 # Quasi Resonant Flyback Duty Cycle Derivation

Quasi Resonant flyback duty cycle will reach maximum when it operates in the transition mode or boundary region. However, as long as the operation will stay at the discontinues mode (DCM), that is the desired region of operation, the Quasi resonant flyback duty cycle will be variable.

## Quasi Resonant Flyback Duty Cycle – Maximum it Can Go

As mentioned above, the maximum duty cycle will happen when the operation reaches the boundary. A boundary mode of operation is when the energy transferred to the secondary during flyback action is just or exactly depleted when the next switching cycle occur. Read Quasi Resonant Flyback Operating Modes for further discussion about operating modes.

By principle, the energy stored in the primary during Ton is ideally equal to the energy transferred to the secondary during Toff. Refer to below figure for the Ton and Toff.

### Energy at Toff

Equating both energy equations above:

The above equation is created in Mathcad. Mathcad has an auto simplify feature that do the task. It gives the quasi resonant flyback duty cycle, D.

By further equation simplification, the duty cycle formula is:

or

Where;

Vref – reflected voltage to the primary

Vin – input voltage to quasi-resonant flyback

Np – number of primary turns

Ns – Number of secondary turns Vsec – secondary voltage

## Variable Quasi Resonant Flyback Duty Cycle

When quasi resonant flyback operates in DCM, the duty cycle will vary. The variation is caused by the amount of dead time on each switching cycle. Refer to below figure where this dead time should happen.

The derivation is still based on the basic principle that says the energy stored on the primary during the Ton is the same energy delivered to the secondary at Toff. However, this time there is a small amount of time added and this is called the dead time (Td in the above figure). The dead time role is to ensure the quasi-resonant flyback will operate in the DCM region.

I did the very long derivations in Mathcad, but I will no longer show it here. Below is the working equation of a quasi resonant flyback duty cycle when the it is in DCM region.

Where;

Vref – reflected voltage to the primary

Vin – input voltage

Fsw – switching frequency

The advantage of using a quasi resonant flyback compared to the ordinary flyback is the semi soft switching on the power switch in courtesy of the quasi resonance. This advantage will be maximized when the quasi resonant flyback is set to operate in the first valley. Read the article How Quasi Resonant Flyback Works – Detailed Operation to know more about valley switching. Below is an illustration of the switch voltage waveform. As you can see, there is an oscillation on the waveform. This is due to the primary inductance and the parasitic capacitance. The goal is to switch on the primary switch on the lowest voltage level. Therefore, the dead time (Td) is from the moment the voltage starts to decay up to the lowest level.

Thus,

Td = 0.5 X T

T = 1 / Fres

Fres = 1 / [ 2π X sqrt ( Lp X Cd ) ]

Td = 0.5 X 2π X sqrt ( Lp X Cd ) = π X sqrt ( Lp X Cd )Td = π X sqrt ( Lp X Cd )

## What is Vref?

This is the voltage reflected to the primary winding during flyback action. This is not the voltage applied across the primary winding if talking with a flyback transformer, unlike with a standard transformer. During the design stage, this can be derived from the equation:

Vref = [ VDS_target / ( 1 + %spike ) ] – Vin

Where;

VDS_target – this is the intended maximum level of the primary switch voltage, should not exceed the actual switch voltage rating.

%spike – added level to anticipate the high frequency spike. Rule of thumb is 20%-30%.

Vin – this is the voltage applied to the primary winding

When all parameters are already set, Vref is can be computed as

Vref = Vsec X Npri / Nsec

Where;

Vsec – voltage across the secondary winding, this is equal to the output voltage plus the diode drop

Npri – this is the primary winding number of turns

Nsec – this is the secondary winding number of turns

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