Resistive Circuits with Steady State Sources In general,can perform the same analysis with any resistive network Voltage Divider-From before: I1R1+I1R2-V。=0 I1=V/R1+R2) v(t) Then: V1=I1R1,V2=I1R2 Vo=Vs R2/(R1+R2) vs(t)=Vp sin(@o t) Inserting the time-varying source: v。(t)=VpR2/(R1+R2】sin(o。t) PAVG VEFF IEFF =⅓V,R2/R1+R2】V。/(R+R2] PAVG =Vp2 [R2 /(R1+R2)2] Basic Electronics-Special Lecture for TIPP 2011 6 Gary Drake,Argonne National Lab-Session 2
Basic Electronics – Special Lecture for TIPP 2011 6 Gary Drake, Argonne National Lab – Session 2 Resistive Circuits with Steady State Sources In general, can perform the same analysis with any resistive network vs(t) = Vp sin( t) Voltage Divider – From before: R1 R2 Vo = ? I1 vs(t) I1 R1 + I1 R2 – Vs = 0 I1 = Vs / (R1 + R2) Then: V1 = I1 R1, V2 = I1 R2 VO = VS R2 / (R1 + R2) Inserting the time-varying source: vo(t) = Vp [R2 / (R1 + R2)] sin(o t) PAVG = VEFF IEFF = ½ Vp [R2 / (R1+R2)] [Vp / (R1+R2)] PAVG = ½ VP2 [R2 / (R1 + R2)2]
Introduction to Steady State (AC)Analysis ■ Sinusoidal Sources Why are sinusoids important? Reason #1:Our electrical grid works on sinusoidal power Famous shoot-out between Nikola Tesla and Thomas Edison, 1893 Worlds Fair in Chicago (Tesla won,but died a pauper...) In the USA: f=60 Hz VEFF=120V Vp 170V Color Key: Substaton Black: Generaton Step Down Subtransmlsslon Blue: Transmlsslon Transfommer Customer Green:Distrbution Transmlsslon llnes 26kV and 69kV 765,500,345,230,and138kW Generadng Staton Prlmary Customer 13kV and 4kV Generating Transmlsslon Customer Secondary Customer 138kV or 230kV 120V and 240V Step Up ▣▣ Transformer Images from Wikipedia Basic Electronics-Special Lecture for TIPP 2011 7 Gary Drake,Argonne National Lab-Session 2
Basic Electronics – Special Lecture for TIPP 2011 7 Gary Drake, Argonne National Lab – Session 2 Introduction to Steady State (AC) Analysis Sinusoidal Sources • Why are sinusoids important? Reason #1: Our electrical grid works on sinusoidal power – Famous shoot-out between Nikola Tesla and Thomas Edison, 1893 Worlds Fair in Chicago (Tesla won, but died a pauper…) In the USA: f = 60 Hz VEFF = 120V VP = 170V Images from Wikipedia
Introduction to Steady State (AC)Analysis Sinusoidal Sources (Continued) Why are sinusoids important(Cont.)? Reason #2:Convenient representation in the Freguency Domain Time Domain Magnitude Frequency Domain 0 00 Angle 0 v(t)=VP cos(oot+φo) Define phasors:F(@)=Mag Angle If:v(t)=Vp cos(@ot+)Phasor form:V(@)=Vp/o Has connection with Complex Number Analysis...More on this later Basic Electronics-Special Lecture for TIPP 2011 8 Gary Drake,Argonne National Lab-Session 2
Basic Electronics – Special Lecture for TIPP 2011 8 Gary Drake, Argonne National Lab – Session 2 Introduction to Steady State (AC) Analysis Sinusoidal Sources (Continued) • Why are sinusoids important (Cont.)? Reason #2: Convenient representation in the Frequency Domain • Define phasors: t v(t) = VP cos(Ot + O) VP T -VP VP O O Time Domain Frequency Domain Phasor form: V() = VP O Magnitude Angle If: v(t) = VP cos(Ot + O) F() = Mag Angle Has connection with Complex Number Analysis… More on this later
Introduction to Steady State (AC)Analysis Sinusoidal Sources (Continued) Why are sinusoids important (Cont.)? Reason #3:Any periodic waveform can be represented as an infinite sum of sinusoids,with frequencies that are multiples of the fundamental frequency 0o>Fourier Series For any periodic waveform V(t),with fundamental frequency oo 00=(2π)/T v(t)=>E edino.),where j=v-1 The coefficients E are given by: T/2 E=1/T/V(t)e-inoot dt -T/2 n@o are the harmonic frequencies of v(t) Basic Electronics-Special Lecture for TIPP 2011 9 Gary Drake,Argonne National Lab-Session 2
Basic Electronics – Special Lecture for TIPP 2011 9 Gary Drake, Argonne National Lab – Session 2 Introduction to Steady State (AC) Analysis Sinusoidal Sources (Continued) • Why are sinusoids important (Cont.)? Reason #3: Any periodic waveform can be represented as an infinite sum of sinusoids, with frequencies that are multiples of the fundamental frequency O Fourier Series – For any periodic waveform V(t), with fundamental frequency O – The coefficients En are given by: t v(t) = En e(jnot) , where j = S -1 n=-h +h T O = (2 ) / T En = 1/T f V(t) e-jnot dt -T/2 T/2 v(t) n0 are the harmonic frequencies of v(t)
Introduction to Steady State (AC)Analysis Sinusoidal Sources (Continued) Why are sinusoids important(Cont.) Example:Fourier Series of a Square Wave Time Domain Frequency Domain 2Vp/π -70 7a0 90 00=(2)/T 十 Vo=∑EeGno.t) wherej=v-1 n=-00 n=-o where E=(2Vp/nπ)sin(nπ/2) Notice that only the odd harmonics of v(t)are present in this particular example... Basic Electronics-Special Lecture for TIPP 2011 10 Gary Drake,Argonne National Lab-Session 2
Basic Electronics – Special Lecture for TIPP 2011 10 Gary Drake, Argonne National Lab – Session 2 Introduction to Steady State (AC) Analysis Sinusoidal Sources (Continued) • Why are sinusoids important (Cont.)? Example: Fourier Series of a Square Wave where En = (2VP / n) sin(n / 2) 2VP / O Time Domain Frequency Domain t VP VP O O O O O O O O O V(t) = En e(jnot) n=-h +h T O = (2 ) / T Notice that only the odd harmonics of v(t) are present in this particular example… where j = S -1 n=-h +h