What is the h parameter in two-port network?
2 Answers
In the context of electrical engineering, particularly in the analysis and design of two-port networks, the \( h \) parameter, or hybrid parameter, is one of several ways to describe the behavior of a two-port network.
### What is a Two-Port Network?
A two-port network is a linear electrical circuit with two input terminals (ports) and two output terminals (ports). The purpose of using a two-port network model is to analyze and design complex circuits by breaking them down into simpler, more manageable parts. Each port of the network can be thought of as having a pair of electrical connections: one for input and one for output.
### The \( h \) Parameters
The \( h \) parameters, also known as hybrid parameters, are a set of four parameters used to describe the relationship between the voltages and currents at the ports of a two-port network. They are particularly useful in describing the behavior of transistor circuits and other components where the network is neither purely resistive nor purely reactive. The \( h \) parameters are defined as follows:
1. **\( h_{11} \) (Input Impedance Parameter):** This parameter represents the input impedance of the network when the output port is open (i.e., no current is flowing through the output port). Mathematically, it is defined as:
\[
h_{11} = \frac{V_1}{I_1} \Bigg|_{V_2 = 0}
\]
where \( V_1 \) is the voltage at the input port, and \( I_1 \) is the current into the input port.
2. **\( h_{12} \) (Reverse Voltage Transfer Parameter):** This parameter represents the change in input voltage \( V_1 \) caused by a unit change in output voltage \( V_2 \), with the input current \( I_1 \) held constant. It is given by:
\[
h_{12} = -\frac{V_1}{V_2} \Bigg|_{I_1 = 0}
\]
Here, \( V_2 \) is the voltage at the output port, and \( V_1 \) is the voltage at the input port.
3. **\( h_{21} \) (Forward Current Gain Parameter):** This parameter measures the change in output current \( I_2 \) due to a unit change in input current \( I_1 \), with the output voltage \( V_2 \) held constant. It is expressed as:
\[
h_{21} = \frac{I_2}{I_1} \Bigg|_{V_2 = 0}
\]
Here, \( I_2 \) is the current through the output port, and \( I_1 \) is the current into the input port.
4. **\( h_{22} \) (Output Admittance Parameter):** This parameter represents the output admittance (the inverse of impedance) when the input port is open. It is given by:
\[
h_{22} = \frac{I_2}{V_2} \Bigg|_{I_1 = 0}
\]
where \( V_2 \) is the voltage at the output port, and \( I_2 \) is the current through the output port.
### How to Use \( h \) Parameters
The \( h \) parameters are used to model and analyze the performance of two-port networks by providing a set of linear equations that relate the input and output voltages and currents. The relationship between the voltages and currents at the ports of the network can be expressed in matrix form:
\[
\begin{bmatrix}
V_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
h_{11} & h_{12} \\
h_{21} & h_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
V_2
\end{bmatrix}
\]
This matrix equation is useful in various applications, such as analyzing transistor amplifiers and other complex electronic circuits. By knowing the \( h \) parameters, you can predict how the network will respond to different input signals and configurations.
In summary, the \( h \) parameters are a set of hybrid parameters used to describe the electrical behavior of a two-port network, offering insights into its impedance, gain, and transfer characteristics.
### What is a Two-Port Network?
A two-port network is a linear electrical circuit with two input terminals (ports) and two output terminals (ports). The purpose of using a two-port network model is to analyze and design complex circuits by breaking them down into simpler, more manageable parts. Each port of the network can be thought of as having a pair of electrical connections: one for input and one for output.
### The \( h \) Parameters
The \( h \) parameters, also known as hybrid parameters, are a set of four parameters used to describe the relationship between the voltages and currents at the ports of a two-port network. They are particularly useful in describing the behavior of transistor circuits and other components where the network is neither purely resistive nor purely reactive. The \( h \) parameters are defined as follows:
1. **\( h_{11} \) (Input Impedance Parameter):** This parameter represents the input impedance of the network when the output port is open (i.e., no current is flowing through the output port). Mathematically, it is defined as:
\[
h_{11} = \frac{V_1}{I_1} \Bigg|_{V_2 = 0}
\]
where \( V_1 \) is the voltage at the input port, and \( I_1 \) is the current into the input port.
2. **\( h_{12} \) (Reverse Voltage Transfer Parameter):** This parameter represents the change in input voltage \( V_1 \) caused by a unit change in output voltage \( V_2 \), with the input current \( I_1 \) held constant. It is given by:
\[
h_{12} = -\frac{V_1}{V_2} \Bigg|_{I_1 = 0}
\]
Here, \( V_2 \) is the voltage at the output port, and \( V_1 \) is the voltage at the input port.
3. **\( h_{21} \) (Forward Current Gain Parameter):** This parameter measures the change in output current \( I_2 \) due to a unit change in input current \( I_1 \), with the output voltage \( V_2 \) held constant. It is expressed as:
\[
h_{21} = \frac{I_2}{I_1} \Bigg|_{V_2 = 0}
\]
Here, \( I_2 \) is the current through the output port, and \( I_1 \) is the current into the input port.
4. **\( h_{22} \) (Output Admittance Parameter):** This parameter represents the output admittance (the inverse of impedance) when the input port is open. It is given by:
\[
h_{22} = \frac{I_2}{V_2} \Bigg|_{I_1 = 0}
\]
where \( V_2 \) is the voltage at the output port, and \( I_2 \) is the current through the output port.
### How to Use \( h \) Parameters
The \( h \) parameters are used to model and analyze the performance of two-port networks by providing a set of linear equations that relate the input and output voltages and currents. The relationship between the voltages and currents at the ports of the network can be expressed in matrix form:
\[
\begin{bmatrix}
V_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
h_{11} & h_{12} \\
h_{21} & h_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
V_2
\end{bmatrix}
\]
This matrix equation is useful in various applications, such as analyzing transistor amplifiers and other complex electronic circuits. By knowing the \( h \) parameters, you can predict how the network will respond to different input signals and configurations.
In summary, the \( h \) parameters are a set of hybrid parameters used to describe the electrical behavior of a two-port network, offering insights into its impedance, gain, and transfer characteristics.
The **h-parameters**, or **hybrid parameters**, are one of the several ways to model and describe a **two-port network**, which is a system with two sets of terminals: input and output. These parameters express the relationship between the voltage and current at the input and output terminals of the network. Two-port networks are widely used in electrical engineering to analyze components like amplifiers, filters, and transmission lines.
### Why "h-parameters"?
The term **hybrid** comes from the fact that the h-parameters mix both voltage and current variables in a specific way. Unlike other parameter sets (like Z-parameters or Y-parameters), which deal solely with impedances or admittances, the h-parameters combine them in a "hybrid" form. Specifically, the input voltage and output current are chosen as dependent variables, making this method particularly useful for transistor modeling and circuits involving amplifiers.
### Defining the h-parameters
The h-parameters describe the relationship between the input voltage, input current, output voltage, and output current through the following two linear equations:
1. **Input equation:**
\[
V_1 = h_{11}I_1 + h_{12}V_2
\]
2. **Output equation:**
\[
I_2 = h_{21}I_1 + h_{22}V_2
\]
Where:
- \(V_1\) = input voltage
- \(I_1\) = input current
- \(V_2\) = output voltage
- \(I_2\) = output current
- \(h_{11}\), \(h_{12}\), \(h_{21}\), \(h_{22}\) are the four h-parameters
### The Meaning of Each h-parameter
Each h-parameter has a physical interpretation and a unit:
1. **\(h_{11}\) (Input impedance):**
\[
h_{11} = \left( \frac{V_1}{I_1} \right)_{V_2=0}
\]
- This represents the **input impedance** of the two-port network when the output is short-circuited (\(V_2 = 0\)).
- Units: **Ohms** (\(\Omega\)).
2. **\(h_{12}\) (Reverse voltage gain):**
\[
h_{12} = \left( \frac{V_1}{V_2} \right)_{I_1=0}
\]
- This represents the **reverse voltage gain**, showing how much of the output voltage \(V_2\) appears at the input when there is no input current (\(I_1 = 0\)).
- It is **dimensionless** because it is a ratio of two voltages.
3. **\(h_{21}\) (Forward current gain):**
\[
h_{21} = \left( \frac{I_2}{I_1} \right)_{V_2=0}
\]
- This represents the **forward current gain** when the output is short-circuited (\(V_2 = 0\)).
- It is also **dimensionless** because it is a ratio of two currents.
- This parameter is particularly important in transistor amplifiers, as it shows the current gain from input to output.
4. **\(h_{22}\) (Output admittance):**
\[
h_{22} = \left( \frac{I_2}{V_2} \right)_{I_1=0}
\]
- This represents the **output admittance** when there is no input current (\(I_1 = 0\)).
- It measures how much the output current changes with respect to the output voltage under open-circuit input conditions.
- Units: **Siemens** (S), which is the unit of admittance (reciprocal of impedance).
### Matrix Representation of h-parameters
For convenience, the two-port network using h-parameters can also be represented in matrix form:
\[
\begin{bmatrix}
V_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
h_{11} & h_{12} \\
h_{21} & h_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
V_2
\end{bmatrix}
\]
This matrix form is useful for performing calculations in circuit analysis and understanding how changes in input current and output voltage affect the system.
### Applications of h-parameters
1. **Transistor modeling**: H-parameters are commonly used to model **transistors**, especially **BJT (Bipolar Junction Transistor)** amplifiers, where input impedance, output admittance, and current gains are important.
2. **Amplifier design**: They help in characterizing and designing amplifier circuits, making it easier to determine how an amplifier responds to changes in input signal and how much amplification it provides.
3. **Hybrid-pi model**: The small-signal model of BJTs (used in AC analysis) also uses h-parameters to represent its performance.
### Advantages of h-parameters
- **Ease of measurement**: They can be measured easily using common test equipment.
- **Flexibility**: They are useful for analyzing both low and high-frequency circuits.
- **Practical for transistors**: Especially useful for transistor modeling and amplifier circuits where currents and voltages are the main factors of interest.
### Disadvantages of h-parameters
- **Non-reciprocal systems**: H-parameters are not as suitable for networks that are **reciprocal** (where input and output can be reversed, like transmission lines), where Z- or Y-parameters might be more appropriate.
- **Limited for passive components**: Other parameters such as Z-parameters are often better suited for purely passive networks (like RLC circuits).
### Conclusion
The **h-parameter** model is a powerful and flexible tool in the analysis of two-port networks, especially for active components like transistors. By breaking down the relationships between input/output currents and voltages into hybrid parameters, this approach simplifies the design and analysis of amplifiers and other circuit elements used in electrical engineering.
### Why "h-parameters"?
The term **hybrid** comes from the fact that the h-parameters mix both voltage and current variables in a specific way. Unlike other parameter sets (like Z-parameters or Y-parameters), which deal solely with impedances or admittances, the h-parameters combine them in a "hybrid" form. Specifically, the input voltage and output current are chosen as dependent variables, making this method particularly useful for transistor modeling and circuits involving amplifiers.
### Defining the h-parameters
The h-parameters describe the relationship between the input voltage, input current, output voltage, and output current through the following two linear equations:
1. **Input equation:**
\[
V_1 = h_{11}I_1 + h_{12}V_2
\]
2. **Output equation:**
\[
I_2 = h_{21}I_1 + h_{22}V_2
\]
Where:
- \(V_1\) = input voltage
- \(I_1\) = input current
- \(V_2\) = output voltage
- \(I_2\) = output current
- \(h_{11}\), \(h_{12}\), \(h_{21}\), \(h_{22}\) are the four h-parameters
### The Meaning of Each h-parameter
Each h-parameter has a physical interpretation and a unit:
1. **\(h_{11}\) (Input impedance):**
\[
h_{11} = \left( \frac{V_1}{I_1} \right)_{V_2=0}
\]
- This represents the **input impedance** of the two-port network when the output is short-circuited (\(V_2 = 0\)).
- Units: **Ohms** (\(\Omega\)).
2. **\(h_{12}\) (Reverse voltage gain):**
\[
h_{12} = \left( \frac{V_1}{V_2} \right)_{I_1=0}
\]
- This represents the **reverse voltage gain**, showing how much of the output voltage \(V_2\) appears at the input when there is no input current (\(I_1 = 0\)).
- It is **dimensionless** because it is a ratio of two voltages.
3. **\(h_{21}\) (Forward current gain):**
\[
h_{21} = \left( \frac{I_2}{I_1} \right)_{V_2=0}
\]
- This represents the **forward current gain** when the output is short-circuited (\(V_2 = 0\)).
- It is also **dimensionless** because it is a ratio of two currents.
- This parameter is particularly important in transistor amplifiers, as it shows the current gain from input to output.
4. **\(h_{22}\) (Output admittance):**
\[
h_{22} = \left( \frac{I_2}{V_2} \right)_{I_1=0}
\]
- This represents the **output admittance** when there is no input current (\(I_1 = 0\)).
- It measures how much the output current changes with respect to the output voltage under open-circuit input conditions.
- Units: **Siemens** (S), which is the unit of admittance (reciprocal of impedance).
### Matrix Representation of h-parameters
For convenience, the two-port network using h-parameters can also be represented in matrix form:
\[
\begin{bmatrix}
V_1 \\
I_2
\end{bmatrix}
=
\begin{bmatrix}
h_{11} & h_{12} \\
h_{21} & h_{22}
\end{bmatrix}
\begin{bmatrix}
I_1 \\
V_2
\end{bmatrix}
\]
This matrix form is useful for performing calculations in circuit analysis and understanding how changes in input current and output voltage affect the system.
### Applications of h-parameters
1. **Transistor modeling**: H-parameters are commonly used to model **transistors**, especially **BJT (Bipolar Junction Transistor)** amplifiers, where input impedance, output admittance, and current gains are important.
2. **Amplifier design**: They help in characterizing and designing amplifier circuits, making it easier to determine how an amplifier responds to changes in input signal and how much amplification it provides.
3. **Hybrid-pi model**: The small-signal model of BJTs (used in AC analysis) also uses h-parameters to represent its performance.
### Advantages of h-parameters
- **Ease of measurement**: They can be measured easily using common test equipment.
- **Flexibility**: They are useful for analyzing both low and high-frequency circuits.
- **Practical for transistors**: Especially useful for transistor modeling and amplifier circuits where currents and voltages are the main factors of interest.
### Disadvantages of h-parameters
- **Non-reciprocal systems**: H-parameters are not as suitable for networks that are **reciprocal** (where input and output can be reversed, like transmission lines), where Z- or Y-parameters might be more appropriate.
- **Limited for passive components**: Other parameters such as Z-parameters are often better suited for purely passive networks (like RLC circuits).
### Conclusion
The **h-parameter** model is a powerful and flexible tool in the analysis of two-port networks, especially for active components like transistors. By breaking down the relationships between input/output currents and voltages into hybrid parameters, this approach simplifies the design and analysis of amplifiers and other circuit elements used in electrical engineering.