-state, and ΔV_O represents the voltage drop of v_o from top to bottom in each half cycle, as illustrated in Fig. 6.8. The averaged value of v_o can be estimated by (6.10). When the value of AVG(v_o) is specified, the peak-to-peak voltage ripple can be determined by (6.11).

One important approximation is made that the off-state period is the half cycle, as expressed by . Following (6.9) and the low ripple of v_o, the output capacitor can be rated by (6.12). When the capacitance is significant, the output DC voltage is maintained in good quality. However, the main issue is from the quality of the input current, i_ac, as illustrated in Fig. 6.9. It shows a significantly high current peak in comparison with the load current, i_o, to balance the power flow from AC to DC. The diodes conduct current only for a short time within each half-cycle. The distorted waveform of i_ac is measured to be more than 100% in total harmonic distortion (THD). Without additional power factor correction, the C-type filter and the peak detection operation can only qualify for low-power applications to avoid significant disturbance to power grids.

FIGURE 6.9 Waveform of full-wave rectifier in operation with C filtering.

where the voltage source is v_ac = V_m sin(2πf_bt).

A case study can demonstrate the design of a full-wave rectifier with C filtering, as shown in Fig. 6.7. The input voltage, v_ac, is rated by 230 V (RMS) and 50 Hz (frequency). The averaged value of the output voltage is specified as 320-V DC without any loss consideration. The load resistance is rated by R = 1024 Ω. The specification is the same as the case study for the half-wave rectifier for comparison. Following (6.11), the peak-topeak ripple of the output voltage, ΔV_O, can be approximated as 10.54 V. The capacitance at the DC side can be rated as C_O = 297 μF by (6.12).

6.2.2  Inductor for Filtering

Distortion at the DC waveform can be reduced by other smoothing components, i.e., inductors. When an inductor is in series with the load resistor, as shown in Fig. 6.10, the current, i_o, is expected to be smooth according to (6.13).

FIGURE 6.10 Circuit of full-wave rectifier in operation with LR.

where v_ac = V_m sin(2πf_bt). When |v_ac| > v_o, the inductor current rises according to (6.13). Otherwise, the level of i_o reduces if |v_ac| < v_o. The steady-state waveform is illustrated in Fig. 6.11, showing low ripples of v_o. The average voltage across L is zero in steady state; therefore, the averaging value of v_o can be derived as the same result as expressed in (6.7) without loss consideration. The average value of i_o can be determined by (6.14). When L is significant in value, the waveform of v_o is flat, represented by the averaged value of v_o, AVG(v_o). Thus, the energy is stored in L and leads to the rise of i_o from bottom to top, which is expressed by (6.15).

FIGURE 6.11 Waveform of full-wave rectifier with L filtering.

where I_top and I_bot refer to the highest and lowest value of i_o in a steady state; T_top and T_bot indicate the moment when i_o = I_top and i_o = I_bot, respectively. The moment of T_top and T_bot can be identified as the phase representation as α and β, which is shown in Fig. 6.13 and can be identified by

A further derivation leads to (6.18) and (6.19). The rating of L can be determined by (6.19) when the peak-to-peak current ripple, ΔI_O, is specified. The ripple percentage of the current is the same as the relative ripple level of v_o when the resistive load is applied.

A case study can demonstrate the design of a full-wave rectifier with L filtering. The input voltage, v_ac, is rated as 230 V (RMS) and 50 Hz (frequency). The nominal load resistance is rated as R = 20.71 Ω. The averaging values of v_o and i_o in the steady state are specified to be 207.1 V and 10 A, respectively. When the peak-to-peak ripple of i_o is designed to be ΔI_O = 20% × AVG(i_o), the same percentage value is applied to the voltage ripple of v_o. The inductance can be determined by (6.19) to be L = 218 mH. In general, the higher the L, the flatter the values of i_o and v_o that can be achieved.

The filtering aims to improve the power quality of the DC current output but leads to the concern of the power quality of the input current. As shown in Fig. 6.11, the waveform of i_ac is seriously distorted from the sinusoidal format. A high level of THD can be expected to measure the waveform of i_ac. When L is significantly high in value, the waveform of i_ac is close to a square waveform.

6.2.3  LC Filter

Figure 6.12 demonstrates the single-phase AC to DC conversion, including an LC filter. The integration is effective since the capacitor maintains the crossing voltage, and the inductor smoothens the through current. The four-switch bridge of diodes only allows v_dc ≥ 0. Therefore, the inductor current can be with either a continuous conduction mode (CCM) or discontinuous conduction mode (DCM). The definitions of CCM and DCM are the same as provided in Chap. 3.

FIGURE 6.12