EMF Equation
\[E = K_e\omega_m\]
Recap from Lec-2A
Key equations derived:
Torque Equation
\[T_e = K_t I_a = K_e I_a\]
Today's focus:
- How does armature circuit behave with resistance and inductance?
- What about the field circuit?
- How to model mechanical dynamics?
- How to combine electrical and mechanical subsystems?
Armature Circuit Modeling
Armature Circuit Components
Physical components in armature circuit:
Resistance:
- Copper winding resistance
- Brush contact resistance
- Combined as \(R_a\)
- Causes power loss (\(I_a^2 R_a\))
Inductance:
- Self-inductance of windings
- Magnetic energy storage
- Denoted as \(L_a\)
- Affects transient response
Back-EMF:
- Generated EMF \(E = K_e\omega_m\)
- Opposes applied voltage (Lenz's law)
- Speed-dependent voltage
- Key coupling between electrical and mechanical systems
Armature Circuit Equivalent
Armature Circuit Equation
Applying Kirchhoff's Voltage Law (KVL) around armature circuit:
\[V = R_a I_a + L_a\frac{dI_a}{dt} + E\]
Substituting \(E = K_e\omega_m\):
Dynamic Armature Equation
\[\boxed{V = R_a I_a + L_a\frac{dI_a}{dt} + K_e\omega_m}\]
This is a first-order differential equation relating:
- Applied voltage \(V\) (input)
- Armature current \(I_a\) (electrical variable)
- Rotor speed \(\omega_m\) (mechanical variable)
Steady-State Armature Equation
In steady state, all variables are constant:
\[\frac{dI_a}{dt} = 0\]
Steady-State Armature Equation
\[\boxed{V = R_a I_a + K_e\omega_m}\]
Solving for armature current:
\[I_a = \frac{V - K_e\omega_m}{R_a} = \frac{V - E}{R_a}\]
Observations:
- As speed increases, back-EMF increases
- Higher back-EMF reduces armature current
- At no load, \(I_a\) is small, \(E \approx V\)
Armature Circuit Time Constant
Rearranging the dynamic equation:
\[L_a\frac{dI_a}{dt} = V - R_a I_a - K_e\omega_m\]
\[\frac{dI_a}{dt} = -\frac{R_a}{L_a}I_a - \frac{K_e}{L_a}\omega_m + \frac{1}{L_a}V\]
Electrical Time Constant
\[\boxed{\tau_a = \frac{L_a}{R_a} \text{ seconds}}\]
Physical significance:
- Determines how fast armature current responds to voltage changes
- Typical values: 5–50 ms (milliseconds)
- Smaller \(\tau_a\) \(\rightarrow\) faster electrical response
- After time \(\tau_a\), current reaches approximately 63% of its final value
Power Flow in Armature Circuit
Multiplying armature equation by \(I_a\):
\[VI_a = R_a I_a^2 + L_a I_a\frac{dI_a}{dt} + K_e\omega_m I_a\]
Power Balance
\[\underbrace{VI_a}_{\substack{\text{Electrical}\\\text{Input Power}}} = \underbrace{R_a I_a^2}_{\substack{\text{Armature}\\\text{Copper Loss}}} + \underbrace{L_a I_a\frac{dI_a}{dt}}_{\substack{\text{Stored in}\\\text{Inductance}}} + \underbrace{K_e\omega_m I_a}_{\substack{\text{Converted to}\\\text{Mechanical}}}\]
In steady state: \(\dfrac{dI_a}{dt} = 0\)
\[\boxed{VI_a = R_a I_a^2 + K_e\omega_m I_a = R_a I_a^2 + EI_a}\]
- \(EI_a\) = Air gap power = Electromagnetic power developed
- \(R_a I_a^2\) = Copper loss (heat dissipated)
Field Circuit Modeling
Field Circuit for Separately Excited Motor
Separately Excited DC Motor:
- Field and armature circuits are independent
- Field supplied from separate DC source
- Allows independent control of flux and torque
Field Circuit Equation
Applying KVL to field circuit:
\[V_f = R_f I_f + L_f\frac{dI_f}{dt}\]
Where:
- \(V_f\) = Field supply voltage (DC)
- \(R_f\) = Field winding resistance (typically large)
- \(L_f\) = Field winding inductance (very large)
- \(I_f\) = Field current
Field Time Constant
\[\boxed{\tau_f = \frac{L_f}{R_f} \text{ seconds}}\]
Typical values: 0.5 to 2 seconds
Key observation: Field circuit is much slower than armature circuit (\(\tau_f \gg \tau_a\))
Flux-Current Relationship
For unsaturated magnetic circuit:
\[\phi = L_{af} I_f\]
Where:
- \(\phi\) = Air-gap flux per pole (Wb)
- \(L_{af}\) = Mutual inductance between armature and field
- \(I_f\) = Field current (A)
Torque constant relationship:
\[K_e = K_t = k\phi = kL_{af}I_f\]
For constant field current, \(K_e\) and \(K_t\) are constants.
Simplified Field Model
For many control applications:
- Field current is kept constant by a controlled supply
- Field dynamics are much slower than armature and mechanical dynamics
- Assumption: \(I_f = \text{constant}\) during transients
Mechanical System Modeling
Mechanical System Components
Rotating mechanical system includes:
Inertia (J):
- Rotor inertia
- Load inertia (reflected to motor shaft)
- Unit: kg·m²
- Resists changes in speed
Friction (B):
- Viscous friction (proportional to speed)
- Bearing friction
- Windage losses
- Unit: N·m·s/rad
Load Torque (\(T_L\)):
- External torque demand from the load
- Can be constant, speed-dependent, or time-varying
- Unit: N·m
Newton's Second Law for Rotation
Applying Newton's law to rotating system:
\[J\frac{d\omega_m}{dt} = \text{Net Torque}\]
Net torque = Electromagnetic torque - Friction torque - Load torque:
Dynamic Mechanical Equation
\[\boxed{J\frac{d\omega_m}{dt} = T_e - B\omega_m - T_L}\]
Where:
- \(T_e = K_e I_a\) = Electromagnetic torque developed by motor
- \(B\omega_m\) = Friction torque (opposes motion)
- \(T_L\) = Load torque (opposes motion)
Mechanical Time Constant
Rearranging the mechanical equation:
\[\frac{d\omega_m}{dt} = \frac{K_e}{J}I_a - \frac{B}{J}\omega_m - \frac{1}{J}T_L\]
Mechanical Time Constant
\[\boxed{\tau_m = \frac{J}{B} \text{ seconds}}\]
Physical significance:
- Determines how fast the motor speed responds to torque changes
- Typical values: 0.1 to 5 seconds
- Slower than electrical time constant: \(\tau_m \gg \tau_a\)
- After time \(\tau_m\), speed reaches approximately 63% of its final value
Steady-State Mechanical Equation
In steady state: \(\dfrac{d\omega_m}{dt} = 0\)
Steady-State Torque Balance
\[\boxed{T_e = B\omega_m + T_L}\]
Interpretation:
- Electromagnetic torque must balance friction and load torque
- At no-load (\(T_L = 0\)): \(T_e = B\omega_m\)
- Higher load torque requires higher electromagnetic torque
Load Torque Characteristics
Types of Load Torque
Load torque can have different speed dependencies:
1. Constant Torque Load
\[T_L = T_0 = \text{constant}\]
Examples: Hoists, cranes, conveyor belts, friction loads
Torque independent of speed
2. Linear (Viscous) Load
\[T_L = k_1\omega_m\]
Examples: Generators with resistive load, some pumps
Torque proportional to speed
3. Quadratic (Fan-type) Load
\[T_L = k_2\omega_m^2\]
Examples: Fans, blowers, centrifugal pumps
Torque proportional to square of speed (fluid dynamics)
4. Constant Power Load
\[T_L = \frac{P_0}{\omega_m}\]
Examples: Machine tools, traction applications
Torque inversely proportional to speed
Impact on Motor Performance
Different load types affect motor behavior differently:
- Constant torque: Motor must develop constant torque regardless of speed
- Fan-type load: Light torque at low speed, increases rapidly at high speed
- Constant power: High torque at low speed, decreases as speed increases
Coupled Electromechanical Equations
Complete System Equations
For a separately excited DC motor with constant field current:
Electrical Subsystem
\[\boxed{V = R_a I_a + L_a\frac{dI_a}{dt} + K_e\omega_m}\]
Mechanical Subsystem
\[\boxed{J\frac{d\omega_m}{dt} = K_e I_a - B\omega_m - T_L}\]
Coupling Equations
\[\boxed{T_e = K_e I_a, \quad E = K_e\omega_m}\]
This is a coupled system of two first-order differential equations
State-Space Representation
Define state variables:
- \(x_1 = I_a\) (armature current)
- \(x_2 = \omega_m\) (rotor speed)
State equations:
State-Space Form
\[\begin{aligned}
\frac{dx_1}{dt} = \frac{dI_a}{dt} &= -\frac{R_a}{L_a}I_a - \frac{K_e}{L_a}\omega_m + \frac{1}{L_a}V\\[0.3cm]
\frac{dx_2}{dt} = \frac{d\omega_m}{dt} &= \frac{K_e}{J}I_a - \frac{B}{J}\omega_m - \frac{1}{J}T_L
\end{aligned}\]
Input: \(V\) (control), \(T_L\) (disturbance)
Output: \(\omega_m\) (typically the variable of interest)
Matrix Form
In compact matrix notation:
\[\frac{d}{dt}\begin{bmatrix} I_a \\ \omega_m \end{bmatrix} =
\begin{bmatrix}
-\dfrac{R_a}{L_a} & -\dfrac{K_e}{L_a} \\[0.4cm]
\dfrac{K_e}{J} & -\dfrac{B}{J}
\end{bmatrix}
\begin{bmatrix} I_a \\ \omega_m \end{bmatrix}
+
\begin{bmatrix}
\dfrac{1}{L_a} & 0 \\[0.4cm]
0 & -\dfrac{1}{J}
\end{bmatrix}
\begin{bmatrix} V \\ T_L \end{bmatrix}\]
Standard form: \(\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}\)
where:
- \(\mathbf{x} = [I_a \quad \omega_m]^T\) = state vector (2\(\times\)1)
- \(\mathbf{u} = [V \quad T_L]^T\) = input vector (2\(\times\)1)
- \(\mathbf{A}\) = system matrix (2\(\times\)2)
- \(\mathbf{B}\) = input matrix (2\(\times\)2)
Block Diagram Representation
Block Diagram: Electrical Subsystem
Transfer function from \((V - E)\) to \(I_a\):
\[\frac{I_a(s)}{V(s) - E(s)} = \frac{1}{R_a + L_a s} = \frac{1/R_a}{1 + \tau_a s}\]
Block Diagram: Mechanical Subsystem
Transfer function:
\[\frac{\omega_m(s)}{T_e(s) - T_L(s)} = \frac{1}{Js + B} = \frac{1/B}{1 + \tau_m s}\]
Complete Block Diagram
Transfer Function: Voltage to Speed
For no-load condition (\(T_L = 0\)):
Using block diagram algebra:
\[\frac{\omega_m(s)}{V(s)} = \frac{\dfrac{K_t}{(R_a + L_a s)(Js + B)}}{1 + \dfrac{K_e K_t}{(R_a + L_a s)(Js + B)}}\]
Simplifying:
Speed Transfer Function
\[\boxed{\frac{\omega_m(s)}{V(s)} = \frac{K_t}{(R_a + L_a s)(Js + B) + K_e K_t}}\]
This is a second-order system with both electrical and mechanical dynamics.
Simplified Model: Neglecting Armature Inductance
If \(L_a\) is very small, we can assume: \(L_a \approx 0\)
Transfer function becomes:
\[\frac{\omega_m(s)}{V(s)} = \frac{K_t}{R_a(Js + B) + K_e K_t} = \frac{K_t}{R_a Js + (R_a B + K_e K_t)}\]
First-Order Approximation
\[\boxed{\frac{\omega_m(s)}{V(s)} = \frac{K_t/(R_a B + K_e K_t)}{1 + \dfrac{R_a J}{R_a B + K_e K_t}s}}\]
DC gain: \(\dfrac{K_t}{R_a B + K_e K_t}\)
Time constant: \(\tau = \dfrac{R_a J}{R_a B + K_e K_t}\)
Summary and Key Takeaways
Summary: Circuit Equations
Armature Circuit (Dynamic)
\[V = R_a I_a + L_a\frac{dI_a}{dt} + K_e\omega_m\]
Time constant: \(\tau_a = L_a/R_a\) (5–50 ms)
Field Circuit (Dynamic)
\[V_f = R_f I_f + L_f\frac{dI_f}{dt}\]
Time constant: \(\tau_f = L_f/R_f\) (0.5–2 s)
Mechanical System (Dynamic)
\[J\frac{d\omega_m}{dt} = T_e - B\omega_m - T_L\]
Time constant: \(\tau_m = J/B\) (0.1–5 s)
Summary: Coupling and Hierarchy
Coupling equations:
\[T_e = K_e I_a, \quad E = K_e\omega_m\]
Time constant hierarchy:
\[\tau_a \ll \tau_m \ll \tau_f\]
- Armature response is fastest (electrical)
- Mechanical response is intermediate
- Field response is slowest
Control implications:
- Armature voltage control gives fast torque/speed control
- Field control is slower but affects flux and torque capability
- In transient analysis, field current often assumed constant
Key Concepts Covered
-
Armature circuit modeling
- \(R_a\), \(L_a\), and back-EMF
- Dynamic and steady-state equations
-
Field circuit modeling
- Separate excitation
- Much slower dynamics than armature
-
Mechanical system modeling
- Inertia, friction, and load torque
- Newton's law for rotation
-
Electromechanical coupling
- Two-way interaction through \(K_e\)
- State-space and block diagram representation
-
Load characteristics
- Different types: constant, linear, quadratic, constant power