1. Resonant Circuit

2. Series Behavior
The behavior of the series RLC circuit is governed by the impedance.
Magnitude and phase i R L v C

3. Perfect Match There is special behavior when XC = XL.
Vectors cancel
Impedance only from resistor
This is called resonance.
VL=IXL
VR=IR
VC=IXC

4. Resonant Frequency
The requirements for resonance come from the reactances.
There is a resonant frequency w0 associated with the circuit.
Angular frequency w
Can be converted into frequency f in Hz

5. Vector Sum The total impedance is the magnitude of Z.
The phase between the current and voltage is the angle f between Z and the x-axis. XL XC Z f R

6. Peak Performance
At resonance the current is at maximum for the voltage.

7. Circuit Example
Find the resonant frequency in the following circuit in Hz.
The problem requires the formula for the frequency f.
Only the inductance and capacitance matter.
1/2p (0.25 H 10-7 F)1/2 = 1 kHz
100 W
250 mH
10 V
0.1 mF

8. Circuit Example
The behavior of the series RLC circuit is governed by the impedance.
Magnitude and phase
100 W
250 mH
10 V
0.1 mF

9. Resonant Reactance next
In the preceding circuit the voltage across each component can be found.
Current due to resistor only
The voltage across the inductor has an amplitude of 158 V.
So does the capacitor
They are each 90° out of phase and cancel out.
next