# Regenerative braking

1. Aug 18, 2013

### MechatronO

I'm doing some calculations on a two quadrant chopped DC motor controller, like below.

[Broken]

Consider the motor to be connected to a vehicle running at an arbitrary vehicle speed V, and that we then start regenerative breaking.

We know that eg can be related to the motors rotation speed and then the vehicle velocity by

eg = k*V

Energy stored in L is

w = L*i^2/2

The current in L, considering that the initial current is zero, is as we know

i = eg/r(1-e^t*L/R)

if we assume eq to be constant in each new PWM cycle

The variable t would be a fraction of a dutycycle and thus the max current can be evaluated by substituting

t = T*D

where T is the PWM period time and D is the PWM duty cycle.

The energy stored in the coil each period is then, by combining the equations

wp(t)= L*(eg/r(1-e^t*L/R)^2/2

The energy in one period can be calculated via

wp,tot = ∫ wp(t) dt

The power output from the motor acting as a generator is then

P = wp,tot*fpwm

where fpwm is the switching frequency.

What I now would like to do is somehow relate the dutycycle and switch frequency to breaking force.

According to the energy principle P must be extracted from the vehicles kinetic energy.

Then

mV^2/2 - wp,tot = F*s = m*a*s

Where s is the distance travelled.

From this average deceleration and such could be calculated, and related to both switch frequency and dutycycle.

However, first of all I have to get rid of s by some neat substitution or so. Suggestions?

Also I wonder if this problem is possible to solve analytically without assuming eq to be constant between each PWM cycle.

Last edited by a moderator: May 6, 2017