RLC Circuits

AC Circuits

• An AC circuit is made up with components.• Power source• Resistors• Capacitor• Inductors

• Kirchhoff’s laws apply just like DC.• Special case for phase

IRVR

tV sin0

R

CX

IXV

C

CC

1

C

LX

IXV

L

LL

L

RLC Circuits

• An RLC circuit (or LCR circuit or CRL circuit or RCL circuit) is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel.

• The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively.

• The circuit forms a harmonic oscillator for current and will resonate in a similar way as an LC circuit will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source.

• This effect of the resistor is called damping. The presence of the resistance also reduces the peak resonant frequency somewhat. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a component. An ideal, pure LC circuit is an abstraction for the purpose of theory.

IMPEDANCE AND THE PHASOR DIAGRAMResistive Elements

• For purely resistive circuit v and i were in phase, and the magnitude:

FIG. 15.1 Resistive ac circuit.

• In phasor form,

IMPEDANCE AND THE PHASOR DIAGRAMResistive Elements

FIG. 15.4 Example 15.2.

FIG. 15.5 Waveforms for Example 15.2.

IMPEDANCE AND THE PHASOR DIAGRAMInductive Reactance

FIG. 15.8 Example 15.3.

FIG. 15.9 Waveforms for Example 15.3.

• for the pure inductor, the voltage leads the current by 90° and that the reactance of the coil XL is determined by ψL.

IMPEDANCE AND THE PHASOR DIAGRAMCapacitive Reactance

FIG. 15.16 Example 15.6.

FIG. 15.17 Waveforms for Example 15.6.

• for the pure capacitor, the current leads the voltage by 90° and that the reactance of the capacitor XC is determined by 1/ψC.

Inductors - how do they work?

L

Start with no current in the circuit. When the battery is connected, the inductor is resistant to the flow of current.

Gradually the current increases to the fixed value V0/R, meaning that the voltage across the inductor goes to zero.

In reality the inductor has a finite resistance since it is a long wire so it will then be more like a pair of series resistances.

dt

dILVL

R

V0

Inductors - time constant L/R Again the behavior of an inductor is seen by analysis with Kirchoff’s laws. Suppose we start with no current.

VR

V0

VL

L

RtVVL exp0

L

RtVVR exp10

There is a fundamental time scale set by L/R, which has units of seconds (=Henry/Ohm)

dt

dILIRVVV LR 0

L

Rt

R

VI exp10then and

Mathematical analysis of a series LRC circuit - bandpass filter

R

Vin

C

L

Vout

First find the total impedance of the circuit

CLiRZ

1

CLiR

R

V

V

in

out

1C

iCi

ZC

1

LiZL

CL

R

1

tan 1

Using a voltage divider

The phase shift goes from 90°to -90°.

Mathematical analysis of a series LRC circuit - bandpass filter (2)

R

Vin

C

L

Vout

The magnitude of the gain, Av, is

22 1

CLR

R

V

VA

in

outv

Note that for high frequencies L is dominant and the gain is R/ L or small. At low frequencies the gain is RC because the impedance of the capacitor is dominant. At 2 = 1/LC the gain is one (assuming ideal components).

Series RLC • A series RLC circuit can be made from each component.

• One loop• Same current everywhere

• Reactances are used for the capacitors and inductors.

• The combination of resistances and reactances in a circuit is called impedance.

R

v L

C

i

Vector Map

• Phase shifts are present in AC circuits.• +90° for inductors• -90° for capacitors

• These can be treated as if on the y-axis.• 2 D vector• Phasor diagram

VL=IXL

VC=IXC

VR=IR

Vector Sum• The current is the same in the loop.

• Phasor diagram for impedance

• A vector sum gives the total impedance.

XL

XC

R

XLXC

R

Z

Vector Sum

XLXC

R

Z

• 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.f

2

2

22

1

CLRZ

XXRZ CL

RC

L

R

XX CL

1

arctan

tan

Phase Changes• The phase shift is different in each component.

Power Factor

• Power loss in an AC circuit depends on the instantaneous voltage and current.• Applies to impedance

• The cosine of the phase angle is the power factor. cos2Zivip

t0

cos20 ZI

P

cos2 ZIIVP rmsrmsrmsrms