Power and Energy Measurement ENE 240 Electrical and Electronic Measurement Class 11, February 4, 2009 werapon.chi@kmutt.ac.th 1 Work, Energy and Power Work is an activity of force and movement in the direction of force (Joules) Energy is the capacity for doing work (Joules) Power is the rate of using energy (Watt) P = W / t, Work done in the time = E / t, Energy transferred in the time 2 1
Electrical Power DC: Instantaneous power at time t P(t) = I(t) V(t) = I 2 (t) R = V 2 (t) / R AC: Mean power per one cycle P avg = 1/T t=0 T P(t) dt = I rms2 R = V rms2 / R 3 Apparent and True Power True power is the power used by the load in Watt unit (W), P = I 2 R = V 2 /R Reactive power in Volt-Amps-Reactive unit (VAR), Q = I 2 X = V 2 /X Apparent power is the power that is applied to the load in Volt-Amps unit (VA), S = I 2 Z = V 2 /Z where impedance Z = R + jx 4 2
Power Factor cos φ = True Power / Apparent Power Pure resistance, cos 0 = 1 perfect Pure inductance, cos 90 = 0 Pure capacitance, cos 90 = 0 Im Reactive Power Apparent Power 0 φ True Power Re 5 Power Factor (Cont d) P = I V cosφ Up to a kind of loads being powered In phase or φ = 0 (resistor) Lagging or current lags voltage, φ > 0 (inductor) Leading or current leads voltage, φ < 0 (capacitor) Lower power factor, higher losses Need to improve power factor 6 3
Power Factor (Cont d) I = sinω(t+ t) = sin(ωt+ω t) = sin(θ+φ) V = sinθ I = sin(θ+φ) t t 7 Wattmeter To measure the true power Two fixed coil is used to provide a magnetic field that is proportional to a current passing through it, instead of using a permanent magnet. B Right Hand s Rule I Pointer Coil Scale I 1 I 2 Spring Moving Coil (Rotatable) Meter Terminal Electrodynamometer 8 4
Wattmeter (Cont d) coils, B I 1 Moving coil, Torgue B I 2 θ = K I 1 I 2 = K I 2, I = I 1 = I 2 P Moving 9 Wattmeter (Cont d) Full-scale deflection current 5-100 ma For larger FSD up to 20 A, it is shunted by a low resistance. Electrodynamometer can be used for both DC and AC (rms) measurement. Moving Shunt 10 5
Single-Phase Wattmeter Based on the principles of the dynamometer The current passing through moving coil and load is proportional to the potential difference across the load. θ = K I 1 I 2 = K I (V/R) = (K/R) (IV) I Current Coil P V Voltage Coil Load 11 Single-Phase Wattmeter (Cont d) Low-current high-voltage load Supply I load Moving P reading = I load2 (R fixed +R load ) P load = I load2 R load Load Wattmeter reads high due to the potential drop across the fixed coil (if neglect the reactance). High-current low-voltage load Supply Moving + V load Load P reading = V load 2 / (R moving R load ) = V load 2 R moving R load (R moving +R load ) P load = V load2 / R load Wattmeter reads high due to the current through 12 the movable coil (if neglect the reactance). 6
Single-Phase Wattmeter (Cont d) Compensated wattmeter Compensating Supply Moving Load To reduce the systematic error, the compensating coil current is, however, in the opposite direction to the load current and cancels out the proportion of the magnetic flux due to the moving coil current. 13 Crossed-Coil Power Factor Meter Two moving coils are crossed and connected in series to an inductor and resistor. For PF = 1, the coil with R will be in phase with the line current (moving), while the coil with L will be out of phase by 90 (no moving). For PF = 0, coil L moves and coil R does not move. Supply 0.9 0.8 Lag Coil L 1 0.9 0.8 Lead Crossed R Load 14 7
Three-Phase Power Distribution The three-phase system uses less conductor material to transmit electric power that the others. Y-Type, V AB = V AN V BN V AB = V AN cos30 + V BN cos30 = 3 V AN, V AN = V BN A V AB 120 N 30 30 V AN V BN C 120 B 120 V BN 15 Three-Phase Loads 3-Phase Transmission Line 3-Phase Generator I Phase Neutral V Phase V Line I Line 3-Phase Loads I Line I Phase Balanced loads, P 1 = P 2 = P 3 = P = I L V P cosφ P total = P 3 = 3 I L V P cosφ = 3 I L (V L / 3 ) cosφ = 3 I L V L cosφ Similarly, Q = 3 I L V L sinφ Star-Type, V Line = 3 V Phase I Line = I Phase Delta-Type, V L = V P I L = 3 I P 16 8
Three-Phase Wattmeter For three-phase system with balanced load, just using single-phase wattmeter, P 3 Current Coil Reading P = 3 /3 I L V L cosφ Voltage Coil 17 Unbalanced-Load 3-Phase Wattmeter P 1 P 2 Using 3 Meters P 3 By using 3 wattmeters, P total = P 1 + P 2 + P 3 18 9
Balanced-Load 3-Phase Wattmeter Brondel s theory, P = I 1 V 1 + I 2 V 2 + I 3 V 3 P 1 Kirchoff s law, V 1 = V 3 + V 1 V 2 = V 3 + V 2 Common V 1 S I 1 V 1 V2 Therefore, P = (I 1 +I 2 +I 3 )V 3 + I 1 V 1 + I 2 V 2 = P 1 + P 2 0 P 2 I 3 V 3 V 2 I 2 19 Balanced-Load 3-Phase Wattmeter (Cont d) P By using 2 wattmeters, 1 R P 1 = I R V RB cosφ 1 I R V RB = V RS V BS Balanced, V RS = V BS φ 1 = 30 φ V BS 120 V BS φ 1 B Y φ lagging 30 60 V RS P 2 S Using 2 Meters I R V RB 20 10
Balanced-Load 3-Phase Wattmeter (Cont d) P 1 = I L V L cos(30 φ) = I L V L ( 3 /2 cosφ + 1/2 sinφ ) Similarly, P 2 = I L V L ( 3 /2 cosφ 1/2 sinφ ) φ V BS 120 φ 2 = 30 + φ 30 I Y V YS V BS V YB Note: cos(θ ± φ) = cosθ cosφ ± sinθ sinφ 21 Balanced-Load 3-Phase Wattmeter (Cont d) Total Power, P total = P 1 + P 2 = 3 I L V L cosφ The same as using 1 meter for balanced load system and P 1 P 2 = I L V L sinφ 22 11
Balanced-Load 3-Phase Wattmeter (Cont d) Reactive Power, Q = 3 I L V L sinφ = 3 (P 1 P 2 ) (P 1 P 2 )/(P 1 +P 2 ) = I L V L sinφ / 3 I L V L cosφ = tanφ / 3 tanφ = 3 (P 1 P 2 ) / (P 1 + P 2 ) 23 Balanced-Load 3-Phase Wattmeter (Cont d) sin 2 φ + cos 2 φ = 1 tan 2 φ + 1 = 1/cos 2 φ 3(P 1 P 2 ) 2 /(P 1 +P 2 ) 2 + 1 = 1/cos 2 φ Power factor, cosφ = 1 / [ 1 + 3(P 1 P 2 ) 2 /(P 1 +P 2 ) 2 ] 24 12
Watt-Hour-Meter To measure the electrical energy supplied to industrial and domestic consumers in unit (kw-hr) Generated Torque by Eddy Current = k g I V cosφ Magnetic Breaking Torque = k b N where N is number of revolution per unit time (gearing to a mechanical counter to count watt-hours) k b N = k g I V cosφ N = (k g /k b ) IVcosφ 25 Watt-Hour-Meter (Cont d) N Gear Wheels for Dial Rotation Voltage Coil S S Magnetic break Eddy Currrent N Current Coil Rotating Aluminium Disk Therefore, number of revolution refers to the energy used by loads. 26 13