Complexes and circuits resemble each other in many ways.
A circuit is an exercise routine where you do one set of an exercise, then move immediately to another exercise. After you finish one set of, or a specific time on, each exercise in the circuit, you rest before starting another round of the same exercises. Your rest times and total number of circuits are determined by your fitness level and the intensity level of the exercises. An example of a circuit are the workouts at Simplefit (a number of pullups, then pushups, then squats); a machine-based circuit is the workout behind Curves.
A complex is a barbell or dumbbell circuit. Instead of going station to station or exercise to exercise, you simply smoothly move from one exercise to another using the same dumbbells or barbell. Once you have finished the proscribed number of reps for each exerice in turn, you rest before doing another round of the same exercises.
Istvan Javorek is famous as the developer, or at least popularizer, of the complex. Many others (including Alwyn Cosgrove and Dan John) have further developed the complex and expanded their usage in the training world.
Here is an example of a circuit: The is "Cindy" from Crossfit.
20 minute AMRAP (As Many Rounds As Possible in 20 minutes)
So for 20 minutes, you do 5 pullups, then 10 pushups, then 15 squats. Then, as soon as you are able (immediately, or after a brief rest), you do another 5 pullups, 10 pushups, 15 squats, etc. You continue until 20 minutes have passed, and see how many rounds you did. Your fitness improvement in this kind of circuit is based on how many rounds you complete.
Here is an example of a barbell complex. This is Istvan Javorek's Complex 1
Barbell Upright Row x 6
Barbell High Pull Snatch x 6
Barbell Behind the Head Squat Push Press x 6
Barbell Behind the Head Good Morning x 6
Barbell Bent Over Row x 6
Each round, you do 30 continuous reps - 6 of each of the five exercises. Then you rest after one round, and then do aother, for as many rounds as you have planned. The weight of the bar will be set by the weakest exercise you do in the circuit, because you must be able to complete all of the reps without changing weight.
What's the difference between a barbell circuit and a barbell complex? Basically, if you're keeping the same weight the whole time, it's a complex. You never set the bar down in a complex (unless your exercises call for doing so, like deadlifts). You don't change weight. In a barbell circuit, you may in fact be setting the bar down, doing a different weight at a different station. For example, Crossfit's workout "Linda" involves bench pressing, cleans, and deadlifts in a circuit. A different weight is used for each exercise, so it's a circuit but not a complex. Compare that to this video below of a complex done with a dumbbells and even weight plates.
What are circuits and complexes good for? Conditioning, mostly. You can't go too heavy on circuits and complexes, and the lack of rest means the early lifts will tire you out for the later lifts. But that very factor makes it great for cardiovascular conditioning - you go a lighter than usual but faster and with less rest. You can get stronger on complexes and circuits, but that's not usually the main goal. The goal is usually to improve your ability to keep going through fatigue. Go heavy enough, and with fewer reps (complexes with 3 reps, for example, done heavier than the 6 reps listed above), and you'll get stronger, too.
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