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--- different_power_gains [2024/09/06 19:43] – admin
+++ different_power_gains [2025/03/28 14:06] (current) – 2a02:1812:40f:9800:ddf3:8f43:6dcc:25a9
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 //Figure 1: A network Z is connected to a <color #00a2e8>generator</color> at the input, and a <color #ed1c24>load</color> at the output//
+We will define three different types of gain for the network Z. We first start by considering two different definitions of input power.
+====Different types of input power====
+===Actual input power $P_{in}$===
+The actual input power $P_{in}$ (usually simply called 'the input power') is the power that the generator delivers to the network $Z$, connected to a load $Z_L$. It is the power dissipated in $R_{in}$ as seen in the following figure.
+{{:different_power_gains-input-power.png|}}
+//Figure 2: Definition actual input power//
+As a result, the input power is dependent not only on the network $Z$, but also on the value of $Z_L$. It is independent on $Z_G$.
+This implies that a load impedance $Z_L$ must be explicitly specified in order to be able to define the actual input power $P_{in}$.
+===Available input power $P_{AG}$===
+The available input power $P_{AG}$ (often called 'the available input power of the generator') is the maximum input power that can be put into the network $Z$ by a generator. This is achieved for a specific value of the load, i.e., if the load value is chosen so that the input impedance $Z_{in}$ of the network is matched (= equal resistance, opposite reactance) to the generator impedance $Z_G$.
+{{:different_power_gains-available-input-power.png|}}
+//Figure 3: Definition available input power//
+The available input power for the network is only dependent on the value of $Z_G$. It is independent on the network $Z$ or $Z_L$ since both are together matched to the generator impedance.
+This implies that the generator impedance must be specified in order to be able to define the available input power.
+====Different types of output power====
+===The power delivered to the load $P_{L}$===
+It is the power dissipated in the load ZLas seen in the following figure.
+{{:different_power_gains-load-power.png|}}
+//Figure 4: Definition power delivered to the load//
+Note that $P_L$ is dependent on the value of the load and not on the generator impedance. This implies that a load impedance $Z_L$ must be explicitly specified in order to be able to define $P_L$.
+===The maximum available load power $P_{A}$===
+The maximum available load power $P_A$ is the maximum power that can be dissipated into the load. This is realized for a specific value of the load, i.e., if the load value is chosen so that the output impedance $Z_{out}$ of the network is matched (= equal resistance, opposite reactance) to the load impedance $Z_L$. In other words, the maximum available load power is the power the network delivers to a load that is matched to its output impedance.
+{{:different_power_gains-maximum-available-load-power.png|}}
+//Figure 5: Definition maximum available load power//
+The maximum available load power $P_A$ for the network is dependent on the value of $Z_G$ and the network $Z$. It is independent on the load $Z_L$.
+This implies that the generator impedance must be specified in order to be able to define $P_A$.
+====Three different power gains====
+It is now possible to define different power gains, depending on which definition of input and output power is chosen. We will limit ourselves to the three most common power gains.
+===The operating power gain $G_{P}$===
+The operating power gain $G_P$ (often simply called 'the power gain' or 'actual gain') is defined as the ratio of the power dissipated in the load $Z_L$ to the power that the generator delivers to the network $Z$, connected to that load $Z_L$:
+$$ G_P=\frac{P_{L}}{P_{in}}$$
+Since both $P_L$ and $P_{in}$ are independent on the generator impedance $Z_G$, the operating power gain $G_P$ is also independent on $Z_G$. The generator impedance does not need to be specified: we just need to know how much power got to the network, not how it got there.
+The gain depends on the network $Z$ and on the load value $Z_L$. Obviously, $G_P$ can only be defined if a value for the load $Z_L$ is specified.
+In systems where the goal is to transfer energy from a source to a load, the operating power gain is often called 'the power conversion efficiency', or simply 'the efficiency" of the system.
+===Available gain $G_{A}$===
+The available gain $G_A$ is defined as the ratio of the maximum available load power $P_A$ to the available input power $P_{AG}$:
+$$ G_A=\frac{P_{A}}{P_{AG}}$$
+Since both $P_A$ and $P_{AG}$ are independent on the load impedance $Z_L$, the available gain $G_A$ is also independent on $Z_L$. The gain depends on the network $Z$ and on the generator impedance value $Z_G$.
+Obviously, $G_A$ can only be defined if a value for the generator impedance $Z_G$ is specified.
+===Transducer gain $G_{T}$===
+The transducer gain $G_T$ is defined as the ratio of the power dissipated in the load $Z_L$ to the available input power $P_{AG}$ of the generator:
+$$ G_T=\frac{P_{L}}{P_{AG}}$$
+It is dependent on both the load impedance $Z_L$ and the generator impedance $Z_G$. Both have to be specified in order to be able to define the transducer gain $G_T$.
+===Overview output power within the gains===
+{{:different_power_gains-overview-output-powers.png|}}
+===Overview input power within the gains===
+{{:different_power_gains-overview-input-powers.png|}}
+====Simulation example====
+As example, let us consider a very simple, purely resistive network, consisting of three resistors:
+{{:different_power_gains-example-network.png|}}
+//Figure 6: Simple, purely resistive, example network//
+We simulate the three gains in LT Spice with a DC voltage source of 12 V, as function of the generator resistance and the load resistance.
+{{:different_power_gains-spice-simulation.png|}}
+//Figure 7: Schematics in LT Spice for the three gains, for the example network, with variable load and/or generator resistance {R}.//
+We first simulate the operating power gain $G_P$ as function of the load resistance. This gain is independent on the generator resistance. In the simulation, a generator resistance of 8.02773 ohm was chosen (see further), but this value doesn't matter for the simulation of the operating power gain $G_P$.
+The graph below (green) shows that a maximum of $G_P$ = 10.9% is reached for a load of 14.5 ohm.
+{{:different_power_gains-simulation-graph.png|}}
+//Figure 8: Simulation results of the gains as function of varying generator or load resistance (logarithmic axis) for the given example.//
+Next, the available gain $G_A$ is simulated as function of varying generator resistance (this gains is independent on the value of the load). We find a maximum of 10.9% at a generator resistance of 8.0 ohm.
+Finally, the transducer gain $G_T$ is simulated, first for varying generator resistance, and next for varying load. The generator/load resistance in each non-varying case is chosen to be the optimal value for the other gains. The graphs are identical to the other gains ([[explanation]]). A maximum of 10.9% is found for the optimal generator and load resistance of 8.0 and 14.5 ohm.
+<color #808080>**Some background references**</color>
+  * <color #808080>Egan,W.F. Practical RF System Design; JohnWiley & Sons: Hoboken, NJ, USA, 2003; pp. 313–315. </color>
+  * <color #808080>Mastri, F.; Mongiardo, M.; Monti, G.; Dionigi, M.; Tarricone, L. Gain Expressions for Resonant Inductive Wireless Power Transfer
+Links with One Relay Element. Wireless Power Transfer 2018, 5, 27–41. </color>
+  * <color #808080>Ben Minnaert, Giuseppina Monti, Alessandra Costanzo and Mauro Mongiardo, [[https://www.mdpi.com/2079-9292/10/6/723/pdf|Gain Expressions for Capacitive Wireless Power Transfer with One Electric Field Repeater]]. Electronics 2021, 10, 723.</color>