Calculating Parallel RLC Resonance Circuit

Like series circuits, parallel RLC circuits (containing inductors and capacitors) are second-order with a resonant frequency.

Both are affected by frequency changes. However in parallel resonance, it is the current through the circuit that reaches a minimum at resonance, not the impedance.

The focus here is, how currents in each branch of the parallel LC tank circuit (L and C in parallel) interact with resonance. (The diagram below shows a parallel RLC circuit. We’ll explore, how branch currents affect resonance behavior.)

Circuit Showing Parallel Connected Resistor, Capacitor and Inductor

Let’s first clarify our existing knowledge on parallel RLC circuits.

• Conductance, G = 1 / R
• Admittance, Y = 1 / Z = √(G² + B²)
• Capacitive Susceptance, B_C = 2πfC
• Inductive Susceptance, B_L = 1 / 2πfL

Parallel Resonance: Current, Not Impedance, Minimizes

In contrast to series resonance, parallel RLC circuits (with resistor R, inductor L, and capacitor C) exhibit “parallel resonance” (or anti-resonance) when the total current aligns in phase with the supply voltage.

At this resonance, a large current circulates between the inductor and capacitor due to oscillating energy transfer.

This results in minimal current drawn from the supply, even though individual branch currents (I_L, I_C) for the inductor and capacitor might be significant.

Since the supply voltage is the same for all branches, it serves as our reference.

Analyzing each branch (similar to series circuits) allows us to determine the total current as the vector sum of individual branch currents and the current through the resistor (I_R).

For analyzing parallel resonance, we have two main options: calculating individual branch currents and summing them, or calculating the admittance of each branch and finding the total current through admittance summation.

Recalling from the series resonance tutorial, resonance occurs when the inductor and capacitor voltages are equal in magnitude but opposite in sign (V_L = -V_C), which happens when their reactances are equal (X_L = X_C).

Lets explore how admittance relates to resonance behavior through the following formula.

Y = G + B_L + B_C

Y = 1 / R + 1 / jωL + jωC

Alternatively, Y can be expressed as:

Y = 1 / R + 1 / 2πfL + 2πfC

Resonance happens when X_L equals X_C, causing the imaginary components of Y to become null. Consequently:

• X_L = X_C
• which gives us:
• 2πfL = 1 / 2πfC
• f² = 1 / (2πfL * 2πfC) = 1 / 4π²LC
• f = √(1 / 4π²LC
• ∴ f_r = 1 / 2π√LC (Hz)
• or, ω_r = 1 / √LC (rads)

Interestingly at resonance, the equation for a parallel RLC circuit becomes identical to that of a series RLC circuit (referring to X_L and X_C values).

This suggests that the placement of the inductor and capacitor (parallel or series), might not significantly affect the resonant frequency itself.

Furthermore during resonance, the parallel LC tank circuit behaves like an open circuit.

This means, minimal current flows through the LC portion, and the total circuit current is primarily determined by the resistor (R).

Meaning, the overall impedance of a parallel resonance circuit at resonance simplifies to just the resistance value (Z = R).

Unlike series resonance, a parallel RLC circuit reaches its maximum impedance at resonance.

This high impedance essentially acts like a high resistance leading to a low current flowing through the entire circuit, and since the impedance behaves purely resistive at resonance, the total current (I) becomes in phase with the supply voltage (V_S).

By adjusting the resistor value (R) but keeping inductance (L) and capacitance (C) constant, we can influence the circuit’s frequency response.

This resonant impedance where Z = R (maximum value) is often referred to as the circuit’s dynamic impedance.

Parallel Resonance Impedance of a Circuit

Intriguingly while the impedance of a parallel RLC circuit peaks at resonance, its admittance (the inverse of impedance) dips to a minimum.

This low admittance restricts the overall current flow in the circuit, which is opposite of series resonance.

Additionally, the resistor in a parallel resonance circuit acts like a damper which reduces the circuit’s bandwidth, making it less selective in filtering specific frequencies.

Voltage and Current Relationship:

Since current remains constant regardless of impedance in a parallel circuit, the voltage waveform across the circuit will mirror the impedance curve. In these circuits the voltage is typically measured across the capacitor.

Resonance and Admittance:

At the resonant frequency (f_r), the circuit’s admittance reaches its minimum and it equals the conductance (G = 1/R).

This occurs because the imaginary component of admittance (susceptance, B) becomes zero.

This zero susceptance is due to the cancellation of inductive susceptance (B_L) and capacitive susceptance (B_C) at resonance, as shown below.

Susceptance during Resonance

Susceptance and Resonance:

The analysis reveals that the inductive susceptance (B_L) exhibits an inverse relationship with frequency, appearing as a hyperbolic curve.

Conversely, the capacitive susceptance (B_C) has a direct proportionality to frequency, hence its representation as a straight line.

The final curve depicts the total susceptance of the parallel resonant circuit plotted against frequency. This curve is derived by subtracting the two individual susceptances.

Dominant Susceptance and Power Factor:

At the resonant frequency (f_r) where the total susceptance curve intersects the horizontal axis, the circuit’s overall susceptance becomes zero.

Below f_r, the inductive susceptance exerts a stronger influence, resulting in a “lagging” power factor.

Above the frequency f_r, the capacitive susceptance becomes dominant, resulting in a “leading” power factor.

Resonance and In-Phase Current:

Significantly, at the resonant frequency (f_r), the current drawn from the power source becomes “in-phase” with the applied voltage.

This is because, the parallel circuit effectively behaves like a pure resistance at resonance, leading to a unity power factor (θ = 0°).

Dynamic Impedance and Resonance:

The impedance of a parallel circuit varies with frequency. This characteristic makes the circuit impedance “dynamic.”

At resonance the current aligns in phase with the voltage because the circuit impedance acts like a resistance.

As observed the impedance of the parallel circuit at resonance is equivalent to the resistance value. This value consequently represents the maximum dynamic impedance (Z_d) of the circuit.

Z_d = L / RC

In a parallel resonant circuit, at a specific frequency, something interesting happens with the current.

• The total opposition to current flow (susceptance) cancels out completely. This means the overall ease with which current flows through the circuit (admittance) reaches its minimum.
• At this special frequency, the only remaining opposition to current is the resistance (conductance is another term for the reciprocal of resistance).
• Because of this cancellation, the total current flowing through the circuit also reaches its minimum value.
• This minimum current occurs because the currents flowing through the inductor and capacitor branches become equal in magnitude (I_L = I_C).
• However, it’s important to remember that these two branch currents are out of sync with each other by 180 degrees.

We know that the total current in a parallel RLC circuit is the combined effect of the currents in each branch. To find the total current for a specific frequency, we need to add these branch currents vectorially, as shown in the following calculations:

• I_R = V / R
• I_L = V / X_L
• = V / 2πfL
• IC = V / XC
• = V * 2πfC

This indicates that I_T is the vector sum of (I_R + I_L + I_C), or

I_T = √[I²_R + (I_L + I_C)²]

At the resonant frequency, the currents through the inductor (IL) and capacitor (IC) become perfectly balanced.

This balancing act cancels out any net reactive current, meaning the total current doesn’t exhibit any lagging or leading behavior. As a result the equation we were using to describe the total current in the circuit simplifies at resonance, as shown below.

I_T = √(I²_R + 0²) = I_R

In a parallel resonant circuit, the current flowing through depends on the voltage divided by the total opposition to current (impedance). However at resonance, something counterintuitive happens.

• The total opposition (impedance Z) actually reaches its highest value, becoming equivalent to just the resistance (R) in the circuit.
• This might seem surprising, but because of this the current flowing through the circuit at this specific frequency dips to its lowest value. This minimum current is simply the voltage divided by the resistance (V/R).

As a consequence, the graph showing how current changes with frequency in a parallel resonant circuit will have a specific shape as illustrated below:

The graph of current versus frequency for a parallel resonance circuit (frequency response curve) reveals some interesting behaviors:

• The graph starts with a high current value.
• As the frequency increases, the current steadily dips and reaches its lowest point at the resonant frequency. Here the current magnitude (I_MIN) is equal to the current that would flow through just the resistor (I_R).
• Beyond the resonant frequency the current magnitude starts to climb again and can reach very high values as the frequency approaches infinity.

It’s important to understand what’s happening with the individual currents in the inductor (L) and capacitor (C) branches.

Even though the total current dips at resonance, the currents flowing through these branches can be much larger than the overall circuit current.

This is because, they are 180 degrees out of phase with each other, meaning they effectively cancel each other out resulting in the low overall current we see in the graph.

Although, a parallel resonant circuit only operates optimally at its resonant frequency, it has a unique ability to filter out or reject specific frequencies. This characteristic earns it the nickname “rejecter circuit.” Here’s why:

• At resonance, the circuits impedance reaches its maximum value.
• This high impedance acts like a barrier, significantly reducing the current that can flow through the circuit at the resonant frequency.
• In essence, the circuit “rejects” current at that particular frequency.

It’s important to note the distinction between resonance in series and parallel circuits.

The calculations and graphical representations used for parallel circuits are similar to those for series circuits. However, the effects of resonance are reversed.

• In a parallel resonant circuit, we experience maximum impedance and minimum current at resonance (opposite to series resonance).
• This is why the term “anti-resonance” is sometimes used to describe resonance in parallel circuits.

The concept of bandwidth in a parallel resonant circuit is identical to that of a series resonant circuit.

We define the upper cut-off frequency (f_upper) and lower cut-off frequency (f_lower) as the half-power points.

At these frequencies, the power dissipated in the circuit is half the power dissipated at the resonant frequency [0.5(I² * R)].

This translates to the same -3dB points where the current has a value equal to 70.7% of its peak resonant value [(0.707 * I)² * R].

Similar to the series circuit, with a constant resonant frequency, a higher quality factor (Q) results in a narrower bandwidth (increased selectivity).

Conversely, a lower Q leads to a wider bandwidth (decreased selectivity) as described by the following equation:

Δf = BW = f_r / Q

or, BW = f_upper – f_lower

Impact of Component Values on Bandwidth and Selectivity

Furthermore, by adjusting the ratio between the inductor (L) and the capacitor (C) or by changing the resistance (R) value we can modify the bandwidth and consequently, the frequency response of the circuit, even while maintaining a constant resonant frequency.

This technique plays a crucial role in tuning circuits for radio and television transmitters and receivers, allowing them to operate on specific frequencies.

Selectivity and the Q-Factor

The selectivity also known as the Q-factor, of a parallel resonant circuit is typically defined as the ratio of the current circulating within the branch containing the inductor and capacitor (combined current) to the total current supplied to the circuit.

This relationship can be expressed mathematically as:

• Quality Factor, Q = R / 2πfL
• Q = 2πfCR
• Q = R√C / L

It’s important to note that the formula used to calculate the Q-factor for a parallel resonant circuit is the inverse of the formula used for a series resonant circuit.

This reflects a fundamental difference in their behavior. In series resonance, the Q-factor indicates the amplification of voltage at resonance, while in parallel resonance, it signifies the amplification of current at resonance.

Diagram Showing Parallel Resonance Circuit Bandwidth

Solving a Parallel Resonance Problem:

We have a circuit with a resistor (65Ω), capacitor (140uF), and inductor (220mH) connected in parallel. It’s hooked up to a power source with a constant 110V output at any frequency. We need to calculate the following:

1. Resonant frequency: The specific frequency where the capacitor and inductor’s effects cancel out.
2. Quality factor (Q): A value indicating how sharp the resonance is.
3. Bandwidth: The range of frequencies where the circuit’s resistance is high.
4. Current at resonance: The amount of current flowing at the resonant frequency.
5. Current magnification: How much the current is amplified at resonance compared to other frequencies.

Solution:

Resonant frequency f_r = 1 / 2π√LC = 1 / 2 * 3.14 * √(220 x 10^-3 * 140 * 10^-6)

f_r = 28.73 Hz

To find Quality factor, we have to first find the Inductive Reactance at resonance, using the following formula:

X_L = 2πf_rL = 2 * 3.14 * 28.73 * 220 * 10^-3

X_L = 39.69 Ω

Therefore, Quality Factor, Q = R / X_L = 65 / 39.69

Q = 1.63

Bandwidth can be calculated as:

Δf = BW = f_r / Q = 28.73 / 1.63

BW = 17.62 Hz

The upper and lower -3dB frequency points, f_L and f_Hcan be calculated as follows:

f_L = f_r – 1/2(BW) = 28.73 – 1 /2 (17.62) = 19.92 Hz

f_H = f_r + 1/2(BW) = 28.73 + 1 /2 (17.62) = 37.54

Next, the Circuit Current at Resonance, I_T , can be calculated as:

I_T = I_R = V / R = 110 / 65 = 1.69 A

Current Magnification can be calculated as:

I_MAG = Q * I_T = 1.63 * 1.69 = 2.75 A

Interestingly, the current supplied by the source (at resonance this is purely resistive) is only 1.69 amps. However the current circulating within the LC circuit itself is larger, at 2.75 amps. This higher circulating current can be verified by calculating the current through the inductor (or capacitor) at resonance.

I_L = V / X_L = V / 2πf_rL = 110 / 2 * 3.14 * 28.73 * 220 * 10^-3

= 2.77 Amps

Conclusion

Understanding Parallel Resonance

This tutorial explored parallel resonance circuits which share similarities with their series counterparts. In a parallel RLC circuit, resonance occurs when the total current aligns (becomes “in-phase”) with the supply voltage. This happens because, the reactive components (inductor and capacitor) cancel each other’s effects.

At resonance, two key things happen:

1. Minimum Admittance: The circuit’s overall ability to conduct current (admittance) reaches its lowest point, matching the conductance (ability of the resistor to conduct current).
2. Minimum Current Draw: The current drawn from the power source also dips to a minimum, determined solely by the value of the parallel resistor.

While the formula for calculating the resonant frequency remains the same as in series circuits there’s a crucial difference. In series RLC circuits the purity of components (ideal vs. real-world) doesn’t affect the resonant frequency calculation. However, for parallel RLC circuits, it does.

Addressing Real-World Imperfections

This tutorial assumed ideal components: a perfect inductor with pure inductance and a perfect capacitor with zero resistance. In reality, inductors have some inherent resistance (denoted by RS) due to the wire used in their construction. This resistance is typically in series with the inductance.

Because of this real-world limitations, the basic equation for calculating the resonant frequency (f_r) of a perfect parallel resonance circuit needs slight modification to account for the series resistance within the inductor.

References:

Parallel Resonance

Parallel and series resonance