Single phase half wave circuit with rl load and freewheeling diode formula

The waveform of load current is in Figure (Single phase half wave controlled rectifier with an rl load) can be improved by connecting a freewheeling (or flywheeling) diode across load as shown in Figure (Single-phase Half-wave Circuit with RL Load and Freewheeling Diode in circuit diagram).

Fig:-Single-phase Half-wave Circuit with RL Load and Freewheeling Diode (a) circuit diagram and (b) wave form

A freewheeling diode is also called by-pass or commutating diode. At ωt = 0, source voltage is becoming positive. At some delay angle α, forward biased SCR is triggered and source voltage appears across load as Vo. At ωt=π, source voltage is zero and just after this instant, as source voltage tenda to reverse,freewheeling diode is forward biased through the conducting SCR. As a result, load current is immediately transferred from SCR to freewheeling diode as source voltage tends to reverse. At the same time, SCR is subjected to reverse voltage and zero current, it is therefore turned off at ωt = π. It is assumed that during freewheeling period load current does not decay to zero until the SCR is triggered again at (2π + α). Voltage drop across freewheeling diode is taken as almost zero, the load voltage is, therefore, zero during the freewheeling period. The voltage variation across SCR is shown as thyristor voltage in Figure.

It is seen from this wave-form that SCR is reverse biased from ωt=π to ωt=2π. Therefore, circuit turn-off time is π/ω second.The source current  and thyristor current  have the same waveform as shown.

Operation of the circuit diagram of figure can be explained in two modes. In the first mode, called conduction mode, SCR conducts from α to π, 2π + α to 3π and so on and FD is reverse biased. The duration of this mode is for [(π-α)/ω) sec. Let the load current at the beginning of mode I be Io. The expression for current io in mode I can be obtained as follows:

Mode I: For conduction mode, the voltage equation is....................... 

_Vmsinωt = Ri₀ + L(di₀/dt)

It's solution,

 i₀ = (V_m/Z)sin(ωt-φ) + Ae^-1(R/L)t

^{At ωt = α,}      i₀ = I_0,     I.e.at t = α/ ω,

_{For A =} [i₀ - V_m/Z sin(α-φ)]e^Rα/ωL

For i₀ = (V_m/Z)sin(ωt-φ) + [i₀ - V_m/Z sin(α-φ)]exp{-R/L(t-α/ω)}

 Note that for mode I, α ≤ ωt ≥ π

Mode II: This mode, called freewheeling mode, extends from π  to 2π+α, 3π to 4π + α and so on. In this mode, SCR is reverse biased from π to 2π, 3π to 4π... as shown by voltage wave form thyristor voltage in Fig. As the load current is assumed continuous, FD conducts from π to (2π+α), 3π to (4π+α) and so on. Let the current at the beginning of mode II be loj as shown. As load current is passing through FD, the voltage equation for mode II is..................

            0 = Ri₀ + L(di₀/dt)

It's solution is         i₀ = Ae^-(R/L)t

^{At ωt = π,}            i₀ = I_01.

It's gives              A = I₀₁e^Rπ/ωL

 For                   i₀ = I₀₁[-R/L(t-π/ω)]

Note that for mode II,  π < ωt ≤ (2π+α)

Average Load Voltage,V₀=V_m/2π(1+ cosα)

Average Load Current,I₀=V₀/R=V_m/2πR(1+cosα)

Note that load current is contributed by SCR from α to π, (2π+α) to 3π and so on and by FD from 0 to α, π to (2π + α) and so on. Thus the waveshape of thyristor current  is identical with the waveshape of load current for at = α to n, (2π + α) to 3π and so on. Similarly, the waveshape of freewheeling diode current is identical with the waveform of load current for  ωt = 0° to α, π to (2π + α) and so on.

In figure single phase half wave controlled rectifier with an rl load formula, load consumes power p1 from source for  α to  π (both load voltage and load current are positive) whereas energy stored in inductance L is returned to the source as power p2 for  π to ẞ (load voltage is negative and load current is positive). As a result, net power consumed by the load is the difference of these two powers p1 and p2.

In figure single-phase Half-wave Circuit with RL Load and Freewheeling Diode, load absorbs power for α to  π, but for  π to (2 π+α),energy stored in L is delivered to load resistance R through the FD. As a consequence, powerconsumed by load is more in figure (Single-phase Half-wave Circuit with RL Load and Freewheeling Diode). It can, therefore, be concluded that power delivered toload, for the same firing angle, is more when FD is used. As volt-ampere input is almost thesame in both Figs ( single phase half wave controlled rectifier with an rl load and single-phase Half-wave Circuit with RL Load and Freewheeling Diode), the input pf(= power delivered to load/input volt-ampere)with the use of FD is improved.

It is also seen from Figs(Single phase half wave controlled rectifier with an rl load) and figs(Single-phase Half-wave Circuit with RL Load and Freewheeling Diode) that load current waveform is improved with FD in Fig(Single-phase Half-wave Circuit with RL Load and Freewheeling Diode). Thus the advantages of using freewheeling diode are

1. input pf is improved
2. load current waveform is improved and 
3. as a result of load performance is better.

It may be seen from Fig(Single-phase Half-wave Circuit with RL Load and Freewheeling Diode) that freewheeling diode prevents the load voltage from becoming negative. Whenever load voltage tends to go negative, FD comes into play. As a result, load current is transferred from main thyristor to FD, allowing the thyristor to regain its forward blocking capability.

It is seen from Figs(Single phase half wave controlled rectifier with an rl load) and fig(Single-phase Half-wave Circuit with RL Load and Freewheeling Diode) that supply current , taken from the source is unidirectional and is in the form of dc pulses. Single phase half-wave converter thus introduces a dc component into the supply line. This is undesirable as it leads to saturation of the supply transformer and other difficulties (harmonics etc.).