RLC circuits
RLC circuits are named after the components that they contain. Resistors (R), Inductors (L) and Capacitors (C). In these circuits, there are two separate reactances, both opposing each other. In series RLC, the inductive voltage (Vl) is leading the current (I) by up to 90 degrees. The capacitive voltage (Vc) is lagging the current by up to -90 degrees. So what we have is two reactances, working to cancel each other out. Luckily enough, this is exactly how we calculate them.
The lower of the two reactances is subtracted from the higher one. Say Xl is 100 ohms, and Xc is 75 ohms, the resulting reactance (Xnet) is 25 ohms, and since the resulting reactance is Xl, we say the circuit is acting inductively. Likewise, if Xc is greater than Xl, after we subtract one from the other, we say the circuit is acting capacitively.
It is easier to remember this by looking at vector diagrams. Current is once again our reference vector, with resistance in phase on the horizontal, inductance is plotted upward, and capacitance plotted downward. Whether we are talking voltages (Vl/Vr/Vc/Vt) or resistance (Xl/R/Xc/Z), in series circuits, the vectors look the same.
When applying the pythagorean theorem to series RLC, we simply subtract the reactances, and use the remainder (Xnet) in the formula. First subtract lesser reactance from the greater.
or
and then
same thing for reactances to calculate impedance.
Xl – Xc or Xc – Xl = Xnet
Z = √(R^2 + Xnet^2)
An example to calculate Impedance in Series RLC:
R = 100 ohms
Xl = 75 ohms
Xc = 60 ohms
75 – 60 = 15 ohms = Xnet
√(100^2 + 15^2) = 101.119 ohms = Z
Tan-1( 15 / 100 ) = 8.53 degrees = Phase angle
Say we had an applied AC voltage of 20 volts @ 100Hz . First calculate total current using Ohm’s law. I = V / Z.
20 / 101.119 = 197.787 mA
From there we can calculate the voltage drops across the components
Vr = .197787 * 100 = 19.7787 volts
Vl = .197787 * 75 = 14.834 volts
Vc = .197787 * 60 = 11.8672 volts
As you can see, there is a lot of voltage here, but Vl and Vc are canceling each other out. Working through it again, we can double check our work.
Vl – Vc = Vnet
14.834 – 11.8672 = 2.9668 volts
√(19.7787^2 + 2.9668^2) = 20 volts = Vt
Tan-1( 2.9668 / 19.7787 ) = 8.53 degrees. It checks out.
Parallel RLC circuits
Once again, the reactances cancel each other, the same way as in series, but in parallel circuits, voltage is our reference vector (since the same voltage flows though each branch) and is plotted horizontally, along with resistive voltage (in phase). Capacitive current leads the voltage by up to 90 degrees (ICE) and inductive current lags the voltage by up to -90 degrees.
An example of a parallel RLC circuit. Let’s keep our 20 volts @ 100Hz and use:
R = 55 ohms
Xl = 225 ohms
Xc = 125 ohms
Say for simplicity, that each component is on it’s own branch. In a parallel circuit, we want to calculate branch currents first.
Ir = Vt /R
Ir = 20 / 55
Ir = 363.636 mA
Ixl = Vt / Xl
Ixl = 20 / 225
Ixl = 88.8889 mA
Ixc = Vt / Xc
Ixc = 20 / 125
Ixc = 160 mA
We subtract Ixl from Ixc to get our net reactance current (Inet):
160 – 88.8889 = 71.1111 mA = Inet
find total current through the pythagorean theorem:
It = √(Ir^2 + Inet^2)
It = √(.363636^2 + .07111111^2)
It = 370.524 mA
use arctan to solve for phase angle:
Tan-1( .0711111 / .363636 ) = 11.0649 degrees = phase angle
and lets finish by calculating impedance from Ohm’s law.
Z = Vt / It
Z = 20 / .370524
Z = 53.9776 ohms
References:
Foundations of Electronics, by Russell L Meade
