**Example I, Maximize Voltage Delivery**

Sometimes there is the need to concatenate two circuits in series. As some
simple example, the source can be a battery and the load can be just a
resistor, or the source is a signal (waveform) generator that can produce
various waveforms (sinusoids, saw tooth, square wave, etc.) and the load
can be an oscilloscope to display such waveforms. Alternatively, as a more
sophisticated example, two **voltage amplifiers** can be cascaded together
so that the output of the first one, the source, is treated as the input
of the second one, the load, for the purpose of amplifying some very weak
signals. Note that the amplifier circuit is *active* in the sense it
contains a voltage source that depends on the input voltage (across
the input impedance), e.g.,
, with being the voltage
gain.

To maximize the output voltage, it is important to consider both the input and output impedances (resistances for now) of the circuits. The output impedance of the first circuit and the input impedance of the second circuit form a voltage divider. For the second circuit to receive maximum voltage from the first one, we want

- The output impedance of the first circuit to be small, so that little voltage drop across it will be caused even if the second circuit draws a large current due to its small input impedance;
- The input impedance of the second circuit to be large, so that it draws little current from the first circuit, causing small voltage drop across its output impedance.

**Example II, Maximize power delivery**

Sometimes we want to maximize the power delivered from the voltage source
to the load:

where is the internal resistance of the voltage source. For example, the

and get , i.e., when the load is equal to the internal resistance, also called the output resistance, of the voltage source, the power it receives is maximal

In this case, the load current is , and the total power delivered by the voltage source is

i.e., the internal resistance consumes the same amount of power as the load .

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

When and the load receives maximal power, but the efficiency is only . Obviously, in order to improve the efficiency, we can increase so that approach 1 when . But in this case the power received by the load is no longer maximal. For example, if , the efficiency becomes:

but the power received by the load is less than the maximum power:

In some 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.

**Example III, Minimize loss in power transmission line:**

In power transmission network, it is more desirable to have high efficiency to avoid wasting energy than delivering maximal power. The power loss along the power transmission line between the power plants and the consumers should be minimized.

We assume the resistance of the power transmission line is and the total load resistance of the power consumers is . We also assume the voltage on the consumer's side of the power line is .

- Power consumption by the load:

- Power dissipation along the transmission line:

**Summary: ** The circuits in the three examples above are essentially
the same, i.e., they all have a voltage source with an internal
resistance (or ), and a load resistance . However, the
circuit will be optimized differently according to different requirements:

**Maximize voltage delivery (or minimize load effect)**: minimize output resistance , maximize input resistance .**Maximize power delivery ( of total power):**match the output and input resistances**Minimize power dissipation in transmission:**increase output voltage .