Many people are not aware that when two circuits are connected together the input and output resistances play a critical part.

For example, there’s a chance that you have noticed that some headphones sound quieter than others with the same media player. Or ever used a headphone splitter resulting in both headphones being quiet? Let’s have a look why.

Source and Load

We have a random device and we’d like to connect something to it: An iPod to a mixing console for example.

  • The music plays from the iPod – so this will be the source.
  • The music signal is fed into the mixing console – so it will be the load.
Source and Load example demonstration with an ipod and mixing console

Thévenin’s Theorem tells us that all complex circuits consisting of voltage sources, current sources and resistors can be theoretically simplified down to a circuit consisting only:

  • An equivalent voltage source (Vth) and an
  • equivalent resistor (Rth) in series
Thevenin's Equivalent Circuit

Above, there is an example of a Thevenin equivalent circuit. Note there is a load RL connected. The rest of the circuit (Vth and Rth) is the source. Consequently, picture RL as the input stage of the mixing console.

A source’s output resistance is simply it’s Thevenin Equivalent Resistance.

Rth = Rs

This is what we would measure at the terminals when there is no load connected. The question is:

Should the load resistance be larger, smaller or equal to the source to have a good quality signal transmission?

3 options of Rs and RL:

  • Rs > RL
  • Rs = RL
  • Rs < RL

Also, don’t forget that these are only the ratios, the values are important too.
So, there is a difference between:

Rs=10kΩ = RL=100kΩ and
Rs=1kΩ = RL=10kΩ .

Consider the Thevenin Equivalent Circuit above where Vth=3.33V will drop across the combination of Rth and RL. The ratios between Rth and RL determine how much of the 3.33V will drop across each.

So for example if RL >> Rs most of the 3.33V will drop across it. You can calculate this using Ohm’s Law. What does this mean? It means we are transferring most of the voltage from the source to the load.

The Source and Load residences’ impact on current

Sometimes we want to make sure we transfer a good amount of current too.

Again, in Thevenin’s circuit, to have a large current transferred to the load we have to have a large current in the entire circuit (RL is in series with Rs).

How do we have a large current in the circuit? By minimising Rs and RL.

If we think design-wise:

  • sometimes Rs is fixed and we are designing the load
    • iPod is a product and we are designing the input stage of the mixing console
  • sometimes RL is fixed and we are designing the source
    • A speaker is fixed 8Ω and we are designing the output stage of the amplifier
  • Rs >> RL is not efficient. We don’t use this.
    • As most of the power would drop in the source.
  • We use Rs << RL or Rs = RL based on what we are designing.
  • We distinguish between two concepts
    • Maximum Power Transfer Theorem
    • Impedance Bridging

Maximum Power Transfer Theorem

  • For this theory, Rs is fixed and we are choosing the best load.
  • We want the maximum power to be dissipated in the load.
  • The power is maximised if Rs and RL are matched:

Rs = RL

  • Used example at radio transmissions, so there is less signal loss through air, and in systems with very high frequencies (GHz).

Max Efficiency vs Max Power

  • So as the calculation tells us, the best option is to have Rs = RL to have maximum power transfer.
  • Efficiency is how much of the source’s power is transferred. For 100% efficiency:
    • we would have all the voltage dropped across RL and
    • maximise the current in the circuit.
  • This is achieved only if Rs = 0. At least very close to 0 Ω.
  • The Max Power Transfer is only 50% Efficient.
  • Efficiency is defined as the ratio between the power dissipated by the load and the total power coming from the source.
    • The source’s power is the sum of the power dissipated in the source resistor and the load resistor.
    • If the efficiency is 1, it means it is 100%
  • Here is the proof:
Equivalent Circuit for demonstrating efficiency between various input and output resistance
Input and Output Resistance Efficiency Calculation
From the above equation we can deduce the following:
  • Rs >> RL: Efficiency will be low
  • Rs = RL: Efficiency is 0.5 or it other words 50%
  • Rs << RL: Efficiency is close to 100%

Difference between Max Power and Max Efficiency?

We said that we have max efficiency when we chose a very large RL to a fixed Rs. We transfer max voltage. But since the series combination of the two resistors will be high, there will be small current in the circuit. In other words, the load would draw only a small current from Vth.

Vth will be supplying a low current. It is efficient yes, as most of the voltage drops across RL, but the power is low hence the small current.

Impedance Bridging

  • For this, we have the load resistor 10x larger than the source resistor as a rule of thumb:

Rs << RL

  • Used in most DC circuits.
    • Very high frequency (GHz) AC circuits – like transmission lines – use maximum power transfer instead
  • Let’s take audio again for an example:
    • In the image below we have 3 components:
      • iPod – the source
      • Amp – the load (for the iPod)
      • Speaker – the load (for the amp).
Impedance bridging example circuit
  • Audio is essentially changes in voltage.
  • Alternating voltages go up and down in value in respect of time.
    • So voltages can go from +3V to -3V. The frequency and the value depends on the audio signal.
    • Voltage moves current – which is another name for a bunch of electrons. By having positive and negative voltages: electrons move forward and back in the cable.
    • Larger voltages (both negative and positive) draw more current.
    • The voltage that the source can supply is limited as it runs on batteries in our cast (iPod).
  • The amplifier amplifies voltage, not current.
    • The amp is connected to the mains, it can draw the current it needs from there.
  • So we are trying to maximise the voltage from the iPod, not the power.
    • The amp amplifies the voltage that appears across RL.
      • this is the input stage of the amp
    • We maximise the voltage across RL by having a large load resistor and a smaller source resistor.
    • Above in the diagram, the iPod’s voltage source’s voltage (Vth) will be dropped entirely across Rsource + Rload.
      • We want most of the voltage to be dropped across RL
      • The ratio between Rs and RL determine this, not their individual values
      • According to Ohm’s Law, more voltage will be dropped across the larger resistor.
      • By rule of thumb we chose Rload to be 10x the value of Rsource.

Amplifier & Loudspeaker Impedances

What is the difference between impedance and resistance?

In this section we considered DC circuits only where we had a voltage or current source in the circuit with some resistors.

AC circuits on the other hand are a bit more complicated.

As mentioned above, here voltages change from positive to negative with time, thus, move the electrons in the wire backward and forward.

  • Frequency is how fast the voltage fluctuates. Example a 10Hz signal does a cycle 10 times a second.
  • Apart from resistors, we have inductors and capacitors in an AC circuit. Their resistance changes with frequency.
  • So impedance is essentially resistance that changes with frequency.
Input impedance example for a loudspeaker

How Loudspeakers Work

Loudspeakers have two important parts:

  • magnet
  • coil

The amp feeds a signal to the speaker (voltage + current). By Faraday’s Law, current in a wire (coil) will induce a magnetic field. This will become an electromagnet.

This is an AC circuit, thus, by the changing current from positive to negative, the electromagnet’s magnetic field will change too.

The permanent magnet inside the speaker has fixed South and North poles. So it will attract than repel the electromagnet as it changes polarity.

The coil is attached to the cone – and this is how the speaker moves.

Impedance Curve of a Loudspeaker

Above there is a typical impedance curve of a loudspeaker. The loudspeaker’s terminals are connected to the coil so it will have impedance.

A typical loudspeaker input impedance is 4-16Ω. This value is measured around 400Hz.

Note, at lower frequencies such as 40Hz the resistance is much higher. This means the amp needs to supply more power to reproduce this frequency.

The amp needs to output a higher voltage to generate the same current at 40Hz than it supplies at 400Hz.

Output Impedance of an Amp

Speakers impedances are usually fixed. We design the output impedance for the amp.

Speakers are not connected to the wall socket, so can’t draw current from there.

Also, Faraday’s Law is about current, not voltage. So, in the case of speakers, we are trying to transfer high current, but we chose the source impedance this time.

High current transfer is achieved if we have very low output impedance from the amp. Usually: Rs < 0.5 Ω.

There is only so much current that the amp can supply, so be careful not to connect speakers that the amp is not designed for otherwise the amp will burn out.

Car amps are designed for speakers with low input impedances () as there is a limited 12V supply. Consequently, we want 12V to be able to draw higher currents to have higher volume.

High vs Low Impedance Headphones

Headphones work the same way as loudspeakers but on a smaller scale. Consider Rs is fixed. We chose the load impedance (headphone).

The thickness of the coil wire determines the headphone impedance.

  • Thicker wire: low impedance, less turns
  • Thiner wire: high impedance, more turns

Thinner wires can be wound up tighter. This results in having less copper volume for the same current that we would get with thicker wires and less turns. This means voice coils with thinner wires are lighter, thus, respond better – less distortion – better sound quality.

As portable devices can’t supply high voltages, headphones used for portable devices have low impedances (20Ω). The smaller the impedance, the smaller the voltage drop across the load.

This means, lower voltages can draw the current needed. Also, low impedance headphones sound louder.

iPhone Jack Example

The iPhone jack had about 6 Ω output impedance. The new lightning to jack adapter has less than 1 Ω . If you had a headphone with 20Ω input impedance, you would notice higher volumes and increased bass. Why? Because there is more current in the circuit as the total impedance is lower. Low end needs more power remember?

With more current there is more power.

optional reading: Success in Electronics book by Tom Duncan