What is the impedance of a parallel resonant LCR circuit?
2 Answers
In a parallel resonant LCR (Inductor-Capacitor-Resistor) circuit, the impedance is a key concept that determines how the circuit responds to different frequencies of an alternating current (AC) signal. Let's break down how to find the impedance of this circuit.
### 1. Components of the Circuit
A parallel resonant LCR circuit consists of:
- **Inductor (L)** with inductance \( L \)
- **Capacitor (C)** with capacitance \( C \)
- **Resistor (R)** with resistance \( R \)
### 2. Impedance of Each Component
The impedance of each component in an AC circuit is given by:
- **Inductor (L):** The impedance of an inductor is \( Z_L = j\omega L \), where \( \omega \) is the angular frequency of the AC signal (\( \omega = 2\pi f \), with \( f \) being the frequency).
- **Capacitor (C):** The impedance of a capacitor is \( Z_C = \frac{1}{j\omega C} \).
- **Resistor (R):** The impedance of a resistor is simply \( Z_R = R \), as it doesn't depend on the frequency.
### 3. Parallel Impedance Calculation
In a parallel resonant circuit, the total impedance \( Z_{total} \) is found using the formula for parallel impedances:
\[ \frac{1}{Z_{total}} = \frac{1}{Z_L} + \frac{1}{Z_C} + \frac{1}{Z_R} \]
Let's plug in the expressions for \( Z_L \), \( Z_C \), and \( Z_R \):
\[ \frac{1}{Z_{total}} = \frac{1}{j\omega L} + \frac{j\omega C}{1} + \frac{1}{R} \]
Combining these terms:
\[ \frac{1}{Z_{total}} = \frac{1}{j\omega L} + j\omega C + \frac{1}{R} \]
To simplify this, you might need to find a common denominator and combine the terms. Let's do it step by step:
1. **Combine Inductive and Capacitive Impedances:**
\[
\frac{1}{Z_{LC}} = \frac{1}{j\omega L} + j\omega C = \frac{-j\omega C + \frac{1}{j\omega L}}{j\omega L \cdot \frac{1}{j\omega C}} = \frac{-j\omega C + \frac{1}{j\omega L}}{1}
\]
Simplify:
\[
\frac{1}{Z_{LC}} = \frac{-j\omega C \cdot j\omega L + 1}{j\omega L}
\]
\[
\frac{1}{Z_{LC}} = \frac{1 - \omega^2 LC}{j\omega L}
\]
2. **Combine with Resistor Impedance:**
\[
\frac{1}{Z_{total}} = \frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}
\]
Finding \( Z_{total} \) from this:
\[
Z_{total} = \frac{j\omega L}{\frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}}
\]
### 4. Resonance Condition
At resonance, the reactances of the inductor and capacitor cancel each other out, meaning \( \omega^2 LC = 1 \). At resonance:
\[ \frac{1}{Z_{total}} = \frac{1}{R} \]
So, the impedance at resonance simplifies to just the resistance \( R \):
\[ Z_{total} = R \]
### Summary
- **At resonance**, the impedance of the parallel LCR circuit is simply the resistance \( R \).
- **Away from resonance**, the impedance is more complex and depends on the frequency of the AC signal, the inductance \( L \), the capacitance \( C \), and the resistance \( R \). The exact impedance can be computed using the formula:
\[
Z_{total} = \frac{j\omega L}{\frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}}
\]
Understanding this impedance behavior is crucial for tuning and designing circuits for specific frequencies and applications.
### 1. Components of the Circuit
A parallel resonant LCR circuit consists of:
- **Inductor (L)** with inductance \( L \)
- **Capacitor (C)** with capacitance \( C \)
- **Resistor (R)** with resistance \( R \)
### 2. Impedance of Each Component
The impedance of each component in an AC circuit is given by:
- **Inductor (L):** The impedance of an inductor is \( Z_L = j\omega L \), where \( \omega \) is the angular frequency of the AC signal (\( \omega = 2\pi f \), with \( f \) being the frequency).
- **Capacitor (C):** The impedance of a capacitor is \( Z_C = \frac{1}{j\omega C} \).
- **Resistor (R):** The impedance of a resistor is simply \( Z_R = R \), as it doesn't depend on the frequency.
### 3. Parallel Impedance Calculation
In a parallel resonant circuit, the total impedance \( Z_{total} \) is found using the formula for parallel impedances:
\[ \frac{1}{Z_{total}} = \frac{1}{Z_L} + \frac{1}{Z_C} + \frac{1}{Z_R} \]
Let's plug in the expressions for \( Z_L \), \( Z_C \), and \( Z_R \):
\[ \frac{1}{Z_{total}} = \frac{1}{j\omega L} + \frac{j\omega C}{1} + \frac{1}{R} \]
Combining these terms:
\[ \frac{1}{Z_{total}} = \frac{1}{j\omega L} + j\omega C + \frac{1}{R} \]
To simplify this, you might need to find a common denominator and combine the terms. Let's do it step by step:
1. **Combine Inductive and Capacitive Impedances:**
\[
\frac{1}{Z_{LC}} = \frac{1}{j\omega L} + j\omega C = \frac{-j\omega C + \frac{1}{j\omega L}}{j\omega L \cdot \frac{1}{j\omega C}} = \frac{-j\omega C + \frac{1}{j\omega L}}{1}
\]
Simplify:
\[
\frac{1}{Z_{LC}} = \frac{-j\omega C \cdot j\omega L + 1}{j\omega L}
\]
\[
\frac{1}{Z_{LC}} = \frac{1 - \omega^2 LC}{j\omega L}
\]
2. **Combine with Resistor Impedance:**
\[
\frac{1}{Z_{total}} = \frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}
\]
Finding \( Z_{total} \) from this:
\[
Z_{total} = \frac{j\omega L}{\frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}}
\]
### 4. Resonance Condition
At resonance, the reactances of the inductor and capacitor cancel each other out, meaning \( \omega^2 LC = 1 \). At resonance:
\[ \frac{1}{Z_{total}} = \frac{1}{R} \]
So, the impedance at resonance simplifies to just the resistance \( R \):
\[ Z_{total} = R \]
### Summary
- **At resonance**, the impedance of the parallel LCR circuit is simply the resistance \( R \).
- **Away from resonance**, the impedance is more complex and depends on the frequency of the AC signal, the inductance \( L \), the capacitance \( C \), and the resistance \( R \). The exact impedance can be computed using the formula:
\[
Z_{total} = \frac{j\omega L}{\frac{1 - \omega^2 LC}{j\omega L} + \frac{1}{R}}
\]
Understanding this impedance behavior is crucial for tuning and designing circuits for specific frequencies and applications.
In a parallel resonant LCR circuit (where L is the inductance, C is the capacitance, and R is the resistance), the impedance is given by:
\[ Z_{parallel} = \frac{R}{1 + jQ} \]
where \( Q \) is the quality factor of the circuit and is defined as:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]
At resonance, the impedance of a parallel LCR circuit is ideally infinite. This is because at resonance, the reactive components of the impedance (the inductive reactance \( X_L \) and the capacitive reactance \( X_C \)) cancel each other out, leaving only the resistance \( R \) in the impedance calculation.
\[ Z_{parallel} = \frac{R}{1 + jQ} \]
where \( Q \) is the quality factor of the circuit and is defined as:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]
At resonance, the impedance of a parallel LCR circuit is ideally infinite. This is because at resonance, the reactive components of the impedance (the inductive reactance \( X_L \) and the capacitive reactance \( X_C \)) cancel each other out, leaving only the resistance \( R \) in the impedance calculation.