Power Electronics 2 Lecture 2 PDF

Power Electronics 2 Lecture 2 Plan for today: Parameters of a Sinusoidal Voltage Using an Oscilloscope (DSO) to measure Sinusoidal parameters Mathematical expressions for a sinewave Average and rms values of a Sinusoidal waveform Calculating Average and Total rms for a Sinewave + DC Offset Powe...

Power Electronics 2 Lecture 2 Plan for today: Parameters of a Sinusoidal Voltage Using an Oscilloscope (DSO) to measure Sinusoidal parameters Mathematical expressions for a sinewave Average and rms values of a Sinusoidal waveform Calculating Average and Total rms for a Sinewave + DC Offset Power Electronics 2 Lecture 2 Measurements of a Sinusiodal Waveform Power Electronics 2 Parameters of a Sinusoidal Voltage Notes: 1. A pure sinusoidal voltage is ‘centred’ about zero volts 2. Vpk-pk = 2.Vpk 3. frequency(Hz) = 1 period (s) Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Display Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Display Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Display Lecture 2 Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Display time/Div Volts/Div Examples: 2V/Div 10ms/Div Power Electronics 2 Know your Units Lecture 2 Power Electronics 2 Digital Sampling Oscilloscope (DSO) Display Example #1: Sinusoidal Voltage Lecture 2 Power Electronics 2 Lecture 2 Digital Sampling Oscilloscope (DSO) Display Example #1a: Calculating the Peak Voltage of a Sinewave 2.4 divisions Vpk 100mV/Div Peak Voltage (Vpk) = 2.4 x 100mV = 240mV Power Electronics 2 Lecture 2 Digital Sampling Oscilloscope (DSO) Display Example #1b: Calculating the period and frequency of a Sinewave 10ms/Div period = 7.4 divisions period = 7.4 x10ms = 74ms frequency (Hz) = 1/period(s) =13.5Hz Power Electronics 2 Lecture 2 Mathematical Expressions for a Sinusoidal Voltage Expressions for Instantaneous Voltage: Example: 50Hz (20ms) AC voltage ω = 2π.f ω = 2π.50 = 314 rad/sec v(q ) = VPKsinq v(t) = VPKsin(w t) where ω = Voltage frequency in rad/sec Note q=w.t Lecture 2 Power Electronics 2 Average value of a Sinusoidal Waveform: Vave = 1t v(q ).dq ò t 0 Vave = Vav 1 2p V ò 2p Sinq.dq pk 0 Graphically the average value is where the area above the line is equal to the area below the line Vpk 2p Vave = [-Cosq ]0 2p Vpk Vave = [-1+1] 2p Vave = 0 Lecture 2 Power Electronics 2 rms value of a Sinusoidal Waveform: rms stands for Root Mean Square, to determine the RMS value for a waveform you perform the following: Square the instantaneous waveform, take the Mean (Average) of this, and finally determine the square-root of the Mean éVpk 2 Vrms = ê êë 2p Mathematically: é1 t 2 ù Vrms = ê ò v (q )dq ú ët 0 û é 1 Vrms = ê ë 2p 2p éVpk 2 Vrms = ê ë 2p òV pk 0 2 2p öù é1 æ 1 ê 2 çq - 2 Sin2q ÷ú øû0 ë è éVpk 2 é 2p ùù Vrms = ê 0 0 + 0 úûú êë 2 p 2 ë û ù sin q .dq ú û 2 ù é1 ù ò0 êë 2 (1- cos2q )úû dq ú û Vpk 2 Vrms = 2 2p Note: this relationship is only true for a sinusoidal waveform Vrms = Vpk 2 ù ú úû Power Electronics 2 Resistance (R): v(t) = i(t).R (Ohms Law) Instantaneous Power: P(t) = v(t).i(t) Pave = Vpk.Ipk 2 Pave = Vpk . Ipk 2 2 Pave = Vrms.Irms Pmax = Vpk.Ipk Pave = Vpk.Ipk 2 Pmin = 0 Voltage Current Lecture 2 Power This is why rms values of voltage and current are important in AC systems Lecture 2 Power Electronics 2 Average Power Loss in a Resistor: DC System: Pave = Pave = 1t t òp .dq inst 0 1p v ò p .i .dq inst inst 0 Pave = VDC.IDC Pave = V ò p S inq.IpkS inq.dq Vpk .Ipk p pk 0 AC System: Pave = Pave = Vrms.Irms 1p p 2 sin ò q .dq 0 Pave = Vpk.Ipk ép ù = Vpk . Ipk p êë2 úû 2 2 In an AC system the rms values of voltage and current across a resistance results in the same power loss (HEAT) as that produced by the same values of voltage and current in a DC system – THAT’S WHY RMS IS IMPORTANT! Power Electronics 2 Sinewave + DC Offset Lecture 2 Lecture 2 Power Electronics 2 Sinewave + DC Offset: Average Voltage Vave = Vdc = V -Vpk _ sin = V max Vave = V max- (V max-V 2 ) min - max Vpk - pk 2 Lecture 2 Power Electronics 2 Sinewave + DC Offset: Total rms Voltage Vrms _ total = (V rms _ sin 2 +Vdc 2 ) where: Vdc = see previous slide Vrms _ sin = Vpk _ sin 2 = Vpk - pk 2 2 = (V max -V 2 2 min ) Power Electronics 2 Sinewave + DC Offset: Average Voltage 𝑉!"# = 12 𝑉 and 𝑉!$% = −3 𝑉 Determine: (1) VDC, (2) Vrms_sin, (3) Vrms_total TTC 2