Resonance in series-parallel circuits

In simple reactive circuits with little or no resistance, the effects of radically altered impedance will manifest at the resonance frequency predicted by the equation given earlier. In a parallel (tank) LC circuit, this means infinite impedance at resonance. In a series LC circuit, it means zero impedance at resonance:

However, as soon as significant levels of resistance are introduced into most LC circuits, this simple calculation for resonance becomes invalid. We'll take a look at several LC circuits with added resistance, using the same values for capacitance and inductance as before: 10 µF and 100 mH, respectively. According to our simple equation, the resonant frequency should be 159.155 Hz. Watch, though, where current reaches maximum or minimum in the following SPICE analyses:

Parallel LC circuit with resistance in series with L.

resonant circuit 
v1 1 0 ac 1 sin
c1 1 0 10u     
r1 1 2 100     
l1 2 0 100m    
.ac lin 20 100 200     
.plot ac i(v1) 
.end   

Resistance in series with L produces minimum current at 136.8 Hz instead of calculated 159.2 Hz

Minimum current at 136.8 Hz instead of 159.2 Hz!

Parallel LC with resistance in serieis with C.

Here, an extra resistor (R_bogus) (Figure below)is necessary to prevent SPICE from encountering trouble in analysis. SPICE can't handle an inductor connected directly in parallel with any voltage source or any other inductor, so the addition of a series resistor is necessary to “break up” the voltage source/inductor loop that would otherwise be formed. This resistor is chosen to be a very low value for minimum impact on the circuit's behavior.