# Right Half Plane Pole

Very few know about the Right Half Plane Pole (not a RHP-Zero) at high duty cycle in a DCM buck with current mode control. Maybe because it is not really a problem.

It is said that this instability starts above 2/3 duty cycle – I think that must be with a resistive load. If loaded with a pure current source, it starts above 50% duty cycle.

Here is a little down-to-earth explanation:

If you run a buck converter at high duty cycle but DCM, it probably works fine and is completely stable. Then imagine you suddenly open the feedback loop, leaving the peak current constant and unchanged. The duty cycle will then rush either back to 50% or to 100% if possible. You now have a system with a negative output resistance – if Voltage goes up, the output current will increase.

You can see it by drawing some triangles on a piece of paper: A steady state DCM current triangle with an up-slope longer than the down-slope and a fixed peak value. Now, if you imagine that the output voltage rises, you can draw a new triangle with the same peak current. The up-slope will be longer, the down-slope will be shorter but the sum of times will be longer than in the steady state case. The new triangle therefore has a larger area than the steady state triangle, which means a higher average output current. So higher output voltage generates higher output current if peak current is constant. Loaded with a current source, it is clear that this is an unstable system, like a flipflop, and it starts becoming unstable above 50% duty cycle.

However, when you close the feedback loop, the system is (conditionally) stable and the loop gain is normally so high at the RHP Pole frequency that it requires a huge gain reduction to make it unstable.

It's like when you drive on your bike. A bike has two wheels and therefore can tilt to either side – it is a system with a low frequency RHPP like a flipflop. If you stand still, it will certainly tilt to the left or to the right because you have no way to adjust your balance back. But if you drive, you have a system with feedback where you can immediately correct imbalance by turning the handlebars. As we know, this system is stable unless you have drunk a lot of beers.

It is said that this instability starts above 2/3 duty cycle – I think that must be with a resistive load. If loaded with a pure current source, it starts above 50% duty cycle.

Here is a little down-to-earth explanation:

If you run a buck converter at high duty cycle but DCM, it probably works fine and is completely stable. Then imagine you suddenly open the feedback loop, leaving the peak current constant and unchanged. The duty cycle will then rush either back to 50% or to 100% if possible. You now have a system with a negative output resistance – if Voltage goes up, the output current will increase.

You can see it by drawing some triangles on a piece of paper: A steady state DCM current triangle with an up-slope longer than the down-slope and a fixed peak value. Now, if you imagine that the output voltage rises, you can draw a new triangle with the same peak current. The up-slope will be longer, the down-slope will be shorter but the sum of times will be longer than in the steady state case. The new triangle therefore has a larger area than the steady state triangle, which means a higher average output current. So higher output voltage generates higher output current if peak current is constant. Loaded with a current source, it is clear that this is an unstable system, like a flipflop, and it starts becoming unstable above 50% duty cycle.

However, when you close the feedback loop, the system is (conditionally) stable and the loop gain is normally so high at the RHP Pole frequency that it requires a huge gain reduction to make it unstable.

It's like when you drive on your bike. A bike has two wheels and therefore can tilt to either side – it is a system with a low frequency RHPP like a flipflop. If you stand still, it will certainly tilt to the left or to the right because you have no way to adjust your balance back. But if you drive, you have a system with feedback where you can immediately correct imbalance by turning the handlebars. As we know, this system is stable unless you have drunk a lot of beers.

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