Problem Solving Part 2

Resonance

Parallel and series circuits do not behave the same way at and around resonance. However, the method for determining their resonant frequency is the same.

Resonant FrequencyWe calculate the resonant frequency of the circuit in Fig. 1-A.

HzHzX

HzX

HzX

FXHLCfr

183910839.1

)10866.0)(28.6(

1100075.028.6

1

)1005.0)(150.0(28.6

1

2

1

3

3

6

6

A parallel circuit at resonance has the following characteristics:

• A phase shift of 00

• An impedance equal to resistance• A resistor current that equals the total current• Equal and opposite currents through the

inductive and capacitive branches• Maximum impedance and minimum current

Effects of Resonance

The best way to consider the effects of resonance on a parallel circuit is to calculate all values and construct a current graph. We first calculate the reactance.

1732

)10150)(1839)(28.6(

23HXHz

fLX L

17325774.0

101

)1005.0)(1839)(28.6(

1

2

1

3

6

X

FXHz

fCXC

The calculations show that one characteristic of resonance exists: Inductive reactance equals capacitive reactance.

We now calculate current. Since both reactances are equal, both currents will be equal. However, the phase angles will be in opposition.

A

X

EI

L

TL

0

0

0

9000462.0

901732

08

A

V

X

EI

C

TC

0

0

0

9000462.0

901732

08

A

AA

III CLXT

0

)9000462.0()9000462.0( 00

A

V

R

EI TR

0

0

0

000364.0

02200

08

A

A

A

AIII XTRT

00364.0

00001325.0

000364.0 22

22

As shown in Fig. 1-3, the currents of the two reactors are equal and opposite, so their vector sum is zero. The net total current in the circuit is that in the resistor. Current flows through one inductor in one direction while flowing through the other inductor in the opposite direction. At resonance, energy transfers back and forth between the two inductors, with any energy loss replaced by the voltage source.

In a parallel resonance, the total current equals the resistor current.

Fig. 1-3. Circuit Currents at Resonance

IC= 0.00462 A

IXT= 0 A

IR = IT = 0.00364 A

IL = 0.00462 A

The current through the inductor lags the applied voltage by 900, while the capacitor current leads it by 900. The inductor and capacitor branch currents are 1800 out of phase with each other. When one has a great supply of electrons, the other has a demand, and vice versa. The net result is that once resonance occurs, these two reactors transfer electrons back and forth. Refer to Fig. 1-4.

Fig. 1-4. Current Paths at Resonance

1

2

IX

!X

IR

Conditions that need to exist before resonance can occur:

• The reactances must be equal for the timing of their fields to be the same.

• The frequency of the applied source must be the same as the resonant frequency for oscillation to occur.

We now calculate impedance. Since little energy is taken from the source at resonance, we expect impedance to be high. The impedance of a parallel RLC circuit is at its highest level during resonance. The phase angle at resonance is 00.

0

0

0

02198

000364.0

08

A

V

I

EZ

T

T

The frequency characteristics of the circuit (Fig. 1-A) which we have evaluated at frequencies above, below and at resonance are summarized in the graph in Fig. 1-5.

Fig. 1-5. Relationships in a Parallel RLC Circuit

38.3

0

-55.3

2198

1724

1252

0.00639

0,00464

0.00364

Quality Factor and Bandwidth

As with series circuits, the quality factor Q is an indication of the relationship between reactance and resistance in an RLC parallel circuit. Q for a parallel circuit is determined by the reactance of the inductor and its internal resistance.

Refer to Fig. 1-6 where RL represents the 100-ohm internal resistance of the inductor, while XL is its 2400-ohm reactance. XL and XC are equal since the circuit is resonant. Q is calculated as:

24100

2400

L

L

R

XQ

Fig. 1-6. Coil Resistance

3

4

2

18V1200 Hz

2400

2400

100

Bandwidth (BW) is inversely related to Q and gives an indication of the frequency response of a circuit. High-Q circuits have a narrow bandwidth, while low-Q circuits have a wide bandwidth. The bandwidth for the circuit is:

HzHz

Q

fBW r 50

24

1200

The half-power points of a circuit are the two frequencies where the voltage drops to 0.707 times the peak, or resonant frequency, value. Since voltage causes current in a circuit, the current also drops to 0.707 of its peak value. Since power equals voltage times current, then the power is now 0.5 of what it was at resonance. The phase shift is normally 450 at the half-power points.

The half-power points for this circuit are:

(1200 Hz) – (50/2 Hz) = 1175 Hz

(1200 Hz) + (50/2 Hz) = 1225 Hz

If a 100-ohm resistor RS was placed in series with the 100 ohms of internal resistance that this inductor has, Q and BW would change, as follows:

HzHz

Q

fBW

RR

XQ

r

SL

L

10012

1200

12200

2400

The effects of this change can be seen in Fig. 1-7. A lower Q results in a broader bandwidth. Values of Q can range from 20 to 100 in a typical RLC circuit.

Fig. 1-7. Circuit Q and BW

1200 Hz

Q = 24

Q = 12

1 250

1 225

1175

1175

Uses for Parallel Resonant Circuits

The most common uses for parallel resonant circuits are in radio and television equipment. Their ability to discriminate among different frequencies makes them useful in the signal selection and rejection process. Inductors and capacitors can be placed in parallel to form a network that allows most frequencies to pass except for those that are close to the resonant frequency.

Refer to Fig. 1-8, which shows a transformer that is tuned to resonate at a particular frequency.

Fig. 1-8. Adjustable Transformer in Radio Circuit

7

8

6

3

Signal InSignal Out

A complex signal of many frequencies is applied to the transformer primary. The LC circuit on the primary side resonates only at the desired frequency, with an amplitude proportional to the size of the desired portion of the incoming signal. Transformer action causes this signal to be reproduced at the secondary side of the transformer.

The desired signal is then developed across the high impedance of the secondary parallel resonant circuit. Undesired signals at frequencies beyond the half-power points of the bandwidth are further suppressed. This selection-discrimination process allows a radio or television circuit to select the desired station and block all others.

Evaluating RLC Circuits

Three parameters are of interest in testing RLC circuits:

• Resonant frequency

• Impedance

• bandwidth

Series Connection

Real inductors have internal series resistance because of their windings.

Real capacitors have internal parallel resistance because of their leakage.

Capacitor leads have inductance.Inductor leads have capacitance.The three components in an actual series RLC

circuit do not each have only on form of opposition but each component has complex oppositions.

The circuit we will evaluate is shown in Fig. 1-9. The objective is to determine the resonant frequency, impedance, and bandwidth.

Fig. 1-9. Series RLC Circuit for Evaluation

L1

2.5mH

4

5

1

C10050uF

R1

47k

2

The following results will be obtained if exact values were used and if the components were ideal.

Hz

HzX

HzX

FXHX

LCfr

243,14

10118.0)(28.6(

1100125.028.6

1

)1005.0)(105.2(28.6

12

1

4

8

63

757.447

6.22347

)105.2)(10243.14)(28.6(

2128.047

10

33

HXHzX

R

XQ

AV

R

E

Z

EI

L

TTT

HzHz

Q

fBW r 944,2

757.4

243,14

The half-power points indicating the bandwidthlimits are:

(14,283 Hz) - (2944/2 Hz) = 12,771 Hz

(14,243 Hz) + (2944/2 Hz) = 15,715 Hz

Bandwidth and impedance are affected by resistance, and the resistor is not the only source of resistance. Inductor winding resistance will be a significant factor. Do not exceed the current capacity of the inductor because excess current will saturate an inductor and cause incorrect results.

The best way to determine resonance is by measuring the resistor voltage which is equal to its maximum at the resonant frequency.

Impedance can be determined by dividing the measured value of total voltage by the measured value of total current. If you knowthe actual value of the resistor you can use resistor voltage to calculate circuit current.

Bandwidth can also be determined by measuring the resistor voltage.

BW limits at ER = (0.707)ERmax

RE

E

I

EZ

R

T

T

T

/

Parallel ConnectionIn evaluating an RLC circuit for the parallel

connection, the resistance branch should be larger, and a series resistance should be added to limit inductor current.

Fig. 1-10. Parallel RLC Circuit for Evaluation

The resonant frequency of the circuit should be the same as the resonant frequency of the series circuit. The impedance of the parallel section of this circuit (if ideal) would be equal to the resistive branch of 1000 ohms.

One way to determine resonance is tomeasure the parallel circuit voltage, the voltage across the parallel network. Thenetwork voltage is 0.707 times the resonantvalue at the upper and lower half-powerpoints.