Gain, Available Gain and Transducer Gain at 10 GHz

Compute the operating power gain \(G_P\), available power gain \(G_A\), and transducer power gain \(G_T\).

The analysis of microwave amplifiers often relies on scattering parameters because they provide a convenient way to characterize high-frequency devices. Once the S-parameters of a transistor or microwave amplifier are known, important gain quantities such as operating power gain, available power gain, and transducer power gain can be evaluated under specified source and load conditions. These gain parameters help engineers understand how effectively power is transferred through the amplifier and how source and load mismatches affect overall performance. In this solved numerical example, the gain characteristics of a microwave amplifier operating at 10 GHz are determined using the given S-parameters together with the specified source and load impedances.

In Calculator

\[ S_{11}=0.84\angle -90^\circ \quad \text{Shift+STR+A} \]

\[ S_{21}=5.1\angle 80^\circ \quad \text{Shift+STR+C} \]

\[ S_{12}=0.3\angle 70^\circ \quad \text{Shift+STR+B} \]

\[ S_{22}=0.62\angle -40^\circ \quad \text{Shift+STR+D} \]

The following calculator settings and functions are useful for the calculation:

\[ \text{Mode + 2} \quad \text{Complex Mode} \]

\[ \text{Conjugate} \quad \Rightarrow \quad \text{Shift + 2} \quad \text{Option 2: conj} \]

\[ \text{Absolute Value} \quad \Rightarrow \quad \text{Shift + hyp} \]

2. Calculation of Source and Load Reflection Coefficients

2.1 Source Reflection Coefficient

The source reflection coefficient is obtained from the source impedance and reference impedance using

\[ \Gamma_S=\frac{Z_S-Z_0}{Z_S+Z_0} \]

\[ \Gamma_S=\frac{20-50}{20+50} \]

\[ \Gamma_S=-0.429 \quad \text{Shift+STR+E} \] 

2.2 Load Reflection Coefficient

Similarly, the load reflection coefficient is

\[ \Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0} \]

\[ \Gamma_L=\frac{30-50}{30+50} \]

\[ \Gamma_L=-0.25 \quad \text{Shift+STR+F} \]

3. Calculation of Input Reflection Coefficient

The input reflection coefficient of the amplifier is calculated from

\[ \Gamma_{in} = S_{11} + \frac{S_{12}S_{21}\Gamma_L} {1-S_{22}\Gamma_L} \]

Substituting the given S-parameters and the calculated value of \(\Gamma_L\),

\[ \Gamma_{in} = 0.773\angle -94.8^\circ \quad \text{Shift+STR+X} \]

This is the effective reflection coefficient seen looking into the input port of the amplifier when terminated by the specified load. 

\[ = A + \frac{B \cdot C \cdot F}{1 - D \cdot F} \] 

4. Calculation of Output Reflection Coefficient

The output reflection coefficient is given by

\[ \Gamma_{out} = S_{22} + \frac{S_{12}S_{21}\Gamma_S} {1-S_{11}\Gamma_S} \]

Substituting the given values,

\[ \Gamma_{out} = 0.141\angle -65.6^\circ \quad \text{Shift+STR+Y} \]

\[ = D + \frac{C \cdot B \cdot E}{1 - A \cdot E} \]

This represents the reflection coefficient seen looking toward the output port when connected to the specified source.

In Calculator

\[ G_P = \frac{\text{abs}(c)^2 \left(1 - \text{abs}(f)^2\right)}{\left(1 - \text{abs}(x)^2\right) \vert{}1 - d \cdot f\vert{}^2} \]

\[ G_A = \frac{\text{abs}(c)^2 \left(1 - \text{abs}(e)^2\right)}{\vert{}1 - a \cdot e\vert{}^2 \left(1 - \text{abs}(y)^2\right)} \]

\[ G_T = \frac{\text{abs}(c)^2 \left(1 - \text{abs}(e)^2\right) \left(1 - \text{abs}(f)^2\right)}{\vert{}1 - e \cdot x\vert{}^2 \vert{}1 - d \cdot f\vert{}^2} \]

5. Calculation of Operating Power Gain

The operating power gain is defined as the ratio of power delivered to the load to the power delivered to the input of the two-port network. The expression is

\[ G_P = \frac{|S_{21}|^2(1-|\Gamma_L|^2)} {(1-|\Gamma_{in}|^2)\,|1-S_{22}\Gamma_L|^2} \]

Substituting the known values,

\[ G_P = \frac{(5.1)^2(1-0.25^2)} {(1-0.773^2)\,|1-S_{22}(-0.25)|^2} \]

\[ G_P \approx 50.8 \]

Therefore, the operating power gain of the amplifier is approximately 50.8.

6. Calculation of Available Power Gain

The available power gain is defined as the ratio of the maximum power available from the network to the maximum power available from the source. The available gain formula is

\[ G_A = \frac{|S_{21}|^2(1-|\Gamma_S|^2)} {|1-S_{11}\Gamma_S|^2(1-|\Gamma_{out}|^2)} \]

Substituting the calculated values,

\[ G_A = \frac{(5.1)^2(1-0.429^2)} {|1-S_{11}(-0.429)|^2(1-0.141^2)} \]

\[ G_A \approx 15.4 \]

Hence, the available power gain is approximately 15.4.

7. Calculation of Transducer Power Gain

The transducer power gain is the ratio of power delivered to the load to the power available from the source. It includes the effects of both source and load mismatches and is therefore one of the most useful gain measures in practical amplifier design. The expression is

\[ G_T = \frac{ |S_{21}|^2 (1-|\Gamma_S|^2) (1-|\Gamma_L|^2) } { |1-\Gamma_S\Gamma_{in}|^2 |1-S_{22}\Gamma_L|^2 } \]

Substituting all calculated values,

\[ G_T = \frac{ (5.1)^2 (1-0.429^2) (1-0.25^2) } { |1-\Gamma_S\Gamma_{in}|^2 |1-S_{22}\Gamma_L|^2 } \]

\[ G_T \approx 13.0 \]

Therefore, the transducer power gain of the amplifier is approximately 13.0.

8. Final Results

After calculating the source and load reflection coefficients and evaluating the corresponding input and output reflection coefficients, the three gain parameters of the microwave amplifier operating at 10 GHz are obtained as follows:

\[ G_P \approx 50.8 \]

\[ G_A \approx 15.4 \]

\[ G_T \approx 13.0 \]

These results show that the operating power gain is the highest because it relates the power delivered to the load to the power entering the amplifier. The available power gain is lower because it considers the maximum available power transfer conditions, while the transducer power gain provides the most practical measure of overall amplifier performance by accounting for both source and load mismatches. This example demonstrates how S-parameter analysis can be used to predict microwave amplifier performance accurately at high frequencies.