# Parallel Resonance Circuit

Parallel resonance occurs when the feeding frequency creates a zero phase difference between the supply voltage and the current that forms a resistant circuit.

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In many ways, **the parallel resonance** circuit is exactly the same as the serial resonance circuit that we examined in the previous lesson.Both are 3-element networks containing two reactive components that make them a circumstial circuit, both are affected by changes in feeding frequency, and both have a frequency point at which two reactive components that affect the properties of the circuit cancel each other out.Both circuits have a resonance frequency point.

However, the difference this time is that a parallel resonance circuit is affected by currents flowing from each parallel branch within the parallel LC tank circuit.A **tank circuit is** a parallel combination of L and C used in filter networks to select or reject AC frequencies.Review the following parallel RLC circuit:

### Parallel RLC Circuit

Let's define what we already know about parallel RLC circuits.

A parallel circuit containing a resistance, R, an inductancy, L and a capacitance, C will produce a parallel resonance (also called **anti-resonance)** circuit when the current passing through the parallel combination is in the same phase as the supply voltage.In resonance, due to the energy of the oscillations, there will be a large circulatory current between the inductor and capacitor, after which parallel circuits produce current resonance.

The parallel resonance circuit stores circuit energy in the magnetic field of the inductor and in the electrical field of the capacitor. This energy is continuously transferred back and forth between the inductor and the capacitor, resulting in zero current and energy being drawn from the feed.

This is because the corresponding instantaneous values of I_{L} and I_{C}will always be equal and opposite, and therefore the current drawn from the feed is the vector sum of these two currents and the current flowing in I_{R.}

In the solution of AC parallel resonance circuits, we know that the supply voltage is common to all branches, so this can be taken as our reference vector.Each parallel branch should be handled separately, as in serial circuits, so that the total feed current received by the parallel circuit becomes the vector sum of individual branch currents.

Then there are two methods available to us in the analysis of parallel resonance circuits.We can calculate the current in each branch and then collect or calculate the amount of each branch to find the total current.

From the previous serial resonance tutorial, we know that resonance occurs when V _{L} = -V _{C,} and when the two reactances are equal, it is X _{L} = X _{C.}The admittan of the parallel circuit is given as follows:

Resonance occurs when X _{L} = X _{is C }and the virtual parts of Y are zero.

Note that the parallel circuit in resonance produces the same equation as the serial resonance circuit.Therefore, it does not matter if the inductor or capacitor is connected parallel or serial.

In addition, the parallel LC tank circuit in resonance acts as an open circuit, where the circuit current is determined only by R resistance.Thus, the total impedance of a parallel resonance circuit in resonance is, as shown, the value of the resistance in the circuit and Z = R.

Thus, the impedance of the parallel circuit in resonance is at its maximum value and equals the resistance of the circuit, creating a highly resistant and low current circuit state.Also in resonance, since the impedance of the circuit is now only the impedance of resistance, the total circuit current will be in the "same phase" as the I supply voltage V_{S.}

By changing the value of this resistance, we can change the frequency response of the circuit. Changing the R value affects the amount of current flowing through the circuit in resonance, if both L and C remain constant. Then the impedance of the circuit in the Z = R_{MAX} resonance is called the "dynamic impedance" of the circuit.

### Impedance in Parallel Resonance Circuit

Note that if the impedance of parallel circuits is at maximum in resonance, as a result, **the admittan of** the circuits must be minimal, and one of the characteristics of a parallel resonance circuit is that the input limiting the circuit current is very low.Unlike the serial resonance circuit, resistance in a parallel resonance circuit has a damping effect on the bandwidth of the circuit that makes the circuit less selective.

In addition, since the circuit current is constant for any impedance value (Z), the voltage on a parallel resonance circuit will have the same shape as the total impedance, and the voltage waveform for a parallel circuit is usually taken from opposite the capacitor.

Now, at the resonance frequency, we know that the admittan of the ε _{r} circuit is minimal and equals the G conductivity given by 1/R, since in a parallel resonance circuit the virtual part of the admittan, that is, susceptance, is zero because B is_{L} = B_{C} as shown.

### Susceptance in resonance

From above, *inductive susceptance* B_{L} is inversely proportional to the frequency represented by the hyperbolic curve. *Capacitive Susceptance*is directly proportional to frequency B

_{C}and is therefore represented by a straight line.The last curve shows a graph of the total sensitivity of the parallel resonance circuit against the frequency and is the difference between the two susceptances.

Then, when the horizontal axis passes at the resonance frequency point, we can see that the total circuit susceptance is zero. Below the resonance frequency point, inductive sensitivity controls the circuit that produces a "delayed" power factor, while capacitive sensitivity above the resonance frequency point is prone to producing a "leading" power factor.

Therefore, at the resonance frequency, the current drawn from the εr feed should be in the "same phase" as the applied voltage, since there is only resistance in the effective parallel circuit, so the power factor becomes one, ( ε = 0 ^{o).}

In addition, since the impedance of the parallel circuit changes with frequency, since the circuit impedance acts as a resistance, since the current in the resonance is in the same phase as the voltage, this makes the circuit impedance "dynamic".Then we found that the impedance of a parallel circuit in resonance is equivalent to the resistance value, and this value should represent the maximum dynamic impedance (Z _{d)} of the circuit, as shown.

## Current in Parallel Resonance Circuit

Since the total suseptans at the resonance frequency is zero, the input is minimal and equals conductivity to G.Therefore, in resonance, the current that passes through the circuit should be minimal because the inductive and capacitive branch currents are equal ( I _{L} = I _{C} ) and 180 is out of ^{phase.}

We remember that the total current flowing in a parallel RLC circuit is equal to the vector sum of separate branch currents and is calculated as follows for a given frequency:

In resonance, I_{L} and I_{C} currents are equal and cancel by giving clear reactive current equal to zero.Then in resonance, this is the equation above.

Since the current flowing through a parallel resonance circuit is the impedance part of the voltage, the impedance in resonance is the maximum value of Z (=R).Therefore, the circuit current at this frequency will be at a minimum V/R value and is given as a graph of the current against the frequency for a parallel resonance circuit.

### Parallel Circuit Current in Resonance

The frequency response curve of the parallel resonance circuit indicates that the magnitude of the current is a function of the frequency, and drawing it on a graph indicates to us that the answer starts from its maximum value, reaches its minimum value at the resonance frequency when I_{MIN} = I R is I_{R,} and rises back to the maximum as ε becomes infinite.

As a result, the inductor, L and capacitor, the size of the current passing through the C tank circuit can be many times larger than the feed current (180 ^{o} out-of-phase) effectively cancel each other out, even in resonance, but because they are equal and opposite.

Since the parallel resonance circuit operates only at the resonance frequency, this type of circuit is also known as a **Rejection Circuit,** since in resonance the impedance of the circuit is at its maximum, thereby suppressing or rejecting the current whose frequency is equal to the resonance frequency.In a parallel circuit, the effect of resonance is also called "current resonance".

The calculations and graphs used above to describe a parallel resonance circuit are similar to those we use for a series of circuits.However, the features and graphics drawn for the parallel circuit are the opposite of serial circuits, where the maximum and minimum impedance of parallel circuits, current and magnification are reversed.Therefore, parallel resonance circuit is also called **anti-resonance** circuit.

## Bandwidth and Selectivity of parallel resonance circuit

The bandwidth of the parallel resonance circuit is defined exactly the same way as the serial resonance circuit. As given the lower and upper segments off the frequencies:_{ε} _{upper} and lower respectively, the power spent in the (0.707 x I )2 R circuit gives us the maximum resonance value of 70.7% equal current value of the same-3dB points.

As with the serial circuit, if the resonance frequency remains constant, an increase in the quality factor(Q) causes a decrease in bandwidth, and in the same way, a decrease in the quality factor, as defined, causes an increase in bandwidth:

BW = ε _{r} /Q or BW = ε _{top} – ε _{bottom}

In addition, the inductor will change the ratio between L and capacitor, C or resistance value, changing R bandwidth and therefore the frequency response of the circuit for a constant resonance frequency.This technique is widely used in tuning circuits for radio and television transmitters and receivers.

Selectivity or **Q factor** for parallel resonance circuitry is usually defined as the ratio of circulating branch currents to the feed current and is given as follows:

Note that the Q factor of the parallel resonance circuit is the opposite of the expression of the Q factor of the serial circuit.In addition, in series resonance circuits, the Q factor gives the voltage magnification of the circuit, while in the parallel circuit it gives the current magnification.

### Bandwidth of parallel resonance circuit

### Parallel Resonance Question Example 1

A parallel resonance network consisting of a resistance of 60Ω, a capacitor of 120uF and an inductor of 200mH is connected to a sinusoidal supply voltage with a constant output of 100 volts at all frequencies.Calculate the resonance frequency, quality factor and bandwidth of the circuit, circuit current in resonance and current magnification.

1. Resonance Frequency, ε _{r}

2. Inductive Reactance in Resonance, X _{L}

3. Quality factor, Q

4. Bandwidth, BW

5. Upper and lower -3dB frequency points, ε _{H} and ε _{L}

6. Circuit Current in Resonance, I _{T}

In resonance, the dynamic impedance of the circuit is equal to R.

BC Available Magnification, I _{mag}

Keep in mind that the current (resistance current) drawn from the feed in resonance is only 1.67 amps, while the current flowing around the LC tank circuit is larger at 2.45 amps.We can control this value by calculating the current passing through the inductor (or capacitor) in resonance.

## Parallel Resonance Summary

We found that **parallel resonance** circuits are similar to serial resonance circuits.In a parallel RLC circuit, resonance occurs when the total circuit current is in the "same phase" with the supply voltage, as the two reactive components cancel each other out.

In resonance, the circuit's admittan is minimal and equals the conductivity of the circuit.In addition, in resonance, the current drawn from the source is minimal and determined by the value of parallel resistance.

The equation used to calculate the resonance frequency point is the same as for the previous serial circuit.However, the use of pure or non-pure components in the serial RLC circuit does not affect the calculation of the resonance frequency, but affects it in a parallel RLC circuit.

In this tutorial on parallel resonance, we assumed that the two reactive components are completely inductive and completely capacitive with zero impedance. However, in reality, the inductor will contain some resistance in series with its inductive coil, since inductors (and solenoids) are wrapped wire coils, usually made of copper, wrapped around a central core.

Therefore, in order to calculate the parallel resonance frequency of a pure parallel resonance circuit, the above basic equation, ε_{r}, will need to be slightly changed to take into account the impurities inductor with a series of resistors.

### Resonance Frequency Using Pure Inductor

Where: L is the inductance of the coil, C is the dc resistance value of parallel capacitance and R _{S} coil.