7.4 Variable Frequency Drives (VFDs), Motor Starters & Ladder Logic / PLC Controls

7.4 Variable Frequency Drives (VFDs), Motor Starters & Ladder Logic / PLC Controls

Executive Overview: Variable Frequency Drives (VFDs), solid-state soft starters, electromechanical motor starters, and Programmable Logic Controllers (PLCs) form the backbone of modern industrial automation, process control, and motor power distribution. On the NCEES PE Electrical and Computer: Power examination, questions evaluate candidates on Volts-per-Hertz ($V/f$) scaling, Field Oriented Control (FOC), high $dv/dt$ reflected wave overvoltages on long cables, motor bearing EDM currents, reduced-voltage starter sizing (Autotransformer, Star-Delta, Soft Starter), and PLC ladder logic programming (seal-in contacts, interlocks, timers, and counters).

---

1. Variable Frequency Drive (VFD) Architecture

A modern low-voltage ($480\text{ V} - 690\text{ V}$) or medium-voltage ($2.3\text{ kV} - 13.8\text{ kV}$) VFD consists of three primary power stages:

                         COMPLETE VFD HARDWARE TOPOLOGY
         +-----------------+      +-----------------+      +-----------------+
         |  AC-DC FRONT    |      |     DC LINK     |      |   DC-AC PWM     |
         |   RECTIFIER     |=====>|    FILTER BUS   |=====>|    INVERTER     |=====> 3-Phase
         +-----------------+      +-----------------+      +-----------------+       Motor
          6-Pulse / 12-Pulse       Capacitor Bank           IGBT H-Bridge
          Active Front End (AFE)   DC Choke / Pre-charge    Sinusoidal PWM / SVPWM
                                   Braking Chopper

+---------------------------------------------------------------------------------------------------+
|                             VFD FUNCTIONAL STAGES & ROLES                                         |
+---------------------------------------------------------------------------------------------------+
| Stage               | Core Components             | Primary Function & Engineering Nuances       |
| :---                | :---                        | :---                                         |
| **1. Rectifier**    | 6-pulse diode bridge,       | Converts 3-phase AC utility power to unregu- |
|                     | 12-pulse bridge, or AFE     | lated DC. Active Front End (AFE) uses IGBTs  |
|                     | (bidirectional IGBTs)       | for 4-quadrant regeneration and low THDi.    |
| **2. DC Link**      | Electrolytic / Film caps,   | Smooths DC voltage ripple; stores energy.    |
|                     | DC link reactor (choke),    | Pre-charge resistor limits inrush at power-up|
|                     | Dynamic braking chopper     | Dynamic braking resistor dissipates regen.   |
| **3. Inverter**     | 6 IGBTs with anti-parallel  | Synthesizes variable-voltage, variable-      |
|                     | fast-recovery diodes        | frequency 3-phase AC using high-speed PWM.   |
+---------------------------------------------------------------------------------------------------+

---

2. VFD Speed & Torque Control Schemes

+---------------------------------------------------------------------------------------------------+
|                              VFD CONTROL REGIME SPECIFICATIONS                                    |
+---------------------------------------------------------------------------------------------------+
| Control Scheme      | Operating Principle                        | Dynamic Performance & Typical Use     |
| :---                | :---                                       | :---                                  |
| **Volts-per-Hertz** | Maintains constant stator flux:            | Simple scalar control, no encoder.    |
| **($V/f$ Scalar)**  | $\Phi \approx \frac{V_s}{f_s} = \text{const}$ up to $60\text{ Hz}$.| Fans, pumps, HVAC, multi-motor setups.|
| **Sensorless Vector**| Decomposes stator current into $I_d$ (flux)| High dynamic torque response, 100:1   |
| **(FOC - Open Loop)**| and $I_q$ (torque) using motor model.      | speed range, full torque down to 0.5 Hz.|
| **Closed-Loop Vector**| FOC with quadrature encoder feedback      | Zero-speed 100% full torque holding;  |
| **(FOC + Encoder)** | from motor shaft.                          | Cranes, hoists, elevators, winders.   |
+---------------------------------------------------------------------------------------------------+

                    VOLTS-PER-HERTZ (V/f) OPERATING REGIMES
       Voltage / Torque / Power
            ^
            |    CONSTANT TORQUE REGION     :    CONSTANT HORSEPOWER REGION
            |   (Stator Flux Φ = Constant)  :    (Field Weakening Regime: V = V_rated)
     V_rated|                          /----+----------------------- Voltage V_s (Clamped)
            |                        /      :
     P_rated|                      /        +----------------------- Rated Power P (kW)
            |                    /          :   \
            |   ----------------+           :    \ ----------------- Continuous Torque (∝ 1/f)
     T_rated|   Rated Torque    :           :     \
            |                   :           :      \ --------------- Breakdown Torque (∝ 1/f²)
            +-------------------+-----------+----------------------> Stator Frequency (f_s)
            0                  Boost     Base (60 Hz)            Max Speed (120 Hz+)

Constant Torque Regime (Below Base Speed: $0 \le f \le 60\text{ Hz}$)

• Voltage ramps linearly with frequency: $\frac{V_s}{f_s} = \frac{460\text{ V}}{60\text{ Hz}} = 7.67\text{ V/Hz} = \text{constant}$.
• Stator magnetic flux remains constant at rated value: $\Phi_m \propto \frac{V_s}{f_s}$.
• Maximum continuous torque capability is constant: $T_{max} = K \Phi_m I_{rated} = \text{constant}$.
• Output mechanical power increases linearly with speed: $P = T \cdot \omega_m \propto f$.
• Low-Frequency Voltage Boost ($V_{boost}$): At low frequencies ($<5\text{ Hz}$), the stator winding resistance voltage drop ($I_s R_s$) becomes significant relative to the applied voltage, causing stator flux to collapse. A small offset voltage boost ($V_{boost} \approx 10 - 25\text{ V}$) is added at $0\text{ Hz}$ to overcome $I R$ drop and restore full starting torque.

Field Weakening Regime (Above Base Speed: $f > 60\text{ Hz}$)

• Applied voltage is clamped at maximum nameplate rating: $V_s = V_{rated} = 460\text{ V}$.
• As frequency increases, the $V/f$ ratio drops: $\Phi_m \propto \frac{V_{rated}}{f} \propto \frac{1}{f}$.
• Mechanical shaft power remains constant at rated nameplate: $P_{max} = \text{constant}$ (Constant Horsepower Region).
• Maximum continuous torque decreases inversely with speed: $T_{continuous} \propto \frac{1}{f}$.
• Breakdown (Pull-Out) Torque Limitation: Motor breakdown torque drops inversely with the square of frequency: $$T_{breakdown} \propto \left(\frac{V_{rated}}{f}\right)^2 \propto \frac{1}{f^2}$$

---

3. Practical VFD Installation Phenomena: Reflected Waves & Bearing Currents

Fast-switching IGBT inverters produce ultra-steep voltage rise times ($t_r = 50 - 200\text{ ns}$), corresponding to $dv/dt$ rates of $5,000$ to $10,000\text{ V}/\mu\text{s}$.

                     REFLECTED WAVE VOLTAGE DOUBLING PHENOMENON
        Voltage at Motor Terminals
             ^
             |                     2x Overvoltage Spike (~1300 V - 1600 V on 480 V Bus)
             |                         +---+
      2*V_dc +                        /     \  <-- Reflected Wave Superimposes on Incident Wave
             |                       /       \ 
             |                      /         +---------------------- Steady-State V_dc (650 V)
        V_dc +                     / 
             |       Incident Wave/
             0-------------------+-----------------------------------> Time (t)
             |<-- Prop Delay t_p>|

1. Reflected Wave Overvoltage on Long Motor Leads

A motor cable acts as a distributed-parameter transmission line with characteristic surge impedance $Z_0 \approx 50 - 150\,\Omega$. The motor terminal surge impedance is much higher ($Z_m \approx 1,000 - 4,000\,\Omega$).

• Voltage Reflection Coefficient ($\Gamma$): $$\Gamma = \frac{Z_m - Z_0}{Z_m + Z_0} \approx \frac{2000 - 100}{2000 + 100} = +0.905 \approx +1.0$$
• Critical Cable Length ($L_{crit}$): When cable length exceeds the critical distance where one-way propagation time ($t_p$) exceeds half the IGBT rise time ($t_p \ge \frac{t_r}{2}$): $$L_{crit} = \frac{v_{prop} \cdot t_r}{2} = \frac{(500\text{ ft}/\mu\text{s}) (0.1\,\mu\text{s})}{2} \approx 25 - 50\text{ ft}$$
• On cables longer than $50 - 100\text{ ft}$, the reflected wave superimposes fully onto the incident wave, creating voltage doubling ($V_{peak} \approx 2 \times V_{dc} \approx 1,300 - 1,600\text{ V}$ on $480\text{ V}$ systems). This destroys standard motor winding insulation via dielectric breakdown and partial discharge.
• Engineering Mitigation:
  • Specify NEMA MG1 Part 31 Inverter-Duty Motors (insulation rated for $1,600\text{ V}$ peak, $t_r \ge 0.1\,\mu\text{s}$).
  • Install a $3\% - 5\%$ Load Reactor at drive output (limits $dv/dt$).
  • Install a $dv/dt$ Filter (resistor-inductor-capacitor network limiting $dv/dt < 200\text{ V}/\mu\text{s}$) or Sine Wave Filter for cable runs $>300 - 1,000\text{ ft}$.

2. Motor Bearing Currents & Shaft Voltages (EDM Fluting)

Fast PWM switching generates high-frequency common-mode voltage ($V_{cm} = \frac{V_a + V_b + V_c}{3} \ne 0$). Parasitic capacitive coupling between stator windings, rotor core, and frame charges the motor shaft. When the shaft voltage exceeds the thin dielectric oil film breakdown threshold ($15 - 30\text{ V}$), it discharges via Electrical Discharge Machining (EDM) arcing through the steel ball bearings.

• Failure Mode: Causes microscopic EDM pitting, frosting, and characteristic washboard "fluting" grooves across bearing raceways, resulting in loud acoustic noise and catastrophic bearing seizure.
• Engineering Mitigation: Install a Shaft Grounding Ring (SGR) (conductive micro-fiber brush) on the drive end and an insulated / ceramic bearing on the non-drive end.

---

4. Industrial Motor Starters: Topologies & Sizing

+---------------------------------------------------------------------------------------------------+
|                         MOTOR STARTING METHOD PERFORMANCE MATRIX                                  |
+---------------------------------------------------------------------------------------------------+
| Starting Method          | Starting Voltage (% V_rated)| Starting Current (% I_LRC) | Starting Torque (% T_LRT) |
| :---                     | :---                        | :---                       | :---                      |
| **Full-Voltage (FVNR)**  | $100\%$                     | $100\%$ ($600\% I_{FLA}$)  | $100\%$ ($150-200\% T_{FLT}$) |
| **Autotransformer 50%**  | $50\%$                      | $25\%$                     | $25\%$                    |
| **Autotransformer 65%**  | $65\%$                      | $42.25\%$                  | $42.25\%$                 |
| **Autotransformer 80%**  | $80\%$                      | $64\%$                     | $64\%$                    |
| **Star-Delta ($Y-\Delta$)**| $57.7\% (1/\sqrt{3})$     | $33.3\% (1/3)$             | $33.3\% (1/3)$            |
| **Solid-State Soft Start**| $30\% - 100\%$ (Ramped)    | $150\% - 300\% I_{FLA}$    | Adjustable Ramp           |
+---------------------------------------------------------------------------------------------------+

1. Across-the-Line Starters (FVNR & FVR)

• Full-Voltage Non-Reversing (FVNR): Consists of a 3-pole magnetic contactor with an integrated thermal or electronic Overload Relay (OLR). Start/stop control uses standard 3-wire control with a normally open (NO) auxiliary holding/seal-in contact.
• Full-Voltage Reversing (FVR): Uses two contactors ($F$ and $R$) that swap two incoming supply phases (e.g., $L_1$ and $L_3$) to reverse motor rotation. Mandatory dual interlocks: a mechanical interlock bar preventing both contactors from physically closing simultaneously, plus electrical cross-interlocking wiring normally closed (NC) auxiliary contacts of each contactor in series with the opposing coil.

2. Reduced-Voltage Autotransformer Starters (Korndörfer Method)

Uses a 3-phase autotransformer with standard factory taps ($50\%$, $65\%$, $80\%$):

$$V_{start} = k \cdot V_{line} \quad (k = 0.50, 0.65, 0.80)$$ $$I_{motor,start} = k \cdot I_{LRC}$$ $$\text{Utility Line Starting Current: } I_{line,start} = k^2 \cdot I_{LRC}$$ $$\text{Starting Torque: } T_{start} = k^2 \cdot T_{LRT}$$

3. Star-Delta ($Y-\Delta$) Starters

Requires a 6-lead motor designed to run in Delta ($\Delta$). During starting, the stator windings are connected in Wye ($Y$), applying $\frac{V_{LL}}{\sqrt{3}}$ across each phase. Once the motor reaches $\sim 80-90\%$ speed, a timer transitions the connection to Delta ($\Delta$):

$$I_{start,Y} = \frac{1}{3} I_{start,\Delta} = 33.3\% \cdot I_{LRC}$$ $$T_{start,Y} = \frac{1}{3} T_{start,\Delta} = 33.3\% \cdot T_{LRT}$$

---

5. Ladder Logic & Programmable Logic Controllers (PLCs)

PLCs replace hardwired relay panels with a solid-state microprocessor executing a deterministic scan cycle.

                           THE DETERMINISTIC PLC SCAN CYCLE
         +-----------------------------------------------------------------+
         | 1. INPUT SCAN: Read physical input modules; write status into  |
         |    the Input Image Memory Table.                                |
         +-----------------------------------------------------------------+
                                          |
                                          v
         +-----------------------------------------------------------------+
         | 2. PROGRAM EXECUTION: CPU executes ladder logic rungs sequen-   |
         |    tially top-to-bottom, left-to-right, solving boolean algebra |
         |    and updating internal memory & Output Image Table.           |
         +-----------------------------------------------------------------+
                                          |
                                          v
         +-----------------------------------------------------------------+
         | 3. OUTPUT SCAN: Write Output Image Table bits to physical       |
         |    output modules (energizing contactors, solenoids, pilots).    |
         +-----------------------------------------------------------------+
                                          |
                                          v
         +-----------------------------------------------------------------+
         | 4. HOUSEKEEPING: Communications, self-diagnostics, watchdog.    |
         +-----------------------------------------------------------------+

+---------------------------------------------------------------------------------------------------+
|                         STANDARD PLC LADDER LOGIC INSTRUCTIONS                                    |
+---------------------------------------------------------------------------------------------------+
| Symbol      | Instruction Name      | Mnemonic | Description & Operational Logic                  |
| :---        | :---                  | :---     | :---                                             |
| `-[ ]-`     | Examine If Closed     | XIC      | Evaluates TRUE if referenced bit address is 1    |
|             | (Normally Open)       |          | (Closed/Energized).                              |
| `-[/]-`     | Examine If Open       | XIO      | Evaluates TRUE if referenced bit address is 0    |
|             | (Normally Closed)     |          | (Open/De-energized).                             |
| `-( )-`     | Output Energize       | OTE      | Sets referenced bit to 1 if rung logic is TRUE;   |
|             |                       |          | resets bit to 0 if rung logic is FALSE.          |
| `-(L)-`     | Output Latch (Set)    | OTL      | Latches referenced bit to 1 when rung is TRUE;   |
|             |                       |          | remains 1 even if rung goes FALSE (Retentive).   |
| `-(U)-`     | Output Unlatch (Reset)| OTU      | Resets latched bit to 0 when rung is TRUE.       |
| **TON**     | Timer On-Delay        | TON      | Begins timing when rung is TRUE; Done bit (.DN)  |
|             |                       |          | sets to 1 when Accumulator (ACC) ≥ Preset (PRE). |
| **TOF**     | Timer Off-Delay       | TOF      | .DN bit sets when rung is TRUE; timer begins     |
|             |                       |          | timing when rung goes FALSE; .DN drops after PRE.|
| **CTU**     | Count Up              | CTU      | Increments ACC on each False-to-True transition; |
|             |                       |          | .DN sets when ACC ≥ PRE.                         |
+---------------------------------------------------------------------------------------------------+

Classic Ladder Logic Diagrams

1. Three-Wire Motor Start/Stop with Seal-In & Overload Interlock

   STOP PB      START PB       M_OL
    (NC)          (NO)         (NC)                           MOTOR COIL
   I:1/0         I:1/1        I:1/2                             O:2/0
----[/]-----------[ ]----------[/]-------------------------------( )----
     |             | 
     |    M_AUX    |
     |    (NO)     |
     +----[ ]------+
          O:2/0

• Pressing START (I:1/1) completes the rung, energizing coil O:2/0. Auxiliary contact O:2/0 closes in parallel with START, maintaining current when START is released (Seal-In / Holding Contact). Pressing STOP (I:1/0) or tripping thermal overload (I:1/2) unseals and de-energizes the coil.

2. Forward / Reverse Motor Starter with Cross-Interlocking

   STOP        FWD_PB      REV_COIL       F_OL
   I:1/0        I:1/1       O:2/1        I:1/3                        FWD_COIL (O:2/0)
----[/]----------[ ]---------[/]----------[/]----------------------------( )----
     |            | 
     |   F_AUX    |
     +---[ ]------+
         O:2/0

   STOP        REV_PB      FWD_COIL       R_OL
   I:1/0        I:1/2       O:2/0        I:1/3                        REV_COIL (O:2/1)
----[/]----------[ ]---------[/]----------[/]----------------------------( )----
     |            | 
     |   R_AUX    |
     +---[ ]------+
         O:2/1

---

6. Comprehensive Step-by-Step Worked Mathematical Example

Problem Statement

A $460\text{ V}$, 3-phase, $60\text{ Hz}$, 4-pole, $100\text{ hp}$ ($74.6\text{ kW}$) squirrel-cage induction motor has a full-load current of $I_{FLA} = 124.0\text{ A}$, full-load speed of $1,770\text{ RPM}$, full-load torque of $T_{FLT} = 296.7\text{ lb}\cdot\text{ft}$, Locked-Rotor Current of $I_{LRC} = 6.0 \times I_{FLA} = 744.0\text{ A}$, and Locked-Rotor Torque of $T_{LRT} = 1.60 \times T_{FLT} = 474.7\text{ lb}\cdot\text{ft}$.

Calculate:

1. The starting current drawn from the utility line and the motor starting torque under Full-Voltage Direct-On-Line (FVNR) starting.
2. The motor terminal voltage, motor starting current, utility line starting current, and starting torque when using a $65\%$ tap Reduced-Voltage Autotransformer Starter.
3. The starting current and starting torque when using a Star-Delta ($Y-\Delta$) Starter.
4. If the motor is driven by a VFD in $V/f$ mode at $30.0\text{ Hz}$ (half base speed), compute the synchronous speed ($n_{sync}$), the fundamental stator terminal voltage ($V_s$), and the maximum continuous horsepower available at this frequency.

=========================================================================================
CALCULATION WORKFLOW & DETAILED STEP-BY-STEP SOLUTION:
=========================================================================================

Step 1: Across-the-Line (FVNR) Starting Parameters
  Starting Voltage: V_start = 460 V (100%)
  Line Starting Current:
    I_line,start,DOL = I_LRC = 744.0 A
  Starting Torque:
    T_start,DOL = T_LRT = 474.72 lb-ft

Step 2: 65% Autotransformer Starter Calculations
  Tap ratio: k = 0.65

  Motor Terminal Voltage during starting:
    V_motor,start = k * V_line = 0.65 * 460 V = 299.0 V

  Motor Starting Current:
    I_motor,start = k * I_LRC = 0.65 * 744.0 A = 483.60 A

  Utility Line Starting Current (reduced by k^2 through autotransformer action):
    I_line,start,auto = k^2 * I_LRC
                     = (0.65)^2 * 744.0 A
                     = 0.4225 * 744.0 A
                     = 314.34 A

  Motor Starting Torque (scales with voltage squared):
    T_start,auto = k^2 * T_LRT
                 = (0.65)^2 * 474.72 lb-ft
                 = 0.4225 * 474.72 lb-ft
                 = 200.57 lb-ft

Step 3: Star-Delta (Y-Δ) Starter Calculations
  Line Starting Current in Wye connection (1/3 of DOL):
    I_line,start,YD = (1/3) * I_LRC
                    = (1/3) * 744.0 A
                    = 248.00 A

  Starting Torque in Wye connection (1/3 of DOL):
    T_start,YD = (1/3) * T_LRT
               = (1/3) * 474.72 lb-ft
               = 158.24 lb-ft

Step 4: VFD Operation at 30.0 Hz (Constant V/f Regime)
  Synchronous speed at 30.0 Hz for a 4-pole motor:
    n_sync = (120 * f) / P = (120 * 30.0) / 4 = 900.0 RPM

  Stator Terminal Voltage (maintaining V/f = 460 V / 60 Hz = 7.6667 V/Hz):
    V_s = (30.0 Hz / 60.0 Hz) * 460 V = 0.50 * 460 V = 230.0 V

  Continuous Horsepower Rating at 30.0 Hz:
    Since torque capability is constant (100% rated torque = 296.7 lb-ft) and speed is halved:
    HP_30Hz = HP_rated * (f / f_base)
            = 100.0 hp * (30.0 Hz / 60.0 Hz)
            = 50.0 hp (37.3 kW)
=========================================================================================

---

7. Common PE Exam Traps & Tactical Pitfalls

• Autotransformer Starter Line Current vs. Motor Current: The current drawn from the utility line is $k^2 \cdot I_{LRC}$, while the current inside the motor winding is $k \cdot I_{LRC}$. Missing the autotransformer primary-to-secondary turns ratio ($1/k$) causes a major error in upstream breaker/feeder sizing.
• VFD Breakdown Torque Reduction Above Base Speed: Assuming motor breakdown torque drops linearly ($1/f$) above $60\text{ Hz}$. While continuous rated thermal torque drops as $1/f$, breakdown (peak) torque drops as $\frac{1}{f^2}$. Running a motor at $120\text{ Hz}$ reduces its breakdown torque to only $\frac{1}{4} = 25\%$ of its $60\text{ Hz}$ value!
• PLC Scan Order and Latch Precedence: In Allen-Bradley and IEC 61131-3 ladder logic, if an OTL (Latch) and OTU (Unlatch) address the same bit in different rungs, the rung executed last in the scan order dictates the final state written to the physical output module.
• Reflected Wave Cable Length Assumption: Assuming reflected wave doubling occurs only on runs over $500\text{ ft}$. With modern ultra-fast IGBTs ($t_r < 100\text{ ns}$), full $2\times$ overvoltage doubling occurs on cable runs as short as $30 - 50\text{ ft}$.