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# Module 10.3

# Parallel Resonance

**After studying this section, you should be able to:**- • Describe the action of LCR parallel circuits above, below and at resonance.
- • Describe current magnification, and dynamic resistance in LCR parallel circuits
- • Use appropriate formulae to carry out calculations on LCR parallel circuits, involving resonance, impedance and dynamic resistance.

Just as in series resonant circuits, there are three basic conditions in a parallel circuit. Dependent on frequency and component values, the circuit will be operating below, above or at resonance. This section describes these three conditions using phasor diagrams involving the current phasors I_{C}, I_{L} and their phasor sum I_{S}, with the reference phasor V_{S}. Note that V_{R} is not shown, but its presence in the circuit is indicated by the variable angle of I_{L} as described in Module 10.2.

## Below Resonance

### Fig. 10.3.1 Below Resonance

Firstly, if the supply frequency is low, below the resonant frequency ƒ_{r} then the condition shown in Fig 10.3.1 exists, and the current I_{L} through L will be large (due to its comparatively low reactance). At the same time the current I_{C} through C will be comparatively small. Because I_{C} is smaller than I_{L} the phase angle θ will be small.
Including I_{S} in the diagram shows that it will be lagging on V_{S} and therefore the circuit will appear to be INDUCTIVE. (Note that this is the opposite state of affairs to the series circuit, which is capacitive below resonance).

## Above Resonance

### Fig 10.3.2 Above Resonance

Fig 10.3.2 shows what happens at frequencies above resonance. Here the current I_{C} through C will be greater than the current I_{L} through L, because the frequency is higher and X_{C} is smaller than X_{L}, θ is greater than in Fig 10.3.1. This gives us the condition where I_{S} (the phasor sum of I_{C} and I_{L}) is leading V_{S} and so the circuit is capacitive.

## At Resonance

### Fig 10.3.3 At Resonance

At resonance the ideal circuit has infinite impedance, but this is not quite the case in practical parallel circuits, although very nearly. Fig 10.3.3 shows the conditions for resonance in a practical parallel LCR circuit. I_{C} is leading V_{S} by 90° but I_{L} is not quite in anti phase (due to the resistance in the circuit´s inductive branch). **In the parallel circuit therefore, resonance must be defined as the frequency where the values of I _{C} and I_{L} are such that I_{S} is IN PHASE with V_{S}.**

## Dynamic Resistance

At resonance, Fig 10.3.3 (above) shows that I_{S} is very small, much smaller than either I_{C} or I_{L} so the impedance across the parallel circuit must be very high at ƒ_{r} and as I_{S} is in phase with V_{S}, the circuit impedance is purely resistive. This pure resistance that occurs only at ƒ_{r} is called the DYNAMIC RESISTANCE (R_{D}) of the circuit and it can be calculated (in ohms) for any parallel circuit from just the component values used, using the formula:

Where R is the total resistance of the circuit, including the internal resistance of L.

## Current Magnification

The other important point shown in Fig 10.3.3 is the size of the phasor for I_{S} compared with I_{C} and I_{L}. The supply current is much smaller than either of the currents in the L or C branches of the circuit. This must mean that more current is flowing within the circuit than is actually being supplied to it!

This condition is real and is known as CURRENT MAGNIFICATION. Just as voltage magnification took place in series circuits, so the parallel LCR circuit will magnify current. The MAGNIFICATION FACTOR (Q) of a parallel circuit can be found using the same formula as for series circuits, namely;

## Adjusting for resonance.

The formula for the resonant frequency of a LCR parallel circuit also uses the same formula for ƒ_{r} as in a series circuit, that is;

### Fig 10.3.4 Parallel LC Tuned Circuits.

It should be noted that this formula ignores the effect of R in slightly shifting the phase of I_{L}. In fact the formula gives an approximate value for ƒ_{r}. However, because the internal resistance of L is usually quite small, so is its effect in shifting the resonant frequency of the circuit. For this reason, the same formula may be used for ƒ_{r} in both series and parallel circuits. In those practical LC circuits designed to operate at high frequencies, and where accurate control over ƒ_{r} is required, it is normal for either L or C to be made adjustable in value.

Notice that the usual formula for resonant frequency ƒ_{r} does not have any reference to resistance (R). Although any circuit containing L must contain at least some resistance, the presence of a small amount of resistance in the circuit does not greatly affect the **frequency** at which the circuit resonates. Resonant circuits designed for high frequencies are more affected by stray magnetic fields, inductance and capacitance in their nearby environment than the very small effects of R, so most high frequency LC resonant circuits will have both screening to isolate them from external effects as much as possible, and be made adjustable over a small range of frequency so they can be accurately adjusted after assembly in the circuit.

However, although this formula is widely used at radio frequencies for both series and parallel resonance, at low frequencies where large inductors, having considerable internal resistance are used, the formula below can be used for ƒ_{r} in low frequency (large internal resistance) parallel resonance calculations.

### Fig 10.3.5 Tuned Transformer in a screening can

The need for careful adjustment after circuit assembly is often a deciding factor for the discontinued use of LC circuits in many applications. They have been widely replaced by solid-state ceramic filters and resonating crystal tuned circuits that need no adjustment. Sometimes however, there may be a problem of multiple resonant frequencies at harmonics (multiples) of the required frequency with solid state filters. A single adjustable LC tuned circuit may then also be included to overcome the problem.

The final values for L and C would be achieved by adjusting one of the two components as shown in Fig. 10.3.4 which would be of a variable type, once the system containing the LC circuit was operating. By this method, not only is the effect of R compensated for, but also any stray inductance or capacitance in the circuit that may also affect the final value of ƒ_{r}. Because, at high frequencies, magnetic fields easily radiate from one component in a circuit to another, LC tuned circuits would also be shielded (screened) by containing them in a metal screening can as shown in Fig 10.3.5.

## 6 Things you need to know about LCR Parallel Circuits.

#### (and that are different to the Series Circuit.)

- 1.
**AT RESONANCE (ƒ**V_{r})_{C}is not necessarily exactly equal to V_{L}but V_{S}and I_{S}are IN PHASE - 2.;
**AT RESONANCE (ƒ**Impedance (Z) is at maximum and is called the Dynamic Resistance (R_{r})_{D}) - 3.
**AT RESONANCE (ƒ**Circuit current (I_{r})_{S}) is at a minimum. - 4.
**AT RESONANCE (ƒ**The circuit is entirely resistive._{r}) - 5.
**BELOW RESONANCE (ƒ**The circuit is inductive._{r}) - 6.
**ABOVE RESONANCE (ƒ**The circuit is capacitive._{r})