Transcript Slide 1
CHAPTER 25 RESONANCE Resonance – What is it? WVSS example Vibration Report Much like pumping your legs on a swing Tacoma Narrows Bridge http://www.youtube.com/watch?v=j-zczJXSxnw Electronic Resonance Electronic Resonance is when the capacitive reactance and inductive reactance cancel each other out, resulting in low total impedance and HIGH CURRENT. Thus when XC = XL , then in series circuits: ZT = R – jXC + jXL = R + 0 = R Therefore, resonance is the frequency at which the circuit phase angle is 0˚. The reason for this is because the circuit is purely resistive when the reactances cancel each other out When the capacitor and inductor have the same reactance, they cancel each other out. AKA they combine to be a short. What is the current at resonance? 50A What is the voltage across the resistor? VR = 50A x 2Ω = 100V What is the voltage across the capacitor? VC = 50A x –j5Ω = 250-90˚V What is the voltage across the inductor? VL = 50A x j5Ω = 25090˚V Kirchoff’s Voltage Law still holds Although it seems like there is no VT = VR + VC + VL voltage left for the capacitor and 100 = 100 + 250-90˚ + 25090˚ inductor to have, they still do. The voltages are at opposite angles so they 100 = 100 + 0 cancel out. Notice that the voltage across the capacitor (and inductor as well) have more voltage than the source voltage! Current What is fo in this circuit? This is what current looks like at resonant frequency. Resonant Frequency is labeled fo. fo On either side of fo impedance increases rapidly so current goes down. Current Explaining Frequency Domain vs. Time Domain Frequency At frequencies greater than fo, the load is Inductive and vice versa. When XC=XL , (fO) the circuit is purely resistive. So what frequency will cause the circuit to resonate??? 𝑋𝐿 = 𝑋𝐶 1 2𝜋𝑓𝑜 𝐿 = 2𝜋𝑓𝑜 𝐶 1 2 𝑓𝑜 = 2𝜋𝐿 ∙ 2𝜋𝐶 2 𝑓𝑜 = Notice R is not used for calculating resonant frequency 𝒇𝒐 = 1 2𝜋𝐿 ∙ 2𝜋𝐶 𝟏 𝟐𝝅 𝑳 ∙ 𝑪 = There is only one frequency that causes XL and XC to be equal to each other. (fo) This is what we are solving for 𝟏 𝟐𝝅 𝟐𝟎𝟎𝒎𝑯 ∙ 𝟏𝟎𝒏𝑭 =3559 Hz Make sure they can all do this in their calculator Calculate the Resonant Frequency 𝒇𝒐 = = 𝟏 𝟐𝝅 𝑳 ∙ 𝑪 𝟏 𝟐𝝅 𝟖𝒖𝑯 ∙ 𝟓𝟎𝟎𝒏𝑭 = 𝟕𝟗. 𝟔𝒌𝑯𝒛 What is the current at resonant frequency? 100𝑉 𝐼𝑇 = = 10𝑚𝐴 10,000Ω Calculate other stuff Calculate VCL at fo 0V Calculate VC at fo + --- 1 𝑉𝐶 = 𝐼𝐶 ∙ 𝑋𝐶 = 10𝑚𝐴 ∙ −𝑗 2𝜋𝑓𝐶 1 𝑉𝐶 = 𝐼𝐶 ∙ 𝑋𝐶 = 10𝑚𝐴 ∙ −𝑗 2𝜋79600 ∙ 500𝑛 = 10𝑚𝐴 ∙ −𝑗 4 Ω Just because the circuit is at resonance, doesn’t mean there will be more voltage across the capacitor or inductor than the source voltage. = 40 − 90˚𝑚𝑉 What is the voltage across the inductor? It must be the same size as VC just in the opposite direction! 𝑉𝐿 = 4090˚𝑚𝑉 Calculate the Resonant Frequency 𝒇𝒐 = = 𝟏 𝟐𝝅 𝑳 ∙ 𝑪 𝟏 𝟐𝝅 𝟑𝑯 ∙ 𝟏𝟎𝒖𝑭 = 𝟐𝟗𝑯𝒛 What is the current at resonant frequency? 400𝑉 𝐼𝑇 = = 20𝐴 20Ω Calculate other stuff Calculate VCL at fo 0V Calculate VL at fo + 𝑉𝐿 = 𝐼𝐿 ∙ 𝑋𝐿 = 20𝐴 ∙ 𝑗 2𝜋𝑓𝐿 = 20𝐴 ∙ 𝑗 2𝜋29 ∙ 3 = 10,93290˚𝑉 ‼‼! --- What is the voltage across the capacitor? Again, it must be the same size as VL just in the opposite direction 𝑉𝐶 = 10,932 − 90˚𝑉 Calculating L and C when given fo Suppose you have a 100uF capacitor and want to have a resonant frequency of 1000Hz. What size inductor must you use? 𝒇𝒐 = 𝟏 𝟐𝝅 𝑳 ∙ 𝑪 L = 253uH Calculating L and C when given fo Suppose you have a 20nF capacitor and want to have a resonant frequency of 50kHz. What size inductor must you use? L = 506uH Calculating L and C when given fo Suppose you have a 14mH inductor and want to have a resonant frequency of 60Hz. What size capacitor must you use? 𝒇𝒐 = 𝟏 𝟐𝝅 𝑳 ∙ 𝑪 C = 503uF RVOTD Random Video of the Day Cutoff Frequency Leave up on white board and see notes