# Duty Cycle Calculation

#### powermachines

Joined Oct 26, 2020
2
I am trying to solve a numerical where the Speed of a DC motor operating at 15V was measured to be 2000 rad/sec. Now a PWM signal is applied to the same motor to get a speed of 400 rad/sec. If the voltage amplitude of the PWM signal is 5V, then what should be the value of duty cycle?
I know the formula of duty cycle is
D=Vout/Vin
D=duty cycle
Vin=15V
Vout=5V
therefore the duty cycle is =5/15
i.e 1/3
therefore duty cycle in percentage is 33.33%
is my calculation right or whether I have to consider the speed of motor at 15V(2000 rad/sec) and at 5V(400 rad/sec)? If yes then how?

#### ericgibbs

Joined Jan 29, 2010
15,342
hi pm,
Welcome to AAC.
E

#### powermachines

Joined Oct 26, 2020
2
i have also posted the calculations

#### Papabravo

Joined Feb 24, 2006
18,079
As I interpret the problem statement:
1. The motor runs at 2000 rad/sec at 15 VDC with 100% duty cycle
2. The motor runs at 400 rad/sec from a 15 VDC power supply modulated with duty cycle D and a measured average voltage of 5V
Is that your understanding as well?

#### WBahn

Joined Mar 31, 2012
26,398
As I interpret the problem statement:
1. The motor runs at 2000 rad/sec at 15 VDC with 100% duty cycle
2. The motor runs at 400 rad/sec from a 15 VDC power supply modulated with duty cycle D and a measured average voltage of 5V
Is that your understanding as well?
I don't read it that way.

If you are told that you have a PWM signal and that "the voltage amplitude of the PWM signal is 5V ", would you not interpret that to mean that the amplitude of the rectangular wave signal is 5 V?

#### WBahn

Joined Mar 31, 2012
26,398
I am trying to solve a numerical where the Speed of a DC motor operating at 15V was measured to be 2000 rad/sec. Now a PWM signal is applied to the same motor to get a speed of 400 rad/sec. If the voltage amplitude of the PWM signal is 5V, then what should be the value of duty cycle?
I know the formula of duty cycle is
D=Vout/Vin
D=duty cycle
Vin=15V
Vout=5V
therefore the duty cycle is =5/15
i.e 1/3
therefore duty cycle in percentage is 33.33%
is my calculation right or whether I have to consider the speed of motor at 15V(2000 rad/sec) and at 5V(400 rad/sec)? If yes then how?
What would you expect the speed of the motor to be if a constant 5 V was applied to it?

#### Papabravo

Joined Feb 24, 2006
18,079
I don't read it that way.

If you are told that you have a PWM signal and that "the voltage amplitude of the PWM signal is 5V ", would you not interpret that to mean that the amplitude of the rectangular wave signal is 5 V?
The sequence was what convinced me:
1. the Speed of a DC motor operating at 15V was measured to be 2000 rad/sec
2. Now a PWM signal is applied to the same motor to get a speed of 400 rad/sec.
Only the second situation involved a PWM. The first situation seemed like a motor that could run off of straight DC without the need for any type of PWM.

#### WBahn

Joined Mar 31, 2012
26,398
The sequence was what convinced me:
1. the Speed of a DC motor operating at 15V was measured to be 2000 rad/sec
2. Now a PWM signal is applied to the same motor to get a speed of 400 rad/sec.
Only the second situation involved a PWM. The first situation seemed like a motor that could run off of straight DC without the need for any type of PWM.
It IS a motor that can run off of straight DC without the need for any type of PWM.

As a first order approximation, the (unloaded) speed of a DC motor is proportional to the average voltage applied to it.

So the first data point gives you the proportionality constant.

They then take that same motor and apply a PWM signal to it in order to obtain a specific speed and we are given that the amplitude of that PWM signal is 5 V. We are then asked what the duty cycle of that PWM signal is (or needs to be) when that same motor is then rotating at that different speed.

#### Papabravo

Joined Feb 24, 2006
18,079
It IS a motor that can run off of straight DC without the need for any type of PWM.

As a first order approximation, the (unloaded) speed of a DC motor is proportional to the average voltage applied to it.

So the first data point gives you the proportionality constant.

They then take that same motor and apply a PWM signal to it in order to obtain a specific speed and we are given that the amplitude of that PWM signal is 5 V. We are then asked what the duty cycle of that PWM signal is (or needs to be) when that same motor is then rotating at that different speed.
OK, so you are saying that in the second case you have a +5VDC signal which is modulated with an unspecified duty cycle and you establish a new speed point and from that you can calculate the duty cycle. If that is the case then I did misread the original post.

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#### WBahn

Joined Mar 31, 2012
26,398
OK, so you are saying that in the second case you have a +5VDC signal which is modulated with an unspecified duty cycle and you establish a new speed point and from that you can calculate the duty cycle. If that is the case then I did misread the original post.
I think so (unless I did).

Clearly the duty cycle is an unknown that the TS is being asked to determine.

But I do see two quasi-reasonable ways to interpret what is being asked.

Your way: The PWM signal that is applied produces and average voltage to the motor of 5 V, what is the duty cycle of that PWM signal? This involves a few things that I think make it quite a bit less reasonable as in interpretation. It assumes that the amplitude of the PWM signal is 15 V just because the original DC voltage that was applied happened to be 15 V. That's not absurdly unreasonable, except to the degree that it is directly stated that the amplitude of the PWM signal is 5 V, not that the average value of the PWM signal is 5 V. The second thing that would seem to make it unreasonable is that notice that it makes the speeds that the motor is turning at immaterial. Certainly questions can contain extraneous information, and so this can't be ruled out, but it would seem more likely that the point of the exercise is to get at the understanding of the relationship between the nominal unloaded speed of a DC motor and the average applied voltage and also the relationship between the average voltage of a PWM waveform and it's amplitude and duty cycle. The final thing that would seem to make this interpretation suspect is that it is then saying that at an average applied voltage of 5 V we get a speed of 400 r/s, but that if we increase the average applied voltage by a factor of three to 15 V, we get an increase in speed of a factor of five. Getting an increase of less than three would seem reasonable, but probably not significantly more.

My way: The PWM signal that is applied produces a speed output of the motor of 400 r/s, what is the duty cycle of the PWM signal? This involves assuming that the speed is proportional to the average voltage applied, an assumption that is almost always used for the unloaded speed of a DC motor over quite a large range of operations. From there is it a matter of determining what the average voltage that needs to be applied to the motor in order to get 400 r/s given the information that with 15 V applied it turns at 2000 r/s. Once the needed voltage is determined, it becomes a matter of determining what the duty cycle of a 5 V amplitude PWM signal needs to be in order to achieve that average.

#### ericgibbs

Joined Jan 29, 2010
15,342
hi PowerM,
Please confirm your first posts sentence....If the voltage amplitude of the PWM signal is 5V

Is it voltage amplitude or average of the PWM,? it is an important difference.

I assume it should be average, which makes sense when using a PWM drive voltage and you are asked to state the Duty Cycle.

E