Source and load

Multiple subsystems (e.g. filters, amplifiers, etc.) can be cascaded to form a multi-stage complex system (a processing pipeline). However, the behavior of each stage will be affected by the next stage as the load. (In quantum theory, we learned that any measurement/observation will inevitably affect the physical process being measured/observed.)

In particular, in electrical circuits, how the next stage as the load affects the previous stage as the source depends on the output impedance $Z_{out}$ of the source and the input impedance $Z_{in}$ of the load.

A signal/energy source can be modeled as either an ideal voltage source $V_0$ in series with an internal impedance $Z_0$ or an ideal current source $I_0$ in parallel with an internal impedance $Z'_0$. It can be shown (see this page) that these two models can be converted to each other by

$$Z_0=Z'_0,\;\;\;\;\mbox{and}\;\;\;\;\;I_0=V_0/Z$$

In the following we consider how a source can deliver maximum voltage or power to a load $Z_L$.

• Maximize voltage delivery

  In some application (e.g., voltage amplification) it is desirable for the load to receive maximum voltage. The source can be modeled as an ideal voltage $V_0$ in series with $Z_0$, as well as the load $Z_L$, i.e., the source and load form a voltage divider and the voltage across the load is
  

  $$V_L=V_0\;\frac{Z_L}{Z_0+Z_L}$$
  
  
  Obviously the larger the load impedance, i.e., the lighter the load, the higher voltage the load receives. In particular,
  
  $$V_L=V_0\;\frac{Z_L}{Z_0+Z_L}\vert _{Z_L\rightarrow \infty} \Longrightarrow V_0$$
  
  

  

• Maximize power delivery

  In some other applications (e.g., power amplification/delivery) it is desirable for the load to receive maximum power (instead of maximum voltage):
  

  $$P_L=I^2 Z_L=\frac{V_0^2}{(Z_0+Z_L)^2} Z_L$$
  
  
  where $Z_0$ is the internal resistance of the voltage source. To find proper $Z_L$ for maximum power, we let
  
  $$\frac{d}{dZ_L} P_L(Z_L) =V_0^2\frac{(Z_0+Z_L)^2-2Z_L(Z_0+Z_L)}{(Z_0+Z_L)^4} =V_0^2\frac{Z_0-Z_L}{(Z_0+Z_L)^3}=0$$
  
  
  Solving this we get $Z_L=Z_0$, i.e., when the load is equal to the internal resistance of the voltage source, the power it receives is maximal
  
  $$P_L=\frac{V_0^2}{(Z_0+Z_L)^2} Z_L\vert _{Z_L=Z_0}=\frac{V_0^2}{4Z_0}$$
  
  
  In this case, the load current is $I=V_0/2Z_0$.

  The efficiency of the circuit is defined as the ratio of the power delivered to the load $Z_L$ and the power generated by the source $Z_0$:
  

  $$\eta=\frac{P_L}{P_0}=\frac{I^2 Z_L}{I^2 (Z_0+Z_L)} =\frac{Z_L}{Z_0+Z_L}$$
  
  
  We see that when $Z_L=Z_0$, the load receives maximal power but the efficiency is only $50\%$. The efficiency will approach 1 when $Z_L \gg Z_0$, but in this case the power received by the load is not maximal. In power network, it is more desirable to have high efficiency to avoid wasting energy than delivering maximal power. But in communication network and other applications with small power, efficiency can be sacrificed to maximize the load power. For example, in an audio system, it is important for the speaker's impedance to match the output impedance of the power amplification circuit, so that the speaker can receive maximum power.