5/11/24, 4:00 PM ChatGPT
Let's start with the model identification.
Model Identification:
Given the first-order model for the motor speed dynamics:
τ dω
dt
= k(V − ke ω )
Where:
ω is the motor speed (rad/s)
V is the applied voltage (V)
k is the motor torque constant (N*m/A)
J is the motor inertia (kg*m^2)
ke is the motor back EMF constant (V/(rad/s))
τ is the motor time constant (s)
We need to determine the values of k , ke , and τ based on the provided motor specifications.
1. Motor Torque Constant (k ):
The motor torque constant, k , relates the motor's torque to the current applied to it. It is usually provided in the
motor specifications. Since we are given the nominal current and torque, we can use the formula:
T
k= I
Where:
T is the nominal torque (N*m)
I is the nominal current (A)
Given:
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5/11/24, 4:00 PM ChatGPT
Nominal Torque (T ): Not provided
Nominal Current (I ): 2A
We don't have the nominal torque, so we can't directly calculate k from the provided data. We'll need to use other
methods or assumptions to estimate it.
2. Motor Back EMF Constant (ke ):
The motor back EMF constant, ke , relates the motor's back EMF to its angular velocity. It's also given in the
motor specifications. It's directly provided as 0.1 V/(rad/s).
ke = 0.1 V/(rad/s)
3. Motor Time Constant (τ ):
The motor time constant, τ , characterizes the time response of the motor. It's given by the ratio of motor inertia (J
) to the motor torque constant (k ).
J
τ= k
Given:
Motor Inertia (J ): 0.02 kg*m^2
Since we don't have the value of k , we'll need to estimate it before calculating τ .
Estimating Motor Torque Constant (k ):
We can estimate k using the nominal voltage and current, assuming they are at nominal operating conditions. The
relationship between voltage, current, and torque in a DC motor is given by:
V = I ⋅ R + ke ω
Given:
Nominal Voltage (V ): 24V
Nominal Current (I ): 2A
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Motor Resistance (R): 6 ohms
Motor Back EMF Constant (ke ): 0.1 V/(rad/s)
Using the equation above, we can solve for k when the motor is operating at nominal conditions.
V −I⋅R
k= ω
24−2⋅6
k= ω
Now, we can calculate τ :
J 0.02
τ= k
= 24−2⋅6
ω
0.02⋅ω
τ= 24−12
0.02
τ= 12
⋅ω
0.00167
τ= ω
These equations will give us the relationships between motor speed and time constant. If we get the value of
motor speed, we can find the value of time constant τ .
You
ok find it
ChatGPT
To find the value of the time constant (τ ), we need to substitute the given values into the equation we derived:
0.02
τ= 12
⋅ω
Given that the nominal voltage is 24V, the nominal current is 2A, and the motor resistance is 6 ohms, we can
calculate the nominal angular velocity (ω ) using the relationship:
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5/11/24, 4:00 PM ChatGPT
V = I ⋅ R + ke ω
24 = 2 ⋅ 6 + 0.1 ⋅ ω
24 = 12 + 0.1 ⋅ ω
12 = 0.1 ⋅ ω
12
ω= 0.1
ω = 120 rad/s
Now, we can substitute ω = 120 into the equation for τ :
0.02
τ= 12 ⋅ 120
0.02⋅120
τ= 12
2.4
τ= 12
τ = 0.2 s
So, the value of the motor time constant (τ ) is 0.2 seconds.
You
give transfer function of above
ChatGPT
To obtain the transfer function from the motor speed dynamics equation, we can use Laplace transforms. The
first-order differential equation describing the motor speed dynamics is:
τ dω
dt
= k(V − ke ω )
Taking the Laplace transform of both sides and assuming zero initial conditions, we get:
τ sΩ(s) = k(V − ke Ω(s))
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5/11/24, 4:00 PM ChatGPT
Where:
Ω(s) is the Laplace transform of motor speed ω(t)
s is the Laplace variable
Rearranging the equation to solve for Ω(s), we get:
kV
Ω(s) = τ s+kke
This equation represents the transfer function of the motor speed dynamics system. Let's substitute the given
values to get the numerical expression for the transfer function. Given:
T
k= I
T (nominal torque) and I (nominal current) are not provided, but we've estimated k from the nominal
voltage and current.
ke = 0.1 V/(rad/s)
τ = 0.2 s
V = 24 V
Substituting these values, we get:
kV
Ω(s) = τ
kk
s+ τ e
2⋅24
Ω(s) = 0.2
2⋅24⋅0.1
s+ 0.2
240
Ω(s) = s+24
So, the transfer function of the motor speed dynamics system is:
240
G(s) = s+24
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