What is the transfer function of the RC circuit?
2 Answers
The transfer function of a circuit describes the relationship between the input and output signals in the frequency domain. For a simple resistor-capacitor (RC) circuit, let's consider a basic setup where a resistor (R) is in series with a capacitor (C), and we want to analyze the transfer function from the input voltage (Vin) to the output voltage (Vout) across the capacitor.
### Step-by-Step Derivation
1. **Circuit Description**: In a series RC circuit, the input voltage \( V_{in} \) is applied across the series combination of a resistor \( R \) and a capacitor \( C \). The output voltage \( V_{out} \) is taken across the capacitor.
2. **Impedance Representation**:
- The impedance of the resistor is \( Z_R = R \).
- The impedance of the capacitor in the frequency domain (using the Laplace transform) is given by:
\[
Z_C = \frac{1}{sC}
\]
where \( s \) is the complex frequency variable \( s = j\omega \), with \( \omega \) being the angular frequency.
3. **Total Impedance**: The total impedance \( Z_{total} \) of the series circuit is:
\[
Z_{total} = Z_R + Z_C = R + \frac{1}{sC}
\]
4. **Voltage Divider Rule**: To find the output voltage across the capacitor, we can use the voltage divider rule:
\[
V_{out} = V_{in} \cdot \frac{Z_C}{Z_{total}}
\]
5. **Substituting Impedances**:
\[
V_{out} = V_{in} \cdot \frac{\frac{1}{sC}}{R + \frac{1}{sC}}
\]
6. **Simplifying**:
First, multiply the numerator and the denominator by \( sC \):
\[
V_{out} = V_{in} \cdot \frac{1}{sRC + 1}
\]
7. **Transfer Function**: The transfer function \( H(s) \), which is defined as the ratio of the output to the input in the Laplace domain, is given by:
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{1}{sRC + 1}
\]
### Summary
The transfer function of a simple RC circuit can be expressed as:
\[
H(s) = \frac{1}{sRC + 1}
\]
This function indicates how the output voltage across the capacitor responds to an input voltage in the frequency domain.
### Interpretation
- **Frequency Response**: The transfer function shows that at low frequencies (where \( s \) approaches 0), \( H(s) \) approaches 1, meaning the output voltage is nearly equal to the input voltage. At high frequencies, \( H(s) \) approaches 0, indicating that the capacitor blocks the signal.
- **Time Constant**: The term \( RC \) is known as the time constant of the circuit. It describes how quickly the circuit responds to changes in input voltage. Specifically, the time constant indicates the time required for the voltage across the capacitor to reach about 63% of its final value after a step change in voltage.
### Conclusion
In essence, the transfer function provides crucial insight into the behavior of the RC circuit across different frequencies, making it a fundamental concept in control systems and signal processing.
### Step-by-Step Derivation
1. **Circuit Description**: In a series RC circuit, the input voltage \( V_{in} \) is applied across the series combination of a resistor \( R \) and a capacitor \( C \). The output voltage \( V_{out} \) is taken across the capacitor.
2. **Impedance Representation**:
- The impedance of the resistor is \( Z_R = R \).
- The impedance of the capacitor in the frequency domain (using the Laplace transform) is given by:
\[
Z_C = \frac{1}{sC}
\]
where \( s \) is the complex frequency variable \( s = j\omega \), with \( \omega \) being the angular frequency.
3. **Total Impedance**: The total impedance \( Z_{total} \) of the series circuit is:
\[
Z_{total} = Z_R + Z_C = R + \frac{1}{sC}
\]
4. **Voltage Divider Rule**: To find the output voltage across the capacitor, we can use the voltage divider rule:
\[
V_{out} = V_{in} \cdot \frac{Z_C}{Z_{total}}
\]
5. **Substituting Impedances**:
\[
V_{out} = V_{in} \cdot \frac{\frac{1}{sC}}{R + \frac{1}{sC}}
\]
6. **Simplifying**:
First, multiply the numerator and the denominator by \( sC \):
\[
V_{out} = V_{in} \cdot \frac{1}{sRC + 1}
\]
7. **Transfer Function**: The transfer function \( H(s) \), which is defined as the ratio of the output to the input in the Laplace domain, is given by:
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{1}{sRC + 1}
\]
### Summary
The transfer function of a simple RC circuit can be expressed as:
\[
H(s) = \frac{1}{sRC + 1}
\]
This function indicates how the output voltage across the capacitor responds to an input voltage in the frequency domain.
### Interpretation
- **Frequency Response**: The transfer function shows that at low frequencies (where \( s \) approaches 0), \( H(s) \) approaches 1, meaning the output voltage is nearly equal to the input voltage. At high frequencies, \( H(s) \) approaches 0, indicating that the capacitor blocks the signal.
- **Time Constant**: The term \( RC \) is known as the time constant of the circuit. It describes how quickly the circuit responds to changes in input voltage. Specifically, the time constant indicates the time required for the voltage across the capacitor to reach about 63% of its final value after a step change in voltage.
### Conclusion
In essence, the transfer function provides crucial insight into the behavior of the RC circuit across different frequencies, making it a fundamental concept in control systems and signal processing.
The transfer function of an RC (resistor-capacitor) circuit describes the relationship between the input and output signals of the circuit in the frequency domain. There are two common configurations for RC circuits: the low-pass filter and the high-pass filter. Here’s how to derive the transfer functions for both:
### 1. RC Low-Pass Filter
**Configuration:**
- **Input** is applied across a series combination of resistor \( R \) and capacitor \( C \).
- **Output** is taken across the capacitor \( C \).
**Transfer Function Derivation:**
1. **Impedances:**
- Resistor impedance: \( Z_R = R \)
- Capacitor impedance: \( Z_C = \frac{1}{j\omega C} \)
2. **Voltage Divider Rule:**
The output voltage \( V_{out} \) is taken across the capacitor, and the input voltage \( V_{in} \) is applied across the series combination of \( R \) and \( C \).
\[
V_{out} = V_{in} \times \frac{Z_C}{R + Z_C}
\]
Substituting \( Z_C = \frac{1}{j\omega C} \):
\[
V_{out} = V_{in} \times \frac{\frac{1}{j\omega C}}{R + \frac{1}{j\omega C}}
\]
3. **Simplify:**
\[
V_{out} = V_{in} \times \frac{1}{j\omega C} \times \frac{1}{R + \frac{1}{j\omega C}}
\]
\[
V_{out} = V_{in} \times \frac{1}{1 + j\omega RC}
\]
4. **Transfer Function:**
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{1}{1 + sRC}
\]
where \( s = j\omega \) (Laplace transform variable).
### 2. RC High-Pass Filter
**Configuration:**
- **Input** is applied across the capacitor \( C \).
- **Output** is taken across the resistor \( R \).
**Transfer Function Derivation:**
1. **Impedances:**
- Resistor impedance: \( Z_R = R \)
- Capacitor impedance: \( Z_C = \frac{1}{j\omega C} \)
2. **Voltage Divider Rule:**
The output voltage \( V_{out} \) is taken across the resistor, and the input voltage \( V_{in} \) is applied across the series combination of \( C \) and \( R \).
\[
V_{out} = V_{in} \times \frac{R}{R + Z_C}
\]
Substituting \( Z_C = \frac{1}{j\omega C} \):
\[
V_{out} = V_{in} \times \frac{R}{R + \frac{1}{j\omega C}}
\]
3. **Simplify:**
\[
V_{out} = V_{in} \times \frac{R}{R + \frac{1}{j\omega C}}
\]
\[
V_{out} = V_{in} \times \frac{j\omega RC}{1 + j\omega RC}
\]
4. **Transfer Function:**
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{sRC}{1 + sRC}
\]
In both cases, \( \omega \) is the angular frequency of the input signal, and \( s = j\omega \) in the Laplace transform domain. These transfer functions provide insight into how the RC circuit behaves in response to different frequencies of input signals, with the low-pass filter attenuating high frequencies and the high-pass filter attenuating low frequencies.
### 1. RC Low-Pass Filter
**Configuration:**
- **Input** is applied across a series combination of resistor \( R \) and capacitor \( C \).
- **Output** is taken across the capacitor \( C \).
**Transfer Function Derivation:**
1. **Impedances:**
- Resistor impedance: \( Z_R = R \)
- Capacitor impedance: \( Z_C = \frac{1}{j\omega C} \)
2. **Voltage Divider Rule:**
The output voltage \( V_{out} \) is taken across the capacitor, and the input voltage \( V_{in} \) is applied across the series combination of \( R \) and \( C \).
\[
V_{out} = V_{in} \times \frac{Z_C}{R + Z_C}
\]
Substituting \( Z_C = \frac{1}{j\omega C} \):
\[
V_{out} = V_{in} \times \frac{\frac{1}{j\omega C}}{R + \frac{1}{j\omega C}}
\]
3. **Simplify:**
\[
V_{out} = V_{in} \times \frac{1}{j\omega C} \times \frac{1}{R + \frac{1}{j\omega C}}
\]
\[
V_{out} = V_{in} \times \frac{1}{1 + j\omega RC}
\]
4. **Transfer Function:**
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{1}{1 + sRC}
\]
where \( s = j\omega \) (Laplace transform variable).
### 2. RC High-Pass Filter
**Configuration:**
- **Input** is applied across the capacitor \( C \).
- **Output** is taken across the resistor \( R \).
**Transfer Function Derivation:**
1. **Impedances:**
- Resistor impedance: \( Z_R = R \)
- Capacitor impedance: \( Z_C = \frac{1}{j\omega C} \)
2. **Voltage Divider Rule:**
The output voltage \( V_{out} \) is taken across the resistor, and the input voltage \( V_{in} \) is applied across the series combination of \( C \) and \( R \).
\[
V_{out} = V_{in} \times \frac{R}{R + Z_C}
\]
Substituting \( Z_C = \frac{1}{j\omega C} \):
\[
V_{out} = V_{in} \times \frac{R}{R + \frac{1}{j\omega C}}
\]
3. **Simplify:**
\[
V_{out} = V_{in} \times \frac{R}{R + \frac{1}{j\omega C}}
\]
\[
V_{out} = V_{in} \times \frac{j\omega RC}{1 + j\omega RC}
\]
4. **Transfer Function:**
\[
H(s) = \frac{V_{out}}{V_{in}} = \frac{sRC}{1 + sRC}
\]
In both cases, \( \omega \) is the angular frequency of the input signal, and \( s = j\omega \) in the Laplace transform domain. These transfer functions provide insight into how the RC circuit behaves in response to different frequencies of input signals, with the low-pass filter attenuating high frequencies and the high-pass filter attenuating low frequencies.