CHAPTER 25 RESONANCE

Resonance – What is it?

WVSS example Vibration Report Much like pumping your legs on a swing Tacoma Narrows Bridge http://www.youtu

be.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?

What is the voltage across the capacitor?

What is the voltage across the inductor?

Kirchoff’s Voltage Law still holds

VR = 50A x 2Ω = 100V

VC = 50A x –j5Ω = 250-90˚V

Although it seems like there is no voltage left for the capacitor and

inductor to have, they still do. The voltages are at opposite angles so they

cancel out.

VL = 50A x j5Ω = 25090˚V

VT = VR + VC + VL

100 = 100 + 250-90˚ + 25090˚

100 = 100 + 0

Notice that the voltage across the capacitor (and inductor as well) have more voltage than the source voltage!

This is what current looks like at resonant frequency.

Resonant Frequency is labeled fo.

What is fo in this circuit?

fo

Current

On either side of fo

impedance increases rapidly so

current goes down.

ExplainingFrequency Domain vs. Time Domain

Frequency

Current

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

2𝜋 𝑓 𝑜 𝐿=1

2𝜋 𝑓 𝑜𝐶

𝑓 𝑜2=

12𝜋 𝐿 ∙2𝜋 𝐶

√ 𝑓 𝑜2=√ 12𝜋 𝐿 ∙2𝜋𝐶

𝒇 𝒐=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 HzMake sure they can all do this in their calculator

Notice R is not used for calculating resonant frequency

Calculate the Resonant Frequency

𝒇 𝒐=𝟏

𝟐𝝅 √𝑳 ∙𝑪

¿𝟏

𝟐𝝅√𝟖𝒖𝑯 ∙𝟓𝟎𝟎𝒏𝑭

¿𝟕𝟗 .𝟔𝒌𝑯𝒛What is the current at resonant frequency?

𝐼𝑇=100𝑉10,000Ω

=10𝑚𝐴

Calculate other stuffCalculate VCL at fo

0V

+

---

Calculate VC at fo

𝑉 𝐶=𝐼𝐶 ∙ 𝑋𝐶=10𝑚𝐴 ∙− 𝑗 ( 12𝜋 𝑓𝐶 )

𝑉 𝐶=𝐼𝐶 ∙ 𝑋𝐶=10𝑚𝐴 ∙− 𝑗 ( 12𝜋 79600 ∙500𝑛 )

¿10𝑚𝐴 ∙− 𝑗 (4 )Ω

¿ 40−90 ˚𝑚𝑉Just because the circuit is at resonance, doesn’t mean there will be more voltage across the capacitor or inductor than the

source voltage.

What is the voltage across the inductor?

It must be the same size as VC just in the opposite direction!

𝑉 𝐿=4090 ˚𝑚𝑉

Calculate the Resonant Frequency

𝒇 𝒐=𝟏

𝟐𝝅 √𝑳 ∙𝑪

¿𝟏

𝟐𝝅√𝟑𝑯 ∙𝟏𝟎𝒖𝑭

¿𝟐𝟗𝑯𝒛What is the current at resonant frequency?

𝐼𝑇=4 00𝑉20Ω

=20 𝐴

Calculate other stuffCalculate VCL at fo

0V

+

---

Calculate VL at fo

𝑉 𝐿=𝐼 𝐿 ∙𝑋 𝐿=20 𝐴 ∙ 𝑗 (2𝜋 𝑓𝐿)¿20 𝐴 ∙ 𝑗 (2𝜋 29 ∙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

𝑉 𝐶=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?

𝒇 𝒐=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?

𝒇 𝒐=𝟏

𝟐𝝅 √𝑳 ∙𝑪

C = 503uF

RVOTD

Random Video of the Day

Cutoff Frequency

Leave up on white board and see notes