Closed-Loop Control of DC Motors
A comprehensive three-part treatment: Foundations · System Design · Modern Control
Part 1
Foundations & Separately Excited DC Motor Control
1. Why Closed-Loop Control?
Problem with Open-Loop Operation
- Speed changes if the firing angle is held constant while load torque increases.
- Maintaining constant speed requires continuous adjustment of the firing angle.
- Open-loop drives cannot do this automatically.
Advantages of Closed-Loop Control
- Greater accuracy
- Improved dynamic response
- Reduced effect of load disturbances
- Enables rapid acceleration/deceleration
- Drive characteristics easily modified
- Built-in circuit protection
"Most industrial drive systems operate as closed-loop feedback systems."
Basic Closed-Loop Speed-Control System
Operating Principle
Load ↑ ⟹ \(N\)↓ ⟹ \(\varepsilon_N\)↑ ⟹ \(E_c\)↑ ⟹ firing angle adjusts ⟹ \(E_a\)↑ ⟹ torque ↑ ⟹ \(N\) restored
2. Separately Excited DC Motor — Overview
Key Features
- Separate excitation makes speed control relatively easy.
- Armature voltage is controlled in a closed-loop feedback system.
- Other protective features such as current limiting are incorporated.
Analysis Goal
- Derive transfer functions of the motor and control components.
- Assess the dynamic response of the drive.
Governing Equations
Voltage loop:
\[ e_a = e_g + R_a i_a + L_a \frac{di_a}{dt} \]
Back-EMF:
\[ e_g = K_a \Phi\, n \]
Torque balance:
\[ t = T_L + Bn + J\frac{dn}{dt} \]
Developed torque:
\[ t = K_a \Phi\, i_a \]
3. Motor Transfer Function
Laplace Domain Equations
\[\begin{aligned}
E_a(s) &= E_g(s) + (R_a + sL_a)\,I_a(s) \\
E_g(s) &= K_a \Phi\, N(s) \\
T(s) &= T_L(s) + (B + Js)\,N(s) \\
T(s) &= K_a \Phi\, I_a(s)
\end{aligned}\]
Armature current in terms of net voltage:
\[ I_a(s) = \frac{E_a(s) - E_g(s)}{R_a + sL_a} = \frac{\bigl[E_a(s) - E_g(s)\bigr]/R_a}{1 + \tau_a s} \]
\(\tau_a = L_a/R_a\) — electrical time constant of the armature circuit.
Mechanical Equation
\[ N(s) = \frac{T(s) - T_L(s)}{B + Js} = \frac{\bigl[T(s) - T_L(s)\bigr]/B}{1 + \tau_m s} \]
\(\tau_m = J/B\) — mechanical time constant of the motor.
A feedback loop exists through the back-EMF \(E_g\), which provides the inherent speed regulation characteristic of a separately excited DC motor.
Two Time Constants
\(\tau_a = L_a/R_a\) (electrical)
\(\tau_m = J/B\) (mechanical)
Typically: \(\tau_a \ll \tau_m\)
Simplified Motor Transfer Function
Neglecting \(T_L\) and using the full block-diagram representation:
\[ \frac{N(s)}{E_a(s)} = \frac{K_a\Phi}{(K_a\Phi)^2 + R_aB\,(1+s\tau_a)(1+s\tau_m)} \]
Since \(\tau_a \ll \tau_m\), the electrical time constant may be neglected:
\[ \frac{N(s)}{E_a(s)} = \frac{K_a\Phi}{(K_a\Phi)^2 + R_aB + s R_aB\tau_m} = \frac{k_m}{1 + s\tau_{m1}} \]
\[ \tau_{m1} = \frac{R_aB}{(K_a\Phi)^2 + R_aB}\,\tau_m \]
\[ k_m = \frac{K_a\Phi}{(K_a\Phi)^2 + R_aB} \]
\(\tau_{m1} < \tau_m\) (effective time constant is reduced by back-EMF feedback)
Two-Block Motor Representation
\[ \frac{I_a(s)}{E_a(s)} = \frac{N(s)}{E_a(s)} \times \frac{I_a(s)}{N(s)} = \frac{k_{m1}(1+s\tau_m)}{1+s\tau_{m1}} \]
\[ k_{m1} = \frac{B}{(K_a\Phi)^2 + R_aB} \]
\[ k_{m2} = \frac{K_a\Phi}{B} \]
\[ k_m = k_{m1}\,k_{m2} \]
4. Closed-Loop Speed Control
- A DC tachogenerator attached to the motor shaft feeds back a speed signal.
- Speed error \(\varepsilon_N(s)\) controls the armature voltage.
- The armature voltage is regulated by a three-phase full converter.
Converter gain (cosine firing — linear relationship):
\[ k_c = \frac{E_a(s)}{E_c(s)} = \frac{3\sqrt{2}\;V_{LL}}{\pi\,\hat{E}_c} \]
where \(\hat{E}_c\) corresponds to 0° firing angle and \(V_{LL}\) is the AC line-to-line RMS voltage.
General closed-loop transfer function:
\[ \frac{N(s)}{E_r(s)} = \frac{G(s)}{1 + G(s)H(s)} \]
Two controllers are considered: Proportional (P) and Proportional-Integral (PI).
Proportional (P) Controller
\[ G(s) = \frac{k_s k_c k_{m1} k_{m2}}{1+s\tau_{m1}}, \quad H(s) = k_t \]
Resulting closed-loop transfer function:
\[ \frac{N(s)}{E_r(s)} = \frac{k_1}{1+s\tau_1} \]
\[ k_1 = \frac{k_s k_c k_{m1} k_{m2}}{k_s k_c k_{m1} k_{m2} k_t + 1} \]
\[ \tau_1 = \frac{\tau_{m1}}{k_s k_c k_{m1} k_{m2} k_t + 1} \]
If \(k_s k_c k_{m1} k_{m2} k_t \gg 1\) (high loop gain):
\[ k_1 \simeq \frac{1}{k_t} \]
\[ \tau_1 \simeq \frac{\tau_{m1}}{k_s k_c k_{m1} k_{m2} k_t} \]
P Controller — Current Response to Step Input
\[ \frac{I_a(s)}{E_r(s)} = \frac{k_1}{k_{m2}}\,\frac{1+s\tau_m}{1+s\tau_1} \]
For a step change in \(E_r\), the time-domain current response is:
\[ I_a(t) = \frac{E_r k_1}{k_{m2}}\left[1 + \frac{\tau_m - \tau_1}{\tau_1}\,e^{-t/\tau_1}\right] \]
Normalising with respect to steady-state (\(\tau_m \gg \tau_1\)):
\[ \frac{I_a(t)}{I_a(\infty)} \simeq 1 + \frac{\tau_m}{\tau_1}\,e^{-t/\tau_1} \]
Problem
A step change in \(E_r\) causes a large transient surge in armature current that decays slowly. This overcurrent is undesirable for converter protection.
5. Current Control (Inner Loop)
Limitation of Pure Speed-Error Control
- Motor voltage is controlled by speed error alone.
- Clamping the speed error limits motor voltage, not current.
- If armature resistance is neglected, clamping limits speed, not current.
Solution
- Construct an inner current-control loop.
- Use the clamped speed-error signal as the current reference \(E_I\).
- Both P and PI controllers applicable to the current loop.
Inner Loop Elements
- \(k_r\) = gain of current transducer (sampling resistor)
- \(k_I\) = gain of current controller (proportional)
Current sensed → compared with \(E_I\) → error drives converter
Current Control — P Controller Transfer Function
\[ \frac{I_a(s)}{E_I(s)} = k_{IC}\,\frac{1+s\tau_m}{1+s\tau_{m2}} \]
\[ K_{IC} = \frac{k_I k_c k_{m1}}{1+k_r k_I k_c k_{m1}} \]
\[ \tau_{m2} = \frac{\tau_m k_r k_I k_c k_{m1} + \tau_{m1}}{1+k_r k_I k_c k_{m1}} \]
For high inner-loop gain (\(k_r k_I k_c k_{m1} \gg 1\)) and since \(\tau_m \gg \tau_{m1}\):
\[ k_{IC} \simeq \frac{1}{k_r} \]
\[ \tau_{m2} \simeq \tau_m \]
Pole-Zero Cancellation
- Numerator zero at \(s = -1/\tau_m\)
- Denominator pole at \(s = -1/\tau_{m2} \approx -1/\tau_m\)
- Pole-zero cancellation is achieved → no overshoot, no time delay
Simplified Result
\[ \frac{I_a(s)}{E_I(s)} = k_{IC} \simeq \frac{1}{k_r} \]
⟹ Current is directly proportional to \(E_I\)
Practical Note
A small delay remains due to:
- Armature electrical time constant \(\tau_a\)
- Converter delay
Key Benefit
A limit on \(E_I\) will effectively limit the armature current \(I_a\).
Speed Control with Inner Current Loop — P Controller
\[ \frac{N(s)}{E_r(s)} = \frac{k_2}{1+s\tau_2} \]
\[ k_2 = \frac{k_s k_{m2} k_{IC}}{1+k_t k_s k_{m2} k_{IC}} \]
\[ \tau_2 = \frac{\tau_m}{1+k_t k_s k_{m2} k_{IC}} \]
For \(k_t k_s k_{IC} k_{m2} \gg 1\):
\( k_2 \simeq \dfrac{1}{k_t} = k_1 \)
\( \tau_2 \simeq \dfrac{\tau_m}{k_t k_s k_{m2} k_{IC}} \)
During Current Limiting (Acceleration / Large Load Change)
\(E_I\) is clamped at \(\hat{E}_I\), current is limited to \(\hat{I}_a = k_{IC}\hat{E}_I\), and speed follows:
\[ N(s) = \frac{\hat{I}_a\, k_{m2}}{s(1+\tau_m s)} \]
Tachogenerator Filter — Effect on Transfer Function
A filter (time constant \(\tau_t\)) is sometimes needed to reduce ripple in the tachogenerator output. The resulting transfer function becomes second-order:
\[ \frac{N(s)}{E_r(s)} = \frac{k_s k_{IC} k_{m2}}{1+k_s k_{IC} k_{m2} k_t} \cdot \frac{1+s\tau_t}{1 + s\,\dfrac{\tau_m+\tau_t}{k^{\prime}} + s^2\,\dfrac{\tau_m\tau_t}{k^{\prime}}} \]
where \(k^{\prime} = (1 + k_s k_{IC} k_{m2} k_t) \simeq k_s k_{IC} k_{m2} k_t\).
Result: The second-order response introduces overshoot in speed and oscillation in armature current.
6. Proportional-Integral (PI) Controller
- A P controller leaves a steady-state speed error.
- Adding integral action eliminates steady-state error and reduces the required forward gain.
PI controller transfer function: \(G_c(s) = \dfrac{k_s(1+\tau_s s)}{\tau_s s}\)
For \(k_t k_s k_{IC} k_{m2} \gg 1\), the overall closed-loop transfer function is:
\[ \frac{N(s)}{E_r(s)} = \frac{1}{k_t}\,\frac{1+\tau_s s}{1+\tau_s s + \tau_s\tau_2 s^2} \]
where \(\tau_2 = \dfrac{\tau_m}{k_t k_s k_{IC} k_{m2}}\).
PI controller gives a second-order response with zero steady-state error.
PI Controller — Current Response & P vs PI Comparison
Current Transfer Function with PI Speed Controller
\[ \frac{I_a(s)}{E_r(s)} = \left(\frac{1}{k_t k_{m2}}\right) \frac{(1+\tau_s s)(1+s\tau_m)}{1+\tau_s s + \tau_s\tau_2 s^2} \]
Comparison: P vs PI Controller
| Feature | P | PI |
|---|---|---|
| Steady-state error | Yes | Zero |
| System order | 1st | 2nd |
| Overshoot | Low | Some |
| Speed recovery | Partial | Full |
Design Insight
The PI controller's second-order poles are shared by both speed and current responses — the poles are a property of the closed-loop system, not the input.
Summary — Part 1
Key Concepts
- Closed-loop is essential for constant-speed drives.
- Motor has two time constants: \(\tau_a\) (electrical) and \(\tau_m\) (mechanical); typically \(\tau_a \ll \tau_m\).
- Motor represented by two transfer-function blocks.
- P controller: first-order, but large transient current.
- Inner current loop essential to limit overcurrent.
- With current loop, pole-zero cancellation gives clean response.
- PI controller: second-order, zero steady-state speed error.
Transfer Function Summary
Motor (simplified):
\[ \frac{N(s)}{E_a(s)} = \frac{k_m}{1+s\tau_{m1}} \]
P controller (speed loop):
\[ \frac{N(s)}{E_r(s)} = \frac{k_1}{1+s\tau_1} \]
With current loop + PI:
\[ \frac{N(s)}{E_r(s)} = \frac{1}{k_t}\cdot\frac{1+\tau_s s}{1+\tau_s s+\tau_s\tau_2 s^2} \]
Part 2
System Design, Load Disturbances & Series Motor
1. Load Torque Disturbance Analysis
Practical Scenario
- In many applications a load is suddenly applied to the motor.
- The closed-loop system must reject this disturbance and restore speed.
- Both P and PI controllers are analysed.
Analysis Approach
- Changes in speed reference \(E_r\) are neglected.
- An expression for current is written in terms of speed change \(N(s)\).
- Full block diagram (with tachogenerator filter) is used.
P Controller — Current–Speed Relation
\[ I_a(s) = \frac{1}{R_a}\!\left\{-K_a\Phi N(s) + k_I k_c\!\left[-k_r I_a(s) - \frac{k_s k_t N(s)}{1+s\tau_t}\right]\right\} \]
Simplified (for \(k_I k_c k_s k_t \gg K_a\Phi\) and \(k_I k_c k_r \gg R_a\))
\[ I_a(s) \simeq -\frac{k_s k_t}{k_r(1+s\tau_t)}\,N(s) \]
P Controller — Speed Response to Load Disturbance
\[ \frac{N(s)}{T_L(s)} = \frac{-\dfrac{1}{B}(1+s\tau_t)}{\left(1+\dfrac{K_a\Phi k_s k_t}{k_r B}\right) \!\left[1+s\dfrac{\tau_m+\tau_t}{k^\prime} +s^2\dfrac{\tau_m\tau_t}{k^\prime}\right]} \]
where \(\;k^\prime = 1 + \dfrac{K_a\Phi k_s k_t}{k_r B} = 1 + k_s k_t k_{IC} k_{m2}\). Since \(\dfrac{K_a\Phi k_s k_t}{k_r B} \gg 1\):
\[ \frac{N(s)}{T_L(s)} \simeq \frac{-k_r}{K_a\Phi k_s k_t} \cdot\frac{1+s\tau_t}{1+s\!\left(\dfrac{\tau_m+\tau_t}{k^\prime}\right) +s^2\!\left(\dfrac{\tau_m\tau_t}{k^\prime}\right)} \]
This expression has the same second-order denominator as the speed-reference-step response ⟹ the poles are identical in both cases.
P Controller — Current Response to Load Disturbance
\[ \frac{I_a(s)}{T_L(s)} = \frac{I_a(s)}{N(s)}\cdot\frac{N(s)}{T_L(s)} = \frac{1}{K_a\Phi} \cdot\frac{1}{1+s\!\left(\dfrac{\tau_m+\tau_t}{k^\prime}\right) +s^2\dfrac{\tau_m\tau_t}{k^\prime}} \]
- Second-order response in both speed and current.
- Speed dips transiently and recovers.
- Current rises to supply the additional load torque.
- A steady-state speed error remains (characteristic of P control).
Experimental Confirmation
Oscillograms confirm the second-order response for a step increase in load torque under P control: speed dips and recovers with damped oscillation; current rises and settles at a higher steady-state value.
PI Controller — Speed Response to Load Disturbance
Replace the proportional gain \(k_s\) with the PI transfer function \(k_s\!\left[\dfrac{1+\tau_s s}{\tau_s s}\right]\). Because the PI controller provides filtering, the tachogenerator filter \(\tau_t\) may be omitted:
\[ \frac{N(s)}{T_L(s)} \simeq \frac{-\tau_s k_r}{K_a\Phi k_s k_t} \cdot\frac{s}{1+\tau_s s + \tau_s\tau_2 s^2} \]
where \(\;\tau_2 = \dfrac{\tau_m k_r B}{K_a\Phi k_s k_t} = \dfrac{\tau_m}{k_{IC}k_{m2}k_s k_t}\).
Current Response (PI)
\[ \frac{I_a(s)}{T_L(s)} = \frac{1}{K_a\Phi} \cdot\frac{1+\tau_s s}{1+\tau_s s+\tau_s\tau_2 s^2} \]
Fundamental Principle
Poles are characteristics of the system, not of the input signal. ∴ The same system gives the same natural response regardless of disturbance type.
Zero Steady-State Speed Change (PI)
The transfer function has a zero at the origin ⟹ for a step change in torque, there is no steady-state speed error.
Current Limiting During Start-Up
- Implemented via the inner current-control loop.
- Speed error is clamped at maximum value \(\hat{E}_I\).
- Current is therefore limited to \(\hat{I}_a = k_{IC}\hat{E}_I\).
- Once desired speed is reached, normal operation resumes.
Benefit: Thyristors are inherently protected. Large starting and transient currents that can damage the converter and motor are eliminated.
Speed During Current-Limited Start-Up
\[ N(t) = \hat{I}_a\,k_{m2}\bigl(1-e^{-t/\tau_m}\bigr) \]
If viscous friction \(B\) is very small:
\[ N(t) = \left(\frac{K_a\Phi\,\hat{I}_a}{J}\right)t \]
Speed increases linearly with time during constant-current starting.
2. Design Procedure for Closed-Loop Speed Control
System: 110 V, 2.5 hp, 1800 rpm separately excited DC motor.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(R_a\) | 1 Ω | \(J\) | 0.093 kg-m² |
| \(L_a\) | 46 mH | \(B\) | 0.008 N-m-s/rad |
| \(I_a\) (rated) | 20 A | \(K_a\Phi\) | 0.55 V-s/rad |
| \(\tau_a = L_a/R_a\) | 46 ms | \(\tau_m = J/B\) | 11.63 s |
| \(\tau_t\) | 0.1 s | \(\tau_{m1}\) | 0.3 s |
| \(k_{m1}\) | 0.0258 A/V | \(k_{m2}\) | 68.75 rad/s-A |
| \(k_t\) | 0.057 V-s/rad | \(k_r\) | 0.5 V/A |
| \(k_c\) | 25 | \(k_{IC} \simeq 1/k_r\) | 2 |
Feedback gains \(k_t\) and \(k_r\) are fixed by hardware. The controller gains \(k_I\), \(k_s\), and the clamping reference \(\hat{E}_I\) are the design variables.
Design Step 1 — Current Controller Gain \(k_I\)
Based on steady-state error \(\varepsilon_I(\infty)\) of the current-control loop:
\[ \varepsilon_I(\infty) = \frac{1}{1+k_I k_c k_{m1} k_r} \quad \Rightarrow \quad k_I = \frac{\dfrac{1}{\varepsilon_I(\infty)}-1}{k_c k_{m1} k_r} \]
Numerical calculation for \(\varepsilon_I(\infty) = 10\%\):
\[ k_I = \frac{10-1}{(25)(0.0258)(0.5)} \simeq \mathbf{28} \]
Design Step 2 — Current Limit Reference \(\hat{E}_I\)
\[ \hat{E}_I = \hat{I}_a \cdot k_r \]
Numerical calculation for a current limit of 25 A:
\[ \hat{E}_I = 25 \times 0.5 = \mathbf{12.5\;\text{V}} \]
Physical Meaning
Whenever the speed error demands more than 12.5 V reference current, the system clamps at this value, limiting armature current to 25 A (1.25× rated).
Note
The current limit is set above the rated value (20 A) to allow sufficient accelerating torque during start-up, while staying within safe converter limits.
Design Step 3a — Speed Controller Gain \(k_s\) (P Controller)
\[ \varepsilon_N(\infty) = \frac{1}{1+k_s k_{IC} k_{m2} k_t} \quad \Rightarrow \quad k_s = \frac{\dfrac{1}{\varepsilon_N(\infty)}-1}{k_{IC}\,k_{m2}\,k_t} \]
Numerical calculation for 0.25% steady-state speed error (\(\varepsilon_N(\infty) = 0.0025\)):
\[ k_s = \frac{400 - 1}{(2)(68.75)(0.057)} \simeq \mathbf{51} \]
Result: With \(k_s = 51\), the drive achieves 0.25% speed regulation under proportional control.
Design Step 3b — Speed Controller Gain \(k_s\) (PI Controller)
For the PI controller, steady-state speed error is ideally zero. Design is based on damping ratio and natural frequency.
Characteristic equation: \(1 + s\tau_s + s^2\tau_s\tau_2 = 0\)
Poles: \(s = \dfrac{1}{2\tau_2}\left[-1 \pm j\sqrt{\dfrac{4\tau_2}{\tau_s}-1}\right]\)
Commonly accepted damping ratio \(= 1/\sqrt{2}\), which requires: \(\tau_s = 2\tau_2\)
\[ \omega_n = \frac{1}{\sqrt{2}\,\tau_2} \]
\[ k_s = \frac{\tau_m}{k_t k_{IC} k_{m2} \tau_2} \]
PI Design — Numerical Calculation
Choose \(\omega_n = 10\;\text{rad/s}\). Then:
\[ \tau_2 = \frac{1}{\sqrt{2}\times 10} = 0.071\;\text{s}, \quad \tau_s = 2\tau_2 = \mathbf{0.142\;\text{s}} \]
\[ k_s = \frac{11.63}{(0.057)(2)(68.75)(0.071)}\simeq \mathbf{21} \]
| Controller | \(k_s\) | Speed Error | Response |
|---|---|---|---|
| P | 51 | 0.25% (residual) | 1st order (with filter: 2nd order) |
| PI | 21 | 0% (ideal) | 2nd order |
Design Conclusion
The gains \(k_I \approx 28\), \(k_s \approx 21\) (PI) with \(\hat{E}_I = 12.5\;\text{V}\) were used in the experimental setup. Oscillograms confirm second-order response and good damping as predicted by theory.
3. Series DC Motor Model
Applications
- Best suited for vehicle, crane, and hoist drives.
- Large starting torques at low speeds are required.
- If the circuit saturates, flux per pole is essentially constant ⟹ behaviour similar to separately excited motor.
Governing Differential Equations (Nonlinear)
\[ e_a = R_a i_a + L_a\frac{di_a}{dt} + K_{af}i_a n + K_{res}n \]
\[ K_{af}i_a^2 = J\frac{dn}{dt} + Bn + T_L \]
Sources of Nonlinearity
- Product term: \(K_{af}i_a n\)
- Square term: \(K_{af}i_a^2\)
- Parameter variation: \(R_a, L_a, K_{af}, B\)
Two Analysis Approaches
- Linearized model — valid for small-signal perturbations around a steady-state operating point.
- Numerical method — rigorous, valid for large disturbances.
Linearized Model
Average-value governing equations (inductance voltage averages to zero):
\[ E_a = I_a R_a + K_{af}I_a N + K_{res}N \]
\[ K_{af}I_a^2 = J\frac{dN}{dt} + BN + T_L \]
Small-signal perturbation around operating point \((I_{ao},\,N_o)\):
\[ \Delta E_a = \Delta I_a(R_a + K_{af}N_o) + \Delta N(K_{res}+K_{af}I_{ao}) \]
\[ 2K_{af}I_{ao}\,\Delta I_a = J\,p\,\Delta N + B\,\Delta N + \Delta T_L \]
\(I_{ao}\) = motor current at operating point \(N_o\) = motor speed at operating point \(p = d/dt\)
Laplace Domain & Block Diagram
\[ \Delta I_a(s) = \frac{\Delta E_a(s) - \Delta N(s)(K_{res}+K_{af}I_{ao})}{R_a + K_{af}N_o} \]
\[ \Delta N(s) = \frac{2K_{af}I_{ao}\,\Delta I_a(s) - \Delta T_L(s)}{B + Js} \]
Step Change in Motor Voltage (\(\Delta T_L = 0\))
\[ \frac{\Delta N(s)}{\Delta E_a(s)} = \frac{2K_{af}I_{ao}}{R_a+K_{af}N_o} \cdot\frac{1}{B_o+sJ} = k_{EN}\,\frac{1}{1+s\tau_o} \]
\[ B_o = B + \frac{2K_{af}I_{ao}(K_{res}+K_{af}I_{ao})}{R_a+K_{af}N_o} \]
\[ \tau_o = \frac{J}{B_o} \]
\[ k_{EN} = \frac{2K_{af}I_{ao}}{B_o(R_a+K_{af}N_o)} \]
Step Change in Load Torque (\(\Delta E_a = 0\))
\[ \frac{\Delta N(s)}{\Delta T_L(s)} = k_{TN}\,\frac{1}{1+s\tau_o} \]
\[ \frac{\Delta I_a(s)}{\Delta T_L(s)} = k_{TI}\,\frac{1}{1+s\tau_o} \]
Both speed and current responses are first-order with the same time constant \(\tau_o = J/B_o\). Note that \(B_o > B\): effective damping is increased by the back-EMF feedback.
4. Numerical Analysis vs. Linearized Model
Why Numerical Analysis?
- Linearized model valid only for small disturbances around the operating point.
- Numerical method solves the nonlinear ODEs directly.
- Gives instantaneous variations of speed and current.
- Valid for small and large disturbances.
Rewritten as First-Order ODEs
\[ pi_a = \frac{e_a - R_a i_a - K_{af}i_a n - K_{res}n}{L_a} = f_1(n,i_a) \]
\[ pn = \frac{K_{af}i_a^2 - Bn - T_L}{J} = f_2(n,i_a) \]
Solution Method: Runge-Kutta 4th-Order
- Read motor and supply parameters.
- Initialise \(\omega t = 0\).
- Compute \(i_a\), \(n\) using R-K method.
- Check if one cycle is complete.
- Compute average/rms currents, speed, and torques.
- Check convergence (accel. torque < 0.001 pu).
- Output results; apply next disturbance.
Comparison Parameters
| Parameter | Value | Parameter / Value |
|---|---|---|
| \(K_{af}\) | 0.027 H | \(K_{res} = 0.0273\) V-s/rad |
| \(R_a\) | 1 Ω | \(L_a = 32\) mH |
| \(J\) | 0.0465 kg-m² | \(B = 0.004\) N-m-s/rad |
| Supply | 1ϕ, 120 V, 60 Hz | Converter: Semi-converter |
| Voltage change | \(\Delta E_a = 5.4\) V ≈ 5% | (\(\alpha\): 25.8° to 0°) |
| Torque change | \(\Delta T = 1\) N-m ≈ 10% | (\(T_L\): 9 to 10 N-m) |
Conclusion
The linearized models predict the response with fair accuracy for both 5% voltage change and 10% torque change.
Validity Range
Linearized model is reliable for small-signal perturbations. For large disturbances, the numerical method must be used.
Summary — Part 2
Key Concepts
- Load torque disturbances produce second-order responses under P control; poles are identical for speed-step and torque-step inputs.
- PI controller gives zero steady-state speed change for a step torque disturbance.
- Current limiting during start-up protects the converter; speed rises linearly under constant current.
- Design procedure: gains \(k_I\), \(k_s\) and limit \(\hat{E}_I\) chosen from steady-state error and damping requirements.
- Series motor equations are inherently nonlinear.
- Linearized model gives fair accuracy when validated against the Runge-Kutta numerical method.
Transfer Function Summary (Series Motor)
Step voltage:
\[ \frac{\Delta N(s)}{\Delta E_a(s)} = k_{EN}\frac{1}{1+s\tau_o} \]
\[ \frac{\Delta I_a(s)}{\Delta E_a(s)} = k_{EI}\frac{1+s\tau_m}{1+s\tau_o} \]
Step torque:
\[ \frac{\Delta N(s)}{\Delta T_L(s)} = k_{TN}\frac{1}{1+s\tau_o} \]
\[ \frac{\Delta I_a(s)}{\Delta T_L(s)} = k_{TI}\frac{1}{1+s\tau_o} \]
where \(\tau_o = J/B_o\), \(B_o > B\)
Part 3
Modern Control — PLL & Microcomputer
1. Limitations of Analog Feedback Control
Shortcomings of Analog Control
- Speed sensing devices are not ideal.
- Other analog components in the feedback circuit are less than ideal.
- At best 0.2% speed regulation achievable.
- Transient speed dip when torque changes, with a delay before steady-state speed is restored.
- Not satisfactory in drives where high-quality products are required.
Two Modern Solutions
1. Phase-Locked-Loop (PLL) Control
- Digital technique; locks motor speed to a reference frequency.
- Speed regulation as low as 0.002% — hundredfold improvement over analog.
2. Microcomputer Control
- Eliminates analog component errors.
- Software-configurable control law.
- High accuracy, fast response, built-in fault diagnostics.
2. Phase-Locked-Loop (PLL) Control
Basic PLL Circuit
- Phase detector: digital IC; compares pulse trains \(f_i\) and \(f\); produces pulse-width modulated output \(E_e\) proportional to the phase difference.
- Loop filter: converts \(E_e\) to a DC level \(E_c\) representing the phase error.
- VCO: output frequency \(f\) changes in response to input voltage \(E_c\).
Phase Detector Output Behaviour
- If \(f_i = f\) and coincident: output is zero.
- If \(f\) lags \(f_i\): positive error \(E_e\).
- If \(f\) leads \(f_i\): negative error \(E_e\).
- If \(f > f_i\): continuous positive \(E_e\).
- If \(f < f_i\): continuous negative \(E_e\).
Phase-Locked Condition
Phase error is self-correcting ⟹ loop acts to maintain \(f = f_i\). When \(f = f_i\): system is said to be phase-locked.
PLL Applied to Motor Speed Control
- The speed encoder generates a speed-dependent pulse train \(f\).
- \(f\) is compared with reference pulse train \(f_i\) (representing desired speed).
- Any speed deviation is detected almost instantaneously by the phase detector.
- As long as \(f = f_i\), steady-state speed is unaffected ⟹ theoretically perfect speed regulation.
Speed Regulation Achieved
Regulation as low as 0.002% — a hundredfold improvement over conventional analog control.
PLL Transfer Function Model
Closed-Loop Transfer Function
\[ \frac{N(s)}{\theta_i(s)} = \frac{G(s)}{1+G(s)H(s)} \]
where \(G(s) = \dfrac{k_\phi k_{cm} F(s)}{1+s\tau_{cm}}\), \(H(s) = \dfrac{k_o}{s}\)
Open-Loop Transfer Function
\[ G(s)H(s) = k\,\frac{F(s)}{s(1+s\tau_{cm})} \]
where \(k = k_\phi\, k_{cm}\, k_o\)
Speed Encoder Modelling
Phase is the time integral of frequency ⟹ encoder is modelled by a pole at the origin (\(k_o/s\)). There are two open-loop poles: at \(s=0\) and \(s=-1/\tau_{cm}\), plus poles and zeros from \(F(s)\).
Converter-Motor Gain and Time Constant
With Inner Current-Control Loop
\[ k_{cm} \simeq k_{IC}\,k_{m2}, \quad \tau_{cm} \simeq \tau_m \]
Both gain and time constant are large.
Without Current-Control Loop
\[ k_{cm} = k_c\,k_m, \quad \tau_{cm} \simeq \tau_{m1} \]
Both gain and time constant are small.
Motor Size Matters
A large motor has a large time constant. A small (fractional-hp) motor may not need a current-control loop; a large (integral-hp) motor must have a current limit incorporated.
Root Locus Approach
System stability is studied using the root locus technique — closed-loop poles move from open-loop poles toward open-loop zeros as loop gain increases.
Filter Transfer Function \(F(s)\) — Options & Stability
| Case | \(F(s)\) | Small motor (\(\tau_{cm}\) small) | Large motor (\(\tau_{cm}\) large) |
|---|---|---|---|
| 1 | \(\dfrac{1}{1+s\tau_1}\) (low-pass) | Stable at low gain; unstable at high gain | Unstable at smaller gain |
| 2 | \(\dfrac{1}{s}\) (integrator) | Unstable/oscillatory for all gains | Unstable/oscillatory for all gains |
| 3 | \(\dfrac{1+s\tau_1}{s}\) (PI) | Stable (leftward pole swing) | Not improved |
| 4 | \(\dfrac{1+s\tau_1}{1+s\tau_2},\;\tau_1>\tau_2\) (phase-lead) | Good; leftward swing | Leftward swing of dominant poles |
| 5 | \(s\) (derivative) | System does not remain phase-locked | System does not remain phase-locked |
| 6 | \(1+s\tau_1\) (PD) | Poles drawn left effectively | Stable with high gain |
Case 3: PI Filter \(F(s)=\dfrac{1+s\tau_1}{s}\)
Adds a zero to the system. Zero causes poles to migrate leftward (stable) as gain increases. Works well for small motor; not improved for large motor.
Case 4: Phase-Lead Filter (\(\tau_1>\tau_2\))
For a large motor, results in leftward swing of dominant poles — a desirable stabilising effect. Corresponds to phase-lead compensation.
Case 5: Derivative \(F(s)=s\)
Pole-zero at origin cancels ⟹ a closed-loop pole remains at zero. System does not remain phase-locked.
Best Choice for Large Motor
A filter with proportional and derivative (PD) characteristics is required to make the system stable and maintain phase-lock.
Problem: Derivative feedback has high noise susceptibility.
Solution to Noise Problem
Derivative of phase = frequency. Derivative of phase error = frequency error. Use precision \(f/v\) converters on both the encoder output and the reference frequency. Their difference = derivative feedback signal with significantly less noise.
Hybrid PLL Drive for Stable Operation
- Filter output ∝ phase error.
- Difference of \(f/v\) outputs ∝ frequency error (derivative of phase error).
- All three filters have approximately the same time constant.
Key Insight
Without the PLL circuit, the system is a conventional analog feedback system. The PLL trims the speed command to compensate for feedback and converter errors ⟹ theoretically zero speed regulation.
PLL Drive — Practical Considerations
Pulses per Revolution
- For fast response, more pulses per revolution are required.
- More pulses allow use of a smaller filter time constant.
- A very large number of pulses results in high loop gain, which may create a stability problem.
- 36 to 120 pulses per revolution have been used in practical applications.
Applications
- Multi-motor synchronisation: conveyor systems.
- Digital clock coordination: computer peripherals.
- High-quality product drives: paper mills, textile mills, printing mills.
Multi-Motor PLL System
Several DC motors, each with its own PLL control, governed by a common master speed command. Motors can run at the same speed or at preset speed ratios using frequency dividers \(\div N_1,\;\div N_2,\;\ldots\) ahead of each PLL.
Cost Advantage
A DC motor with PLL control costs substantially less than a synchronous motor controlled by a thyristor inverter, while providing equivalent synchronous speed behaviour.
3. Microcomputer Control
Disadvantages of Analog Control
- Nonlinearity in analog speed transducer.
- Difficulty in accurately transmitting analog signals.
- Errors due to temperature and component aging.
- Drift and offset of analog components.
- Extraneous disturbances.
- Cannot handle complex, adaptive control functions.
Digital Control is Free from All These Disadvantages
Speed reference: set digitally.
Speed sensing: digital tachometer produces pulse train; frequency ∝ motor speed; fed to a digital counter.
Speed comparison: two digital counts compared in a gated comparator.
Speed error: drives an up/down counter; proportionality constant = gain of digital controller.
Basics of Microprocessors and Microcomputers
A microcomputer is a bus-oriented control unit interconnecting several LSI chips.
- Microprocessor (CPU): performs calculations and controls functions; executes instructions stored in memory.
- ALU: performs logical and arithmetic operations (addition, subtraction, bit manipulation).
- Data registers: intermediate data storage; reduce memory transfers.
- Address registers: store memory addresses; used for data transfer to/from memory.
- Control unit: supervises instruction execution; contains the clock.
Three System Buses
| Data bus | 8 lines (8-bit µP) |
| Address bus | 16 lines |
| Control bus | 6 lines |
Memory Types
ROM — Instructions stored permanently (non-volatile); stores look-up tables and fixed programs.
RAM — Information can be written and read; stores data variables and changeable program portions.
I/O Interface, Buses and Software
Input-Output (I/O) Interface
- Permits communication between CPU and the outside world.
- Transfers data between CPU and external devices.
- Converts external data into a form usable by the microcomputer.
Software
- Microprocessor understands only binary codes.
- Programmer writes in assembly language.
- Assembler program converts symbolic source-program into binary object-program.
- Firmware: program held in ROM; not changeable by user.
Greatest Advantage of Microcomputer Control
Flexibility. Drive characteristics can be changed or new functions added simply by changing the software (or firmware) with practically no change to the hardware. Controllers can be standardised.
Microcomputer Control of Reversing Drives
- Speed input via DC tachometer + A/D converter or digital tachometer + digital counter.
- Motor current fed via fast A/D converter.
- Synchronising circuit: microprocessor synchronises firing pulses with supply line frequency.
- Gate pulse generator: receives firing signal from microcomputer; external logic generates firing pulses in proper sequence.
- Changeover logic: manages forward/reverse thyristor converter switching.
Most early µPs were not fast enough to generate gate pulses by software alone. A required firing angle value was fed to an external logic circuit instead.
Stability and Performance
- Stability depends partly on the operating speed of the microprocessor.
- Mainly depends on how the standard analog controller scheme (P or PI) is implemented in software.
- Once I/O hardware is designed, a desired response is obtained by simply changing constants in the software control equations.
Example Applications
- Battery-powered electric vehicle: optimum torque control, regenerative and friction braking, battery charging, fault monitoring.
- Rolling mills, paper mills: high accuracy, good speed resolution, fast response.
Complex Control Tasks Better Performed by µP
- Blending of regenerative and friction braking.
- Programmed battery charging.
- Fault monitoring and diagnostics.
- Nonlinear functions stored as look-up tables.
- Supervisory monitoring from a central computer via high-speed digital transmission.
4. State of the Art — Microprocessor-Based Drives
Historical Milestone
- 1972: Intel first announced microprocessors.
- Most widely used: 8-bit microprocessors.
- Clock frequency range: 0.1 to 10 MHz.
Examples of 8-bit microprocessors:
Intel 8080 · Motorola 6800 · Fairchild F-8 · TI 9980 · RCA 1802 · Rockwell RPS-8 · Signetics 2650 · Zilog Z80
Advantages of Microprocessor-Based Drives
- Flexibility: change control scheme by modifying software only.
- Fully digital: decreased sensitivity to external disturbances.
- Fewer components: improved reliability and less wiring.
- Built-in fault-finding programs: diagnostic programs isolate faulty boards.
- Nonlinear functions: stored as look-up tables in memory.
- High accuracy, better time response, better speed regulation.
- Remote monitoring via digital high-speed transmission.
Future Outlook
Cost Trend
As the price of microprocessors and their associated peripherals continues to fall, microprocessor-based control systems will become cost competitive for a wider range of industrial applications.
Future Incorporation in Industrial Drives
- Drives that require high levels of performance.
- Drives that receive commands from central computers.
- Or both.
Hardwired Logic vs. Microcomputer
| Hardwired Logic | Microcomputer |
|---|---|
| Simple control only | Complex control |
| Not flexible | Highly flexible |
| Difficult to upgrade | Software update only |
| No self-diagnosis | Built-in diagnostics |
| Analog errors | Fully digital |
Summary — All Three Parts
PLL Control — Key Points
- Motor speed converted to a digital pulse train; locked to a reference frequency.
- Phase detector + loop filter + power converter + motor + speed encoder form the closed loop.
- Filter \(F(s)\) choice is critical: PI filter for small motor; PD or phase-lead for large motor.
- Hybrid PLL + analog scheme gives theoretically zero speed regulation.
- Speed regulation as low as 0.002%; hundredfold improvement over analog.
- Suitable for multi-motor synchronised drives.
Microcomputer Control — Key Points
- Eliminates all analog component errors.
- Same outer speed + inner current loop structure, implemented in software.
- Digital tachometer, A/D converter, digital counter for speed and current sensing.
- Stability governed by how P/PI is implemented in software.
- Handles complex tasks: nonlinear functions via look-up tables, fault diagnostics, remote monitoring.
- As cost falls, will become standard in high-performance industrial drives.
Closed-Loop Control of DC Motors — Electric Drives