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Resonant Circuits and Mutual Inductances

This handout considers the properties of resonant circuits and circuits containing mutual

inductances. The material in this handout will be useful for the writeup of the Laboratories.

1 Resonant Circuits

A resonant circuit is simply an RLC circuit. Usually the effect of the resistance is smallrelative to the size of the inductance and capacitance. This leads to highly resonantbehaviour. Usually resonant circuits are considered to by excited by a sinusoidal voltageor current source of frequency . In this handout, we will concentrate on a particularparallel RLC circuit. However, most of the ideas extend to any other configuration. Moredetails concerning resonant circuits can be found in Chapter 17 of the text book.

Consider the sinusoidal current driven parallel RLC circuit as shown in Figure 1.

Iin Z

+

-

Vout R L C

Figure 1: Parallel RLC circuit.

The applied current is assumed to be a sinusoidal current with frequency ; i.e., Iin(t) =Icos(t). As you saw last year, the best way to analyze the steady state behaviour of sucha circuit is via the use of phasor/complex variable methods. In particular, we are interestedin the total impedance of this circuit as a function of frequency:

Z(j) =Vout(j)

Iin(j).

That is, the impedance is the transfer function from the current input to the voltage output.This can be calculated from the impedances of the individual components:

Resistor RInductor jLCapacitor 1

jC

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Hence, we obtain the total impedance:

Z(j) =1

1

R+ jC + 1

jL

.

From this, we obtain the following expression for the magnitude of the impedance function

|Z(j)| = 11

R

2

+

C 1L

2

.

A typical plot of the impedance function of a resonant circuit is shown in Figure 2.

w0

|Z(j)|

R

B

R2

1 2m

Figure 2: Magnitude Frequency Response.

Resonant Frequency The resonant frequency m of a resonant circuit is the frequencycorresponding to the peak value of the transfer function. From the above expression for|Z(j)|, we can see that the peak value for the impedance transfer function of this circuit

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occurs when

m

C

1

mL= 0

2m =1

LC

m = 1LC

Bandwidth The bandwidth B of a resonant circuit is defined as follows: The frequen-cies 1 and 2 are defined as the frequencies at which the transfer function magnitude fallsto a factor of 1

2time its peak value; see Figure 2. Then

B = 2

1.

For the above circuit, we obtain

2 =1

2RC+

1

2RC

2+

1

LC

1 = 12RC

+

1

2RC

2+

1

LC

Hence we obtain

B =1

RC.

Q factor The Q of a resonant circuit is defined as

Q =m

B.

For the above circuit, we obtain

Q = mRC = R

C

L.

2 Mutual InductanceWe now look at some basic properties of transformers and coupled inductors. More detailson this material can be found in Chapter 18 of the text book.

When two inductors are in close proximity (or wound on the same core), they becomemagnetically coupled and exhibit mutual inductance. This can be indicated on a circuitdiagram as shown in Figure 3.

A transformer is an example of magnetically coupled inductors in which both inductorsare wound on the same core to achieve tight magnetic coupling.

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i1

L1

+

-

v1

M

i2

L2

+

-

v2

Figure 3: Coupled Inductors.

The voltage and current relations for the coupled inductors shown in Figure 3 are asfollows:

v1(t) = L1di1

dt+ M

di2

dt

v2(t) = Mdi1

dt+ L2

di2

dt

The quantity M is the mutual inductance between the two inductors and is measuredin Henrys. Note that the mutual inductance is the same in both directions. It follows fromenergy considerations that the mutual inductance M can be no bigger than

L1L2. As an

alternative to defining M for a pair of mutually coupled inductors, one can also define thecoefficient of coupling k

k =ML1L2

.

This is a dimensionless quantity in the range

0 k 1.However, the case of k = 1 corresponds to perfect coupling which is impossible to achievein practice.

Ideal Transformer An idea transformer is one for which the following relationship holds

(referring to Figure 3)v1(t)

v2(t)=

N1

N2

where N1N2

= a is a constant referred to as the turns ratio. An ideal transformer is notpossible in practice but could be achieved by coupled inductors such that L1, L2, Mand k = 1. A real transformer (ignoring energy dissipation effects) with self inductancesL1 and L2 and mutual inductance M can be represented by an equivalent circuit involvingan ideal transformer as shown in Figure 4.

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Ideal

M:L2

k2L1

(1-k2)L1

Figure 4: Transformer equivalent circuit.

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