Before we start, we need to know what we mean by "circuit" and  "impedance."
A circuit is almost any component and its surrounding wiring. A circuit may be as simple as a globe, length of wire and a switch. It may be an oscillator circuit consisting of 15 components, or it may be the power supply for a 15 layer PC board.
A circuit is not just a globe or resistor. We call these components.
A circuit must include the wiring as this can create problems such as picking up spikes or reduce the current-flow. And when the circuit contains more than one component plus trackwork on a PC board, the situation becomes very complex.
The following figure shows three typical circuits:

Fig 1: Some typical circuits.

WHAT IS IMPEDANCE?
IMPEDANCE is very similar to RESISTANCE. They both have the same unit: OHM.
A low impedance circuit is very similar to saying a low resistance circuit however we use the term impedance because the actual resistance of the circuit may change according to the frequency at which it is operating. The term impedance takes this into account.
If the circuit only operates at DC (this is a steady state such as a globe in a torch and is equal to a frequency of 0Hz - zero cycles per second) we can use the term resistance. But most circuits have waveforms and sometimes noise or ripple, so the term IMPEDANCE is used.
The term LOW IMPEDANCE and HIGH IMPEDANCE is a relative term. If we are working on a one-watt amplifier circuit for example, a low impedance may be 10 ohms or one ohm. If we are working on a 1,000 watt amplifier, a low impedance may be 0.1ohms or 0.001 ohms!
A high impedance in an automotive circuit may be 100 ohms or 1k, whereas a high impedance in an oscillator circuit may be 100k. A high impedance OP-AMP or CMOS circuit may be 1M or 10M.
The following figure shows some stages we will be covering (also called "gates" or "blocks."). Each line has been identified as "high-impedance" or "low-impedance."
An output line is identified as low impedance as it is either HIGH or LOW (the two stages of a digital gate) and the line will be held in this state by the action of a transistor (or transistors) inside the chip. If the impedance of the line is measured (in either state), it will be LOW.
An input line is classified as a high-impedance line. But if it is directly connected to the output of a gate, it will measure as a LOW-impedance line.

Fig 2: Some high and low impedance "blocks."
We show how to test 3 of them.

One of the reasons we use the term "IMPEDANCE" is to remind you that the circuit has a particular feature (measured in ohms) at a particular frequency.
We also use the term "impedance" when non-resistive components are involved in working out the final value - such as capacitors (and electrolytics) as these have different effects at different frequencies.
For instance, a power supply must have a LOW IMPEDANCE so that the output voltage is maintained over its full range of current. That's why the output nearly always contains a large-value electrolytic.
The circuitry in a CMOS project must be HIGH IMPEDANCE if you want it to consume the lowest current. The whole purpose of using CMOS technology is to consume the lowest current.

With any circuit, there are three things you need to know:
1. Is the circuit HIGH IMPEDANCE or LOW IMPEDANCE?
2. How to test a HIGH or LOW IMPEDANCE circuit.
3. How to keep a circuit HIGH IMPEDANCE or LOW IMPEDANCE  -  when designing or
        modifying it.

1. IS A CIRCUIT HIGH OR LOW IMPEDANCE?
We have already covered this point. Impedance is a relative term. An engineer who designs CMOS projects will consider any line less than 1M as low impedance. An audio technician will consider any speaker-line above 1 ohm as high impedance!

2. TESTING
We will take "stages" or "blocks" from the circuits above and explain how to test them.

FIRST CIRCUIT:
The first "block" or "gate" to be discussed is shown in fig 3 below. Both inputs of the gate are high impedance. These high-impedance lines are arrowed.
Since this is a digital circuit (the voltages on the inputs will be LOW (very near 0v) or HIGH (very near rail voltage). It is very easy to measure the voltage on a digital line and determine if the circuit is working - providing the measuring equipment does not upset the voltage.
The output of the NOR Gate is HIGH when both inputs are LOW.
Suppose output pin 4 is not changing.
The reason may be a faulty chip  -  change the chip.
If the fault still persists, here is the procedure for testing the circuit:
When pin 3 goes HIGH, the voltage is passed to pin 5 via a 1M2 resistor.

Fig 3: A high-impedance circuit

The 1M2 is not needed as pin 5 is a high impedance input and is capable of receiving a voltage equal to that delivered by pin 3. The 1M2 serves NO PURPOSE. It can be removed.
The person who designed the circuit did not go though it and work out if each component is absolutely necessary. This is one of the final points we teach when designing a circuit.
If you are not sure about the effect of removing a resistor, it can be replaced with a lower value, such as 10k. This is called a "safety resistor" as it will prevent any damage, in case the output and input do not match up perfectly. If the 10k does not affect the circuit it can be replaced with 1k and finally removed altogether.
The 680k can also be replaced with a 10k resistor and finally removed.
This will reduce the input impedance of the gate to a "low impedance" state and allow you to test the inputs. The following diagram shows how to reduce the value of a resistor while testing the circuit to see if the change has made any difference in the operation. Before removing a resistor completely, you need to take a current reading through the resistor. If no current flows (i.e: no voltage is developed across the resistor) it can be removed safely.

Fig 4: Reducing the value of the resistors

Fig 5: Removing the resistors to
produce low-impedance inputs

With the two resistors removed, the waveform on the inputs can be probed with a CRO. The waveforms will be a "square wave." This means the rise and fall will be very fast, with a flat top and bottom to the shape. The excursions will be as close as possible to the positive rail and 0v rail. The waveforms will not be "square" but "rectangular." The term "square wave" simply means the sides are parallel, the top and bottom are flat and the corners are sharp. (There are waveshapes such as sinusoidal, triangular and exponential to describe other waveforms).

Fig 6: The waveforms on the gate.

The input lines are now low-impedance and the CRO will not load the inputs of the gate. The waveforms will be as shown in figure 6. If a waveform has any of the characteristics shown in figure 7, the gate will not detect the change from a low-to-high or high-to-low and the output will not change. In Fig 7, the first waveform does not rise high enough for the input of a gate to detect a HIGH. The waveform must rise to about 55% of rail voltage to be detected by a digital gate as a HIGH. If the gate is a Schmitt Trigger, the waveform must rise above 66% of rail voltage.
The second waveform does not fall below 45% to be detected by a digital gate as a LOW and if the gate is a Schmitt Trigger, the waveform must fall below 33% of rail voltage to be detected as a LOW. To make sure a waveform is detected, it must be as large as possible.

Fig 7: Faulty waveforms - these
waveforms will not trigger an input

SECOND CIRCUIT:
The next circuit to be discussed is an oscillator made up of a NOR gate and a coil. The input (pins 12 and 13 are connected together) takes time to rise and fall due to the charging/discharging of the 10n capacitor. The frequency is also set by the waveform produced by the inductor L1. The waveform adds or detracts from the voltage produced by pin 11 and this alters the charge-time for the 10n and thus alters the frequency of the circuit. The circuit is self-starting as the NOR gate is wired as an inverter and when the input is low, the output is high. This high is transferred to the input to change the state of the gate. The 10n capacitor slows down the time for the input to change state and this creates the frequency for the circuit (as well as the effect of the waveform from the inductor - as mentioned above).  If the inductance of L1 is changed (by placing a metal object near the coil) the amplitude produced by the coil is altered.

Fig 8: A difficult circuit to test

If the output of the circuit above does not produce a waveform as shown in fig 9,

Fig 9: The output of the gate

the fault could lie in the gate, the coil (inductor L1), the 10n capacitor, the 4n7 capacitor, or a dry joint on the printed circuit board.
The first thing to do is build the circuit with new components on a bread-board or a "birds-nest," to prove the circuit will work. You can then swap the 3 components and the chip from the printed circuit board to the birds-nest, one component at a time, to locate the faulty item. If the circuit still does not work, the fault will lie in the track-work on the printed circuit board or the voltage to the chip.
You cannot test any of the circuit with a multimeter as the input and output is low-impedance and the circuit is oscillating at a frequency from 10kHz to 150kHz, depending on the inductance of L1 and the effect of both capacitors.  This makes resistance tests impossible to provide any answers and you have to use the "comparison" approach of transferring the components.
A CRO placed on the output will show a waveform similar to that in fig 9. The sharpness of the waveform will be modified by the effect of the 4n7 capacitor and the inductor, as well as the 10n capacitor.


THIRD CIRCUIT:
The third circuit is shown in Fig:10. It drives a piezo diaphragm. There are two types of piezo's. One has the driving circuit included in the plastic case and will operate from a DC supply. This is sometimes called a piezo sounder, piezo buzzer or piezo siren. The other type has only a diaphragm and requires a driving voltage (AC voltage). This is the type we are using.
The second gate in Fig: 10 is a NAND gate wired as an INVERTER. The two gates form an oscillator with the 100k and 100n as timing components.
Both the input and output of the second gate are low impedance and the piezo diaphragm is driven from the output of the first gate and also the second gate. These two lines are out-of-phase and the diaphragm sees a voltage in one direction and then the other.  The change in voltage "dishes" the diaphragm in one direction, then the other to create sound.

Fig 10: The oscillator circuit

When input pin 1 is taken HIGH, the first NAND gate will function as an INVERTER - the output will always be the opposite of the input.  This means the HIGH output on pin 3 will pass to input pin 2 via the 100k resistor. This will make the output LOW via the action of the gate to create a HIGH output. Normally, the gate will oscillate very quickly when the output is connected to the input. But the circuit contains a 100n on input pin 2.  This capacitor slows down the rise and fall on the input pin and creates the frequency for the oscillator.
If the circuit does not work, you can take pin 2 HIGH via a jumper lead and measure pin 3 to see if it is LOW. Take pin 2 LOW and see if pin 3 is HIGH.
This is called "forcing the circuit to work."

FOURTH CIRCUIT:
The next discussion is a high-impedance circuit:

Fig 11: A high-impedance input

The waveform on pin 6 will depend on the frequency of the signal on pin 3 as well as the shape of the signal. If the frequency is too high or the Mark-Space ratio is not suitable, the 10n will not have sufficient time to charge (or discharge) and thus the voltage level will not be sufficient to be detected by the gate. The following diagrams show the results of two unsuitable waveforms and a correct waveform:

Fig 12: Detecting different waveforms

In the third circuit of Fig:12, the frequency is ok and the waveform from pin 3 is detected by pin 6. The waveform is only just sufficient to be detected by the gate.
If you place a CRO on pin 6, the added resistance (impedance) of the CRO may reduce the waveform and prevent pin 6 detecting the signal. This is shown in fig 13:

Fig 13: The load of the CRO will "freeze" the circuit.

The CRO has a resistance (impedance) of approx 1M and this creates a voltage divider. The voltage on the 10n will never rise above half-rail voltage and the gate will not detect a high. Thus the output of the gate will not change.  The CRO will "freeze" the operation of the circuit.

FIFTH CIRCUIT:
Fig 14 is a power supply made up of a battery and capacitor (or electrolytic).

Fig 14: Low-impedance power supply

The purpose of a capacitor (or electrolytic) across a battery is to improve (increase) the current on the POWER RAIL while keeping the voltage on the POWER RAIL as high as possible.
It effectively turns an "old" battery into a "new" battery.
The additional current comes from the energy stored in the capacitor and this can only be delivered in the form of "spikes" or "bursts" as the capacitor does not hold a lot of energy.
All batteries have a limited life. As they get "used" the output voltage reduces. This is due to microscopic bubbles of gas developing inside the cell and increasing the resistance as well as the active chemicals being converted to inactive compounds that have a higher resistance and this does two things:
The output voltage of the cell reduces and the current capability is reduced.
A 9v battery is one of the best examples of this problem. It contains 6 cells. If an old battery is tested, the output voltage may be say 8v. This reading does not tell you anything. It does not tell you if the battery is old, very old or almost completely dead. You cannot test it by taking a voltage reading.
A piece of equipment uses a battery entirely differently to a "test."
Equipment requires current in bursts (such as the audio from a radio or beeps from a gold detector).
An "old" battery cannot supply "bursts of current" (due to the increased internal resistance of each cell) and the voltage on the "supply rail" reduces. The lower voltage causes the circuit to draw a lower current and on top of this, the battery can only supply a low current.
This creates distortion or a number of other annoying faults.
To reduce the problem, a capacitor or electrolytic is placed across the supply rails.
The value of the capacitor can be worked out by trial and error. Simply select a high value and view the waveform of the supply rail on a CRO. Reduce the value until the waveform become noticeable (distorted audio etc) or the operation of the circuit is upset.
The addition of the capacitor is said to "tighten up the rails" or "reduce the impedance of the supply rails" - the circuit becomes LOW IMPEDANCE.
This type of low impedance circuit is not easy to measure. The capacitor provides an "effect" rather than a measurable impedance and you can readily hear the "effect" by listening to an audio amplifier.
The effect of the electrolytic is to create a 9v battery capable of delivering a high current.
The battery charges the electrolytic when the current required by the circuit is low and delivers it in a burst when needed.

SIXTH CIRCUIT:
Fig 15 shows a simple transistor stage.

The only way to work on the biasing is to fit a resistor and observe the results. Listen to the output, measure the voltage on the collector or place a CRO on the collector and view the signal.

LOAD
The LOAD of a stage is the value of the LOAD RESISTOR as shown in Fig: 21.

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